Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 901625 12 pageshttpdxdoiorg1011552013901625

Research ArticleControllability of Neutral Fractional Functional Equations withImpulses and Infinite Delay

R Ganesh1 R Sakthivel2 Yong Ren3 S M Anthoni1 and N I Mahmudov4

1 Department of Mathematics Anna University Regional Centre Coimbatore 641 047 India2Department of Mathematics Sungkyunkwan University Suwon 440-746 Republic of Korea3 Department of Mathematics Anhui Normal University Wuhu 241000 China4Department of Mathematics Eastern Mediterranean University Gazimagusa Mersin 10 Turkey

Correspondence should be addressed to R Sakthivel krsakthivelyahoocom

Received 15 March 2013 Accepted 11 July 2013

Academic Editor Abdelaziz Rhandi

Copyright copy 2013 R Ganesh et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We examine the controllability problem for a class of neutral fractional integrodifferential equations with impulses and infinitedelay More precisely a set of sufficient conditions are derived for the exact controllability of nonlinear neutral impulsive fractionalfunctional equation with infinite delay Further as a corollary approximate controllability result is discussed by assuming compa-ctness conditions on solution operatorThe results are established by using solution operator fractional calculations and fixed pointtechniques In particular the controllability of nonlinear fractional control systems is established under the assumption that thecorresponding linear control system is controllable Finally an example is given to illustrate the obtained theory

1 Introduction

Control theory is an area of application-oriented mathe-matics which deals with the analysis and design of controlsystems In particular the concept of controllability plays animportant role in various areas of science and engineeringMore precisely the problem of controllability deals with theexistence of a control function which steers the solution ofthe system from its initial state to a final state where theinitial and final states may vary over the entire space Controlproblems for various types of deterministic and stochasticdynamical systems in infinite dimensional systems have beenstudied in [1ndash6]

On the other hand the impulsive differential systemscan be used to model processes which are subject to abruptchanges and which cannot be described by the classicaldifferential systems [7] Moreover impulsive control whichis based on the theory of impulsive equations has gainedrenewed interests due its promising applications towardscontrolling systems exhibiting chaotic behavior Thereforethe controllability problem for impulsive differential and

integrodifferential systems in Banach spaces has been studiedextensively (see [8] and the references therein) Moreoverfractional calculus has received great attention because frac-tional derivatives provide an excellent tool for the descriptionof memory and hereditary properties of various processes[9] Also the study of fractional differential equations hasemerged as a new branch of applied mathematics whichhas been used for construction and analysis of mathematicalmodels in various fields of science and engineering [10]Therefore the problem of the existence of solutions forvarious kinds of fractional differential systems has beeninvestigated in [11ndash13] Very recentlyDabas andChauhan [14]studied the existence uniqueness and continuous depen-dence of mild solution for an impulsive neutral fractionalorder differential equation with infinite delay by using thefixed point technique and solution operator on a complexBanach space

Recently many authors pay their attention to studythe controllability of fractional evolution systems [15 16]Wang and Zhou [17] investigated the complete controllability

2 Abstract and Applied Analysis

of fractional evolution systems without involving the com-pactness of characteristic solution operators Kumar andSukavanam [18] derived a new set of sufficient conditionsfor the approximate controllability of a class of semilineardelay control systems of fractional order by using contractionprinciple and the Schauder fixed point theorem Sakthivelet al [19] studied the controllability results for a class offractional neutral control systems with the help of semigrouptheory and fixed point argument The minimum energycontrol problem for infinite-dimensional fractional-discretetime linear systems is discussed in [20] Debbouche andBaleanu [21] derived a set of sufficient conditions for thecontrollability of a class of fractional evolution nonlocalimpulsive quasilinear delay integrodifferential systems byusing the theory of fractional calculus and fixed pointtechnique

However controllability of impulsive fractional inte-grodifferential equations with infinite delay has not beenstudied via the theory of solution operator Motivated by thisconsideration in this paper we investigate the exact control-lability of a class of fractional order neutral integrodifferentialequations with impulses and infinite delay in the followingform

119863119902

119905[119909 (119905) + 119892 (119905 119909119905)] = 119860 [119909 (119905) + 119892 (119905 119909119905)]

+ 1198691minus119902

119905[119861119906 (119905) + 119891 (119905 119909119905 119867119909 (119905))]

119905 isin 119869 = [0 119887] 119905 = 119905119896

Δ119909 (119905119896) = 119868119896 (119909 (119905minus

119896)) 119896 = 1 2 119898

1199090 = 120601 isinBℎ

(1)

where 119863119902

119905is the Caputo fractional derivative of order 119902 0 lt

119902 lt 1 119860 119863(119860) sub 119883 rarr 119883 is an infinitesimal generator ofthe solution operator 119878119902(119905)119905ge0 is defined on a Banach space119883 with the norm sdot 119883 the control function 119906(sdot) is given in1198712(119869 119880) 119880 is a Banach space 119861 is a bounded linear operator

from 119880 into 119883 the histories 119909119905 (minusinfin 0] rarr 119883 are definedby 119909119905(120579) = 119909(119905 + 120579) belongs to an abstract phase space Bℎ

defined axiomatically and 119868119896 119883 rarr 119883 119896 = 1 2 119898 arebounded functions Also the fixed times 119905119896 satisfy 0 le 1199050 lt1199051 lt 1199052 lt sdot sdot sdot lt 119905119898 lt 119905119898+1 le 119887 Δ119909(119905119896) = 119909(119905

+

119896) minus 119909(119905

minus

119896)

and 119909(119905+119896) = limℎrarr0119909(119905119896 + ℎ) and 119909(119905

minus

119896) = limℎrarr0119909(119905119896 minus ℎ)

denote the right and left limits of 119909(119905) at 119905 = 119905119896 respectivelyFurther 119892 119869 times Bℎ rarr 119883 119891 119869 times Bℎ times 119883 rarr 119883

are given functions the term 119867119909(119905) is given by 119867119909(119905) =int119905

0119866(119905 119904)119909(119904)119889119904 where 119866 isin 119862(119863 119877+

) is the set of all positivecontinuous functions on119863 = (119905 119904) isin 1198772

0 le 119904 le 119905 le 119887Themain aim of this paper is to obtain some suitable sufficientconditions for the controllability results corresponding toadmissible control sets without assuming the semigroup iscompact Further we address the approximate controllabilityissue for the considered fractional systems In order to provethe controllability results we follow a technique similar tothat of [14 19] with some necessary modifications

2 Preliminaries

In this section we will recall some basic definitions andlemmas which will be used in this paper Let 119871(119883) denotethe Banach space of bounded linear operators from 119883 into119883 with the norm sdot 119871(119883) Let 119862(119869119883) denote the space of allcontinuous functions from 119869 into 119883 with the norm 119909 =

sup119905isin119869119909(119905)

Now we present the abstract space Bℎ [7] Let ℎ

(minusinfin 0] rarr (0 +infin) be a continuous function with 119897 =int0

minusinfinℎ(119905)119889119905 lt +infin For any 119886 gt 0 define B = 120593

[minus119886 0] rarr 119883 such that 120593(119905) is bounded and measurableand equip the space B with the norm 120593

[minus1198860]=

sup119904isin[minus1198860]

120593(119904) 120593 isin B Further define the space Bℎ =

120593 (minusinfin 0] rarr 119883 for any 119888 gt 0 120593|[minus1198880] isin B with 120593(0) =0 andint0

minusinfinℎ(119904)120593

[1199040]119889119904 lt +infin If Bℎ is endowed with the

norm 120593Bℎ = int0

minusinfinℎ(119904)120593

[1199040]119889119904 120593 isinBℎ then (Bℎ sdot Bℎ

)

is a Banach spaceWe assume that the phase space (Bℎ sdot Bℎ

) is a semi-normed linear space of functions mapping (minusinfin 0] into 119883and satisfying the following fundamental axioms [22]

(A1) If 119909 (minusinfin 119887] rarr 119883 119887 gt 0 is continuous on 119869and 1199090 isin Bℎ then for every 119905 isin 119869 the followingconditions hold

(i) 119909119905 isinBℎ(ii) 119909(119905) le 119871119909119905Bℎ (iii) 119909119905Bℎ le 1198621(119905)sup0le119904le119905119909(119904) + 1198622(119905)1199090Bℎ

where 119871 gt 0 is a constant 1198621 [0 119887] rarr [0infin)

is continuous 1198622 [0infin) rarr [0infin) is locallybounded and 1198621 1198622 are independent of 119909(sdot)

(A2) For the function 119909(sdot) in (A1) 119909119905 is a Bℎ-valuedfunction on [0 119887]

(A3) The spaceBℎ is complete

Definition 1 (see [10]) The Caputo derivative of order 119902 for afunction 119891 [0infin) rarr 119877 can be written as

119863119902

119905119891 (119905) =

1

Γ (119899 minus 119902)int

119905

0

(119905 minus 119904)119899minus119902minus1

119891(119899)(119904) 119889119904 = 119868

119899minus119902119891119899(119905)

(2)

for 119899 minus 1 lt 119902 lt 119899 119899 isin 119873 If 0 lt 119902 le 1 then

119863119902

119905119891 (119905) =

1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902119891(1)(119904) 119889119904 (3)

The Laplace transform of the Caputo derivative of order 119902 gt 0is given as

119871 119863119902

119905119891 (119905) 120582 = 120582

119902119891 (120582) minus

119899minus1

sum

119896=0

120582119902minus119896minus1

119891(119896)(0)

119899 minus 1 lt 119902 lt 119899

(4)

Abstract and Applied Analysis 3

The Mittag-Leffler type function in two arguments isdefined by the series expansion

119864119902119901 (119911) =

infin

sum

119896=0

119911119896

Γ (119902119896 + 119901)=1

2120587119894int119862

120583119902minus119901119890120583

120583119902 minus 119911119889120583

119902 119901 gt 0 119911 isin C

(5)

where 119862 is a contour which starts and ends at minusinfin andencircles the disc 120583 le |119911|12 counter clockwise The Laplacetransform of the Mittag-Leffler function is given as follows

int

infin

0

119890minus120582119905119905119901minus1119864119902119901 (120596119905

119902) 119889119905 =

120582119902minus119901

120582119902 minus 120596

Re 120582 gt 1205961119902 120596 gt 0

(6)

and for more details (see [14])

Definition 2 (see [9]) A closed and linear operator 119860 issaid to be sectorial if there are constants 120596 isin 119877 120579 isin

[1205872 120587] and 119872 gt 0 such that the following two conditionsare satisfied

(i) 120588(119860) sub sum(120579120596)

= 120582 isin 119862 120582 = 120596 |arg(120582 minus 120596)| lt 120579

(ii) 119877(120582 119860)119871(119883) le 119872|120582 minus 120596| 120582 isin sum(120579120596)

Definition 3 (see [11]) Let 119860 be a linear closed operator withdomain119863(119860) defined on119883 One can call119860 the generator of asolution operator if there exist120596 ge 0 and strongly continuousfunctions 119878119902 R

+rarr 119871(119909) such that 120582119902 Re 120582 gt 120596 sub 120588(119860)

and

120582119902minus1(120582

119902119868 minus 119860)

minus1119909 = int

infin

0

119890minus120582119905119878119902 (119905) 119909 119889119905

Re 120582 gt 120596 119909 isin 119883

(7)

In this case 119878119902 is called the solution operator generated by119860

Consider the space

B119887 = 119909 (minusinfin 119887] 997888rarr 119883 such that 119909|119869119896 isin 119862 (119869119896 119883)

and there exist 119909 (119905minus119896) and 119909 (119905+

119896)

with 119909 (119905119896) = 119909 (119905minus

119896) 1199090 = 120601 isinBℎ

119896 = 0 1 2 119898

(8)

where 119909|119869119896 is the restriction of 119909 to 119869119896 = (119905119896 119905119896+1] 119896 =

0 1 2 119898 Let sdot B119887 be a seminorm inB119887 defined by

119909B119887= sup

119904isin119869

119909 (119904) +10038171003817100381710038171206011003817100381710038171003817Bℎ 119909 isinB119887 (9)

Lemma 4 (see [14]) If the functions 119892 119869 times Bℎ rarr 119883119891 119869 timesBℎ times 119883 rarr 119883 satisfy the uniform Holder conditionwith the exponent 120573 isin (0 1] and 119860 is a sectorial operator then

a piecewise continuously differentiable function 119909 isin B0

119887is a

mild solution of

119863119902

119905[119909 (119905) + 119892 (119905 119909119905)] = 119860 [119909 (119905) + 119892 (119905 119909119905)]

+ 1198691minus119902

119905119891 (119905 119909119905 119867119909 (119905))

119905 isin 119869 = [0 119887] 119905 = 119905119896

Δ119909 (119905119896) = 119868119896 (119909 (119905minus

119896)) 119896 = 1 2 119898

1199090 = 120601 isinBℎ

(10)

if 119909 is a solution of the following fractional integral equation119909 (119905)

=

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1))

+119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus119892 (1199051 1199091199051)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus 119892 (119905119894 119909119905119894)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(11)where 119878119902(119905) is the solution operator generated by 119860 given by

119878119902 (119905) = 1198641199021 (119860119905119902) =

1

2120587119894int119861119903

119890120582119905 120582

119902minus1

120582119902 minus 119860119889120582 (12)

where 119861119903 denotes the Bromwich path

Let 119909119887(120601 119906) be the state value of system (1) at terminaltime 119887 corresponding to the control 119906 and the initial value120601 isin Bℎ Introduce the set R(119887 120601) = 119909119887(120601 119906)(0) 119906(sdot) isin1198712(119869 119880) which is called the reachable set of system (1) at

terminal time 119887

Definition 5 The fractional control system (1) is said to beexactly controllable on the interval 119869 ifR(119887 120601) = 119883

Assume that the linear fractional differential controlsystem

119863119902

119905119909 (119905) = 119860119909 (119905) + (119861119906) (119905) 119905 isin [0 119887]

119909 (0) = 120601 (0)

(13)

4 Abstract and Applied Analysis

is exactly controllable It is convenient at this point tointroduce the controllability operator associated with (13) as

Γ119887

0= int

119887

0

119878119902 (119887 minus 119904) 119861119861lowast119878lowast

119902(119887 minus 119904) 119889119904 (14)

where 119861lowast denotes the adjoint of 119861 and 119878lowast119902(119905) is the adjoint

of 119878119902(119905) It is straightforward that the operator Γ1198870is a linear

bounded operator [19]

Lemma 6 If the linear fractional system (13) is exactlycontrollable if and only then for some 120574 gt 0 such that ⟨Γ119887

0119909 119909⟩ ge

1205741199092 for all 119909 isin 119883 and consequently (Γ119887

0)minus1 le 1120574

In order to define the concept ofmild solution for the con-trol problem (1) by comparison with the impulsive neutralfractional equations given in [14] we associate problem (1) tothe integral equation

119909 (119905)

=

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1))

+119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus119892 (1199051 1199091199051)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(15)

Definition 7 A function 119909 (minusinfin 119887] rarr 119883 is said to bea mild solution for the system (1) if for each 119906 isin 1198712(119869 119880)1199090 = 120601 isin Bℎ on (minusinfin 0] Δ119909|119905=119905119896 = 119868119896(119909(119905

minus

119896)) 119896 = 1 119898

the restriction of 119909(sdot) to the interval [0 119887) 1199051 119905119898 iscontinuous and the integral equation (15) is satisfied

3 Controllability Results

In this section we formulate and prove a set of sufficientconditions for the exact controllability of impulsive neutralfractional control differential system (1) by using the solutionoperator theory fractional calculations and fixed pointargument To prove the controllability result we need thefollowing hypotheses

(H1) There exists a constant119872 gt 0 such that

10038171003817100381710038171003817119878119902 (119905)

10038171003817100381710038171003817119871(119883)le 119872 forall119905 isin [0 119887] (16)

(H2) The function 119892 119869 times Bℎ rarr 119883 is continuous andthere exists a constant 119871119892 gt 0 such that

1003817100381710038171003817119892 (1199051 1205951) minus 119892 (1199052 1205952)1003817100381710038171003817119883le 119871119892 (

10038161003816100381610038161199051 minus 11990521003816100381610038161003816 +10038171003817100381710038171205951 minus 1205952

1003817100381710038171003817Bℎ)

119905119894 isin 119869 120595119894 isinBℎ 119894 = 1 2

(17)

(H3) There exist constants 1205831 gt 0 and 1205832 gt 0 such that

1003817100381710038171003817119891 (119905 120593 119909) minus 119891 (119905 120595 119910)1003817100381710038171003817119883le 1205831

1003817100381710038171003817120593 minus 1205951003817100381710038171003817Bℎ

+ 12058321003817100381710038171003817119909 minus 119910

1003817100381710038171003817119883

119905 isin 119869 120593 120595 isinBℎ 119909 119910 isin 119883

(18)

(H4) 119868119896 isin 119862(119883119883) and there exist constants 120588 gt 0 suchthat

1003817100381710038171003817119868119896 (119909) minus 119868119896 (119910)1003817100381710038171003817119883le 1205881003817100381710038171003817119909 minus 119910

1003817100381710038171003817119883 119909 119910 isin 119883

for each 119896 = 1 119898(19)

(H5) The linear fractional system (13) is exactly control-lable

Theorem 8 Assume that the hypotheses (H1)ndash(H5) are sat-isfied then the fractional impulsive system (1) is exactlycontrollable on 119869 provided that

= (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588)

+119871119892119862lowast

1+119872119887 (1205831119862

lowast

1+ 1205832119867

lowast) ] lt 1

(20)

where 119862lowast

1= sup

0lt120591lt1198871198621(120591) and 119867lowast

= sup119905isin[0119887]

int119905

0119866(119905 119904)119889119904 lt

infin

Proof For an arbitrary function 119909(sdot) choose the feedbackcontrol function as follows

Abstract and Applied Analysis 5

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) (Γ

1199051

0)minus1

times[1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

) minus int

1199051

0

119878119902 (1199051 minus 119904) times 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) (Γ

1199052

0)minus1

times [1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)] minus 119878119902 (1199052 minus 1199051) 1198681 (119909 (119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

)) minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

) minus int

1199052

0

119878119902 (1199052 minus 119904) times 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) (Γ

119887

0)minus1

times[119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)] minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin (119905119898 119887]

(21)

and define the operator Φ B119887 rarr B119887 by

(Φ119909) (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894)) +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(22)

It should be noted that the control (21) transfers the system(1) from the initial state 120601 to the final state 119909119887 provided thatthe operator Φ has a fixed point In order to prove the exactcontrollability result it is enough to show that the operatorΦhas a fixed point inB119887

Define the function 119910(sdot) (minusinfin 119887] rarr 119883 by

119910 (119905) = 120601 (119905) 119905 isin (minusinfin 0]

0 119905 isin [0 119887] (23)

then 1199100 = 120601 For each 119911 isin 119862(119869 119877) with 119911(0) = 0 let thefunction 119911 be defined by

119911 (119905) = 0 119905 isin (minusinfin 0]

119911 (119905) 119905 isin [0 119887] (24)

If 119909(sdot) satisfies (15) then we can decompose 119909(sdot) as 119909(119905) =119910(119905) + 119911(119905) for 119905 isin 119869 which implies that 119909119905 = 119910119905 + 119911119905 for 119905 isin 119869and the function 119911(sdot) satisfies

6 Abstract and Applied Analysis

119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) times 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119910 (119905minus

1) + 119911 (119905

minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) minus 119892 (119905 119910119905 + 119911119905)]

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(25)

where

119906119910+119911 (119905)

=

119861lowast119878lowast

119902(1199051 minus 119905) (Γ

1199051

0)minus1

times[1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199101199051

+ 1199111199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)) 119889119904)] 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) (Γ

1199052

0)minus1

times [1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)] minus 119878119902 (1199052 minus 1199051) 1198681 (119910 (119905

minus

1) + 119911 (119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)]

+119892 (1199052 1199101199052+ 1199111199052

) minus int

1199052

0

119878119902 (1199052 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) (Γ

119887

0)minus1

times [119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)] minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) + 119892 (119887 119910119887 + 119911119887)

minusint

119887

0

119878119902 (119887 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (119905119898 119887]

(26)

LetB0

119887= 119911 isinB119887 1199110 = 0 isinBℎ For any 119911 isinB

0

119887 we get

119911B0119887

= sup119904isin119869

119911(119904)119883 +100381710038171003817100381711991101003817100381710038171003817Bℎ

= sup119904isin119869

119911(119904)119883 119911 isinB0

119887

(27)

Thus (B0

119887 sdot B0

119887

) is a Banach space Define the operatorΠ B0

119887rarr B0

119887by

Abstract and Applied Analysis 7

Π119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119911 (119905minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119911 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

)]

minus119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(28)

It can be easily seen that the operator Φ has a fixed point ifand only ifΠ has a fixed point Now we will prove thatΠ hasa unique fixed point In order to prove this we show thatΠ isa contraction mapping Let 119911 119911lowast isin B0

119887 for all 119905 isin [0 1199051] we

have

1003817100381710038171003817(Π119911) (119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(1199051 minus 120578) (Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [10038171003817100381710038171003817119892 (1199051 1199101199051

+ 1199111199051) minus 119892 (1199051 1199101199051

+ 119911lowast

1199051)10038171003817100381710038171003817119883

+ int

1199051

0

10038171003817100381710038171003817119878119902 (1199051 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus 119891 (119904 119910119904 + 119911lowast

119904

119867 (119910 (119904) + 119911lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le 119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+1

120574119872

2

119861119872

2int

119905

0

[119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

times1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus 119867 (119910 (119904)

minus119911lowast(119904))1003817100381710038171003817119883) 119889119904] 119889120578

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus119867(119910(119904) minus 119911lowast(119904))1003817100381710038171003817119883)119889119904

le (1 +1

120574119872

2

119861119872

2) [119871119892119862

lowast

1+119872119887 (1205831119862

lowast

1+ 1205832119867

lowast)]

times1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(29)Similarly for 119905 isin (119905119896 119905119896+1] 119896 = 1 2 119898 we can obtain1003817100381710038171003817(Π119911) (119905) minus (Π119911

lowast) (119905)

1003817100381710038171003817119883

le

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894 (119911 (119905minus

119894)) minus 119868119894 (119911 (119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [

10038171003817100381710038171003817100381710038171003817

119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))

10038171003817100381710038171003817100381710038171003817119883

8 Abstract and Applied Analysis

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(119905119896+1 minus 120578)(Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [

119896

sum

119894=1

10038171003817100381710038171003817119878119902(119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894(119911(119905minus

119894)) minus 119868119894(119911(119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [100381710038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894+ 119868119894 (119910119905minus

119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))100381710038171003817100381710038171003817119883

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+10038171003817100381710038171003817119892 (119905119896+1 119910119905119896+1

+ 119911119905119896+1) minus 119892 (119905119896+1 119910119905119896+1

+ 119911lowast

119905119896+1)10038171003817100381710038171003817119883

+ int

119905119896+1

0

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891(119904 119910119904 + 119911lowast

119904 119867(119910(119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le (1 +1

120574119872

2

119861119872

2) [119896119872120588 + 119896119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(30)

Thus for all 119905 isin [0 119887] we get the estimate

1003817100381710038171003817(Π119911)(119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

le 1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(31)

Thus we have (Π119911)(119905) minus (Π119911lowast)(119905)119883 le 119911 minus 119911lowastB0119887

for all119905 isin [0 119887] Since lt 1 this implies that Π is a contractionmapping and hence Π has a unique fixed point 119911 isin B0

119887

Thus the system (1) is exactly controllable on [0 119887]The proofis complete

However the concept of exact controllability is verylimited for many dynamic control systems and the app-roximate controllability is more appropriate for these con-trol systems instead of exact controllability Taking thisinto account in this paper we will also discuss the app-roximate controllability result of the nonlinear impul-sive fractional control system (1) The control system issaid to be approximately controllable if for every initialdata 120601 and every finite time horizon 119887 gt 0 an admis-sible control process can be found such that the corr-esponding solution is arbitrarily close to a given squareintegrable final condition Further approximate controllablesystems are more prevalent and often approximate con-trollability is completely adequate in applications In rec-ent years for deterministic and stochastic control systemsincluding delay term there are several papers devoted tothe study of approximate controllability [23ndash25] Suka-vanam and Kumar [26] obtained a set of conditions whichensure the approximate controllability of a class of semi-linear fractional delay control systems Recently Sakthivelet al [23] formulated and proved a new set of sufficientconditions for approximate controllability of fractional dif-ferential equations by using the fractional calculus theory andsolutions operators

Definition 9 The fractional control system (1) is said to beapproximately controllable on [0 119887] if the closure of thereachable setR(119887 120601) is dense in119883 that isR(119887 120601) = 119883

Remark 10 Assume that the linear fractional control system

119863119902

119905119909 (119905) = 119860119909 (119905) + (119861119906) (119905) 119905 isin [0 119887]

119909 (0) = 120601

(32)

is approximate controllable Let us now introduce the oper-ators associated with (32) as Γ119887

0= int

119887

0S119902(119887 minus 119904)119861119861

lowastSlowast

119902(119887 minus

119904)119889119904 119877(120572 Γ1198870) = (120572119868+Γ

119887

0)minus1 for 120572 gt 0 It should be mentioned

that the approximate controllability of (32) is equivalent tothe convergence of function 120572119877(120572 Γ119887

0) to zero as 120572 rarr 0

+

in the strong operator topology (see [23 27] and referencestherein)

Theorem 11 Assume that conditions (H1)ndash(H4) hold and thatthe family 119878119902(119905) 119905 gt 0 is compact In addition assumethat the function 119891 is uniformly bounded and the linear systemassociated with the system (1) is approximately controllablethen the nonlinear fractional control system with infinite delay(1) is approximately controllable on [0 b]

Proof For each 120572 gt 0 define the operatorΨ B119887 rarr B119887 byΨ119909(119905) = 119911(119905) where

Abstract and Applied Analysis 9

119911 (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909 (sdot)) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

11990521199051) 119901 (119909 (sdot)) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

119905119898) 119901 (119909 (sdot)) 119905 isin (119905119898 119887]

(33)

where

119901 (119909 (sdot))

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

)

minusint

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909 (119905minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(34)

One can easily show that for all 120572 gt 0 the operator Ψ hasa fixed point by employing the technique used in Theorem 8with some changes

Let 119909120572(sdot) be a fixed point of Ψ Further any fixed point ofΨ is a mild solution of (1) under the control

120572(119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(35)

and satisfies

119909120572(1199051) = 1199091199051

minus 120572119877 (120572 Γ1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119909120572(1199052) = 1199091199052

minus 120572119877 (120572 Γ1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119909120572(119887) = 119909119887 minus 120572119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(36)

Moreover by the assumption that 119891 is uniformly boundedthere exists119873 gt 0 such that

int

119887

0

1003817100381710038171003817119891(119904 119909120572

119904 119867119909

120572(119904))1003817100381710038171003817

2119889119904 le 119887119873

2 (37)

and consequently the sequence 119891(119904 119909120572119904 119867119909

120572(119904)) is bounded

in 1198712(119869 119883) Then there is a subsequence still denoted

10 Abstract and Applied Analysis

by 119891(119904 119909120572119904 119867119909

120572(119904)) that converges weakly to say 119891(119904) in

1198712(119869 119883) Now we define

119908

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)]

+119892 (1199051 119909120572

1199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909120572(119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 119909120572

1199051+ 1198681 (119909

120572

119905minus

1

))

minus119892 (1199051 119909120572

1199051)] + 119892 (1199052 119909

120572

1199052)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909120572(119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909120572

119905119894+ 119868119894 (119909

120572

119905minus

119894

))

minus119892 (119905119894 119909120572

119905119894)] + 119892 (119887 119909

120572

119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904) 119889119904 119905 isin (119905119898 119887]

(38)

Now for 119905 isin [0 1199051] we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199051

0

119878119902 (1199051 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin[01199051]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(39)

Also for 119905 isin [119905119894 119905119894+1] 119894 = 1 119898 we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

119905119894+1

0

119878119902 (119905119894+1 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin(119905119894 119905119894+1]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(40)

By using infinite-dimensional version of the Ascoli-Arzelatheorem it is easy to show that an operator 119897(sdot) rarr int

sdot

0119878119902(sdot minus

119904)119897(119904)119889119904 1198711(119869 119883) rarr 119862(119869119883) is compact Hence for all

119905 isin [0 119887] we obtain that 119901(119909120572) minus 119908 rarr 0 as 120572 rarr 0+

Moreover from (36) we get for 119905 isin [0 1199051]10038171003817100381710038171003817119909120572(1199051) minus 1199091199051

10038171003817100381710038171003817le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817

+10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0)10038171003817100381710038171003817

1003817100381710038171003817119901 (119909120572) minus 119908

1003817100381710038171003817

le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817+1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

(41)

It follows from Remark 10 and (39) that 119909120572(1199051) minus 1199091199051 rarr 0

as 120572 rarr 0+ Similarly in the view of (40) for 119905 isin (119905119894 119905119894+1]

119894 = 1 11989810038171003817100381710038171003817119909120572(119905119894+1) minus 119909119905119894+1

10038171003817100381710038171003817997888rarr 0 as 120572 997888rarr 0

+ (42)

Thus for all 119905 isin [0 119887] we get 119909120572(119887) minus 119909119887 rarr 0 as 120572 rarr 0+

This proves the approximate controllability of (1) The proofis completed

Example 12 Now we present an example to illustrate theabstract results of this paper which do not aim at generalitybut indicate how our theorem can be applied to concreteproblems Let 119883 = 119871

2[0 120587] Define 119860 119883 rarr 119883

by 119860119911 = 11991110158401015840 with domain 119863(119860) = 119911 isin 119883

119911 1199111015840 are absolutely continuous 11991110158401015840 isin 119883 119911(0) = 119911(120587) = 0

Then 119860 generates an analytic semigroup 119879(119905) 119905 gt 0 in 119883and it is given by [14]

119879 (119905) 119911 =

infin

sum

119899=1

119890minus1198992119905(119911 119890119899) 119890119899 119911 isin 119883 (43)

where 119890119899(119910) = radic2120587 sin 119899119910 119899 = 1 2 3 is the orthogonalset of eigenvectors of 119860 Also define an infinite dimensionalcontrol space 119880 by 119880 = 119906 | 119906 = sum

infin

119899=2119906119899119890119899 with sum

infin

119899=21199062

119899lt

infin with norm defined by 119906119880 = (suminfin

119899=21199062

119899)12 Define a

continuous linear map 119861 from 119880 to 119883 as 119861119906 = 211990621198901 +

suminfin

119899=2119906119899119890119899 for 119906 = sum

infin

119899=2119906119899119890119899 isin 119880

Note that the subordination principle of solution operatorimplies that 119860 is the infinitesimal generator of a solutionoperator 119878119902(119905)119905ge0 Since 119878119902(119905) is strongly continuous on[0infin) by the uniformly bounded theorem there exists aconstant119872 gt 0 such that 119878119902(119905)119871(119883)

le 119872 for 119905 isin [0 119887] [14]Consider the following fractional partial integrodifferen-

tial equation with infinite delay and control in the followingform

119888119863

119902

119905[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

=1205972

1205971199102[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

+1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902

times [120583 (119904 119910) + int

119904

minusinfin

119870 (119904 119909 120585 minus 119904)

times1198762 (119909 (120585 119910)) 119889120585

+ int

119904

0

119892 (120585 119904) 119890minus119909(120585119910)

119889120585] 119889119904

119905 isin 119869 = [0 1] 119910 isin [0 120587] 119905 = 119905119896

(44)

with 119909(119905 0) = 119909(119905 120587) = 0 119909(119905 119910) = 120601(119905 119910) 119905 isin (minusinfin 0] 119910 isin[0 120587] Δ119909(119905119894)(119910) = int

119905119894

minusinfin119902119894(119905119894 minus 119904)119909(119904 119910)119889119904 and 119910 isin [0 120587]

Abstract and Applied Analysis 11

where 120597119902119905is the Caputo fractional partial derivative of order

0 lt 119902 lt 1 120583 [0 1] times [0 120587] rarr [0 120587] is continuous and in119905 119902119894 119877 rarr 119877 are continuous 0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899 lt 119887 areprefixed numbers and 120601 isinBℎ

Let ℎ(119904) = 1198902119904 119904 lt 0 then 119897 = int

0

infinℎ(119904)119889119904 = 12 Let

Bℎ be the phase space endowed with the norm 120593Bℎ=

int0

minusinfinℎ(119904)sup

120579isin[1199040]120593(120579)

1198712119889119904 Let 119909(119905)(119910) = 119909(119905 119910) and define

the bounded linear operator 119861 119880 rarr 119883 by (119861119906)(119905)(119910) =120583(119905 119910) 0 le 119910 le 120587 119892(119905 120601)(119910) = int0

minusinfin119886(119905 119910 120579)1198761(120601(120579)(119910))119889120579

119868119896(119909(119905minus

119894))(119910) = int

119905119894

minusinfin119902119894(119905119894minus119904)119909(119904 119910)119889119904 and119891(119905 120601119867119909(119905))(119910) =

int0

minusinfin119870(119905 119910 120579)1198762(120601(120579)(119910))119889120579 + 119867119909(119905)(119910) where 119867119909(119905)(119910) =

int119905

0119892(119904 119905)119890

minus119909(119904119910)119889119904 and 120601(120579)(119910) = 120601(120579 119910) (120579 119910) isin (minusinfin 0] times

[0 120587] Moreover the linear fractional control system corre-sponding to (44) is exactly controllable Further if we imposesuitable conditions on 119886 119870 1198761 1198762 119892 119902119894 and 119861 to verifyassumptions onTheorem 8 then the system (44) can be writ-ten in the abstract form of (1)Therefore all the conditions ofTheorem 8 are satisfied and hence the nonlinear fractionalcontrol system (44) is exactly controllable on [0 119887]

Acknowledgment

The work of Yong Ren is supported by the National NaturalScience Foundation of China (no 11371029)

References

[1] P Muthukumar and P Balasubramaniam ldquoApproximate con-trollability of mixed stochastic Volterra-Fredholm type inte-grodifferential systems in Hilbert spacerdquo Journal of the FranklinInstitute vol 348 no 10 pp 2911ndash2922 2011

[2] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009

[3] J Klamka ldquoConstrained controllability of semilinear systemswith delaysrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 169ndash177 2009

[4] J Klamka ldquoStochastic controllability of systems with multipledelays in controlrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 39ndash47 2009

[5] J Klamka ldquoLocal controllability of fractional discrete-timesemilinear systemsrdquo Acta Mechanica et Automatica vol 15 pp55ndash58 2011

[6] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[7] J Dabas A Chauhan and M Kumar ldquoExistence of the mildsolutions for impulsive fractional equations with infinite delayrdquoInternational Journal of Differential Equations vol 2011 ArticleID 793023 20 pages 2011

[8] J Wang M Feckan and Y Zhou ldquoRelaxed controls fornonlinear fractional impulsive evolution equationsrdquo Journal ofOptimization Theory and Applications vol 156 no 1 pp 13ndash322013

[9] M Haase The Functional Calculus for Sectorial Operators vol169 of Operator Theory Advances and Applications BirkhauserBasel Switzerland 2006

[10] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[11] R P Agarwal B de Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[12] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functional evo-lution equations with infinite delayrdquo Applied Mathematics andComputation vol 216 no 1 pp 61ndash69 2010

[13] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications A vol 74 no 5pp 2003ndash2011 2011

[14] J Dabas and A Chauhan ldquoExistence and uniqueness of mildsolution for an impulsive neutral fractional integro-differentialequation with infinite delayrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 754ndash763 2013

[15] R Sakthivel and Y Ren ldquoComplete controllability of stochasticevolution equations with jumpsrdquo Reports on MathematicalPhysics vol 68 no 2 pp 163ndash174 2011

[16] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters of Rapid Publication vol 26 no 4 pp 418ndash4242013

[17] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[18] S Kumar and N Sukavanam ldquoApproximate controllability offractional order semilinear systems with bounded delayrdquo Jour-nal of Differential Equations vol 251 no 11 pp 6163ndash6174 2012

[19] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[20] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends inNanotechnology and Fractional Calculus D Baleanu Z BGuvenc and J A T Machado Eds pp 503ndash509 Springer NewYork NY USA 2010

[21] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[22] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[23] R Sakthivel S Suganya and S M Anthoni ldquoApproximate con-trollability of fractional stochastic evolution equationsrdquo Com-puters amp Mathematics with Applications vol 63 no 3 pp 660ndash668 2012

[24] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[25] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[26] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

of fractional evolution systems without involving the com-pactness of characteristic solution operators Kumar andSukavanam [18] derived a new set of sufficient conditionsfor the approximate controllability of a class of semilineardelay control systems of fractional order by using contractionprinciple and the Schauder fixed point theorem Sakthivelet al [19] studied the controllability results for a class offractional neutral control systems with the help of semigrouptheory and fixed point argument The minimum energycontrol problem for infinite-dimensional fractional-discretetime linear systems is discussed in [20] Debbouche andBaleanu [21] derived a set of sufficient conditions for thecontrollability of a class of fractional evolution nonlocalimpulsive quasilinear delay integrodifferential systems byusing the theory of fractional calculus and fixed pointtechnique

However controllability of impulsive fractional inte-grodifferential equations with infinite delay has not beenstudied via the theory of solution operator Motivated by thisconsideration in this paper we investigate the exact control-lability of a class of fractional order neutral integrodifferentialequations with impulses and infinite delay in the followingform

119863119902

119905[119909 (119905) + 119892 (119905 119909119905)] = 119860 [119909 (119905) + 119892 (119905 119909119905)]

+ 1198691minus119902

119905[119861119906 (119905) + 119891 (119905 119909119905 119867119909 (119905))]

119905 isin 119869 = [0 119887] 119905 = 119905119896

Δ119909 (119905119896) = 119868119896 (119909 (119905minus

119896)) 119896 = 1 2 119898

1199090 = 120601 isinBℎ

(1)

where 119863119902

119905is the Caputo fractional derivative of order 119902 0 lt

119902 lt 1 119860 119863(119860) sub 119883 rarr 119883 is an infinitesimal generator ofthe solution operator 119878119902(119905)119905ge0 is defined on a Banach space119883 with the norm sdot 119883 the control function 119906(sdot) is given in1198712(119869 119880) 119880 is a Banach space 119861 is a bounded linear operator

from 119880 into 119883 the histories 119909119905 (minusinfin 0] rarr 119883 are definedby 119909119905(120579) = 119909(119905 + 120579) belongs to an abstract phase space Bℎ

defined axiomatically and 119868119896 119883 rarr 119883 119896 = 1 2 119898 arebounded functions Also the fixed times 119905119896 satisfy 0 le 1199050 lt1199051 lt 1199052 lt sdot sdot sdot lt 119905119898 lt 119905119898+1 le 119887 Δ119909(119905119896) = 119909(119905

+

119896) minus 119909(119905

minus

119896)

and 119909(119905+119896) = limℎrarr0119909(119905119896 + ℎ) and 119909(119905

minus

119896) = limℎrarr0119909(119905119896 minus ℎ)

denote the right and left limits of 119909(119905) at 119905 = 119905119896 respectivelyFurther 119892 119869 times Bℎ rarr 119883 119891 119869 times Bℎ times 119883 rarr 119883

are given functions the term 119867119909(119905) is given by 119867119909(119905) =int119905

0119866(119905 119904)119909(119904)119889119904 where 119866 isin 119862(119863 119877+

) is the set of all positivecontinuous functions on119863 = (119905 119904) isin 1198772

0 le 119904 le 119905 le 119887Themain aim of this paper is to obtain some suitable sufficientconditions for the controllability results corresponding toadmissible control sets without assuming the semigroup iscompact Further we address the approximate controllabilityissue for the considered fractional systems In order to provethe controllability results we follow a technique similar tothat of [14 19] with some necessary modifications

2 Preliminaries

In this section we will recall some basic definitions andlemmas which will be used in this paper Let 119871(119883) denotethe Banach space of bounded linear operators from 119883 into119883 with the norm sdot 119871(119883) Let 119862(119869119883) denote the space of allcontinuous functions from 119869 into 119883 with the norm 119909 =

sup119905isin119869119909(119905)

Now we present the abstract space Bℎ [7] Let ℎ

(minusinfin 0] rarr (0 +infin) be a continuous function with 119897 =int0

minusinfinℎ(119905)119889119905 lt +infin For any 119886 gt 0 define B = 120593

[minus119886 0] rarr 119883 such that 120593(119905) is bounded and measurableand equip the space B with the norm 120593

[minus1198860]=

sup119904isin[minus1198860]

120593(119904) 120593 isin B Further define the space Bℎ =

120593 (minusinfin 0] rarr 119883 for any 119888 gt 0 120593|[minus1198880] isin B with 120593(0) =0 andint0

minusinfinℎ(119904)120593

[1199040]119889119904 lt +infin If Bℎ is endowed with the

norm 120593Bℎ = int0

minusinfinℎ(119904)120593

[1199040]119889119904 120593 isinBℎ then (Bℎ sdot Bℎ

)

is a Banach spaceWe assume that the phase space (Bℎ sdot Bℎ

) is a semi-normed linear space of functions mapping (minusinfin 0] into 119883and satisfying the following fundamental axioms [22]

(A1) If 119909 (minusinfin 119887] rarr 119883 119887 gt 0 is continuous on 119869and 1199090 isin Bℎ then for every 119905 isin 119869 the followingconditions hold

(i) 119909119905 isinBℎ(ii) 119909(119905) le 119871119909119905Bℎ (iii) 119909119905Bℎ le 1198621(119905)sup0le119904le119905119909(119904) + 1198622(119905)1199090Bℎ

where 119871 gt 0 is a constant 1198621 [0 119887] rarr [0infin)

is continuous 1198622 [0infin) rarr [0infin) is locallybounded and 1198621 1198622 are independent of 119909(sdot)

(A2) For the function 119909(sdot) in (A1) 119909119905 is a Bℎ-valuedfunction on [0 119887]

(A3) The spaceBℎ is complete

Definition 1 (see [10]) The Caputo derivative of order 119902 for afunction 119891 [0infin) rarr 119877 can be written as

119863119902

119905119891 (119905) =

1

Γ (119899 minus 119902)int

119905

0

(119905 minus 119904)119899minus119902minus1

119891(119899)(119904) 119889119904 = 119868

119899minus119902119891119899(119905)

(2)

for 119899 minus 1 lt 119902 lt 119899 119899 isin 119873 If 0 lt 119902 le 1 then

119863119902

119905119891 (119905) =

1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902119891(1)(119904) 119889119904 (3)

The Laplace transform of the Caputo derivative of order 119902 gt 0is given as

119871 119863119902

119905119891 (119905) 120582 = 120582

119902119891 (120582) minus

119899minus1

sum

119896=0

120582119902minus119896minus1

119891(119896)(0)

119899 minus 1 lt 119902 lt 119899

(4)

Abstract and Applied Analysis 3

The Mittag-Leffler type function in two arguments isdefined by the series expansion

119864119902119901 (119911) =

infin

sum

119896=0

119911119896

Γ (119902119896 + 119901)=1

2120587119894int119862

120583119902minus119901119890120583

120583119902 minus 119911119889120583

119902 119901 gt 0 119911 isin C

(5)

where 119862 is a contour which starts and ends at minusinfin andencircles the disc 120583 le |119911|12 counter clockwise The Laplacetransform of the Mittag-Leffler function is given as follows

int

infin

0

119890minus120582119905119905119901minus1119864119902119901 (120596119905

119902) 119889119905 =

120582119902minus119901

120582119902 minus 120596

Re 120582 gt 1205961119902 120596 gt 0

(6)

and for more details (see [14])

Definition 2 (see [9]) A closed and linear operator 119860 issaid to be sectorial if there are constants 120596 isin 119877 120579 isin

[1205872 120587] and 119872 gt 0 such that the following two conditionsare satisfied

(i) 120588(119860) sub sum(120579120596)

= 120582 isin 119862 120582 = 120596 |arg(120582 minus 120596)| lt 120579

(ii) 119877(120582 119860)119871(119883) le 119872|120582 minus 120596| 120582 isin sum(120579120596)

Definition 3 (see [11]) Let 119860 be a linear closed operator withdomain119863(119860) defined on119883 One can call119860 the generator of asolution operator if there exist120596 ge 0 and strongly continuousfunctions 119878119902 R

+rarr 119871(119909) such that 120582119902 Re 120582 gt 120596 sub 120588(119860)

and

120582119902minus1(120582

119902119868 minus 119860)

minus1119909 = int

infin

0

119890minus120582119905119878119902 (119905) 119909 119889119905

Re 120582 gt 120596 119909 isin 119883

(7)

In this case 119878119902 is called the solution operator generated by119860

Consider the space

B119887 = 119909 (minusinfin 119887] 997888rarr 119883 such that 119909|119869119896 isin 119862 (119869119896 119883)

and there exist 119909 (119905minus119896) and 119909 (119905+

119896)

with 119909 (119905119896) = 119909 (119905minus

119896) 1199090 = 120601 isinBℎ

119896 = 0 1 2 119898

(8)

where 119909|119869119896 is the restriction of 119909 to 119869119896 = (119905119896 119905119896+1] 119896 =

0 1 2 119898 Let sdot B119887 be a seminorm inB119887 defined by

119909B119887= sup

119904isin119869

119909 (119904) +10038171003817100381710038171206011003817100381710038171003817Bℎ 119909 isinB119887 (9)

Lemma 4 (see [14]) If the functions 119892 119869 times Bℎ rarr 119883119891 119869 timesBℎ times 119883 rarr 119883 satisfy the uniform Holder conditionwith the exponent 120573 isin (0 1] and 119860 is a sectorial operator then

a piecewise continuously differentiable function 119909 isin B0

119887is a

mild solution of

119863119902

119905[119909 (119905) + 119892 (119905 119909119905)] = 119860 [119909 (119905) + 119892 (119905 119909119905)]

+ 1198691minus119902

119905119891 (119905 119909119905 119867119909 (119905))

119905 isin 119869 = [0 119887] 119905 = 119905119896

Δ119909 (119905119896) = 119868119896 (119909 (119905minus

119896)) 119896 = 1 2 119898

1199090 = 120601 isinBℎ

(10)

if 119909 is a solution of the following fractional integral equation119909 (119905)

=

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1))

+119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus119892 (1199051 1199091199051)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus 119892 (119905119894 119909119905119894)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(11)where 119878119902(119905) is the solution operator generated by 119860 given by

119878119902 (119905) = 1198641199021 (119860119905119902) =

1

2120587119894int119861119903

119890120582119905 120582

119902minus1

120582119902 minus 119860119889120582 (12)

where 119861119903 denotes the Bromwich path

Let 119909119887(120601 119906) be the state value of system (1) at terminaltime 119887 corresponding to the control 119906 and the initial value120601 isin Bℎ Introduce the set R(119887 120601) = 119909119887(120601 119906)(0) 119906(sdot) isin1198712(119869 119880) which is called the reachable set of system (1) at

terminal time 119887

Definition 5 The fractional control system (1) is said to beexactly controllable on the interval 119869 ifR(119887 120601) = 119883

Assume that the linear fractional differential controlsystem

119863119902

119905119909 (119905) = 119860119909 (119905) + (119861119906) (119905) 119905 isin [0 119887]

119909 (0) = 120601 (0)

(13)

4 Abstract and Applied Analysis

is exactly controllable It is convenient at this point tointroduce the controllability operator associated with (13) as

Γ119887

0= int

119887

0

119878119902 (119887 minus 119904) 119861119861lowast119878lowast

119902(119887 minus 119904) 119889119904 (14)

where 119861lowast denotes the adjoint of 119861 and 119878lowast119902(119905) is the adjoint

of 119878119902(119905) It is straightforward that the operator Γ1198870is a linear

bounded operator [19]

Lemma 6 If the linear fractional system (13) is exactlycontrollable if and only then for some 120574 gt 0 such that ⟨Γ119887

0119909 119909⟩ ge

1205741199092 for all 119909 isin 119883 and consequently (Γ119887

0)minus1 le 1120574

In order to define the concept ofmild solution for the con-trol problem (1) by comparison with the impulsive neutralfractional equations given in [14] we associate problem (1) tothe integral equation

119909 (119905)

=

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1))

+119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus119892 (1199051 1199091199051)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(15)

Definition 7 A function 119909 (minusinfin 119887] rarr 119883 is said to bea mild solution for the system (1) if for each 119906 isin 1198712(119869 119880)1199090 = 120601 isin Bℎ on (minusinfin 0] Δ119909|119905=119905119896 = 119868119896(119909(119905

minus

119896)) 119896 = 1 119898

the restriction of 119909(sdot) to the interval [0 119887) 1199051 119905119898 iscontinuous and the integral equation (15) is satisfied

3 Controllability Results

In this section we formulate and prove a set of sufficientconditions for the exact controllability of impulsive neutralfractional control differential system (1) by using the solutionoperator theory fractional calculations and fixed pointargument To prove the controllability result we need thefollowing hypotheses

(H1) There exists a constant119872 gt 0 such that

10038171003817100381710038171003817119878119902 (119905)

10038171003817100381710038171003817119871(119883)le 119872 forall119905 isin [0 119887] (16)

(H2) The function 119892 119869 times Bℎ rarr 119883 is continuous andthere exists a constant 119871119892 gt 0 such that

1003817100381710038171003817119892 (1199051 1205951) minus 119892 (1199052 1205952)1003817100381710038171003817119883le 119871119892 (

10038161003816100381610038161199051 minus 11990521003816100381610038161003816 +10038171003817100381710038171205951 minus 1205952

1003817100381710038171003817Bℎ)

119905119894 isin 119869 120595119894 isinBℎ 119894 = 1 2

(17)

(H3) There exist constants 1205831 gt 0 and 1205832 gt 0 such that

1003817100381710038171003817119891 (119905 120593 119909) minus 119891 (119905 120595 119910)1003817100381710038171003817119883le 1205831

1003817100381710038171003817120593 minus 1205951003817100381710038171003817Bℎ

+ 12058321003817100381710038171003817119909 minus 119910

1003817100381710038171003817119883

119905 isin 119869 120593 120595 isinBℎ 119909 119910 isin 119883

(18)

(H4) 119868119896 isin 119862(119883119883) and there exist constants 120588 gt 0 suchthat

1003817100381710038171003817119868119896 (119909) minus 119868119896 (119910)1003817100381710038171003817119883le 1205881003817100381710038171003817119909 minus 119910

1003817100381710038171003817119883 119909 119910 isin 119883

for each 119896 = 1 119898(19)

(H5) The linear fractional system (13) is exactly control-lable

Theorem 8 Assume that the hypotheses (H1)ndash(H5) are sat-isfied then the fractional impulsive system (1) is exactlycontrollable on 119869 provided that

= (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588)

+119871119892119862lowast

1+119872119887 (1205831119862

lowast

1+ 1205832119867

lowast) ] lt 1

(20)

where 119862lowast

1= sup

0lt120591lt1198871198621(120591) and 119867lowast

= sup119905isin[0119887]

int119905

0119866(119905 119904)119889119904 lt

infin

Proof For an arbitrary function 119909(sdot) choose the feedbackcontrol function as follows

Abstract and Applied Analysis 5

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) (Γ

1199051

0)minus1

times[1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

) minus int

1199051

0

119878119902 (1199051 minus 119904) times 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) (Γ

1199052

0)minus1

times [1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)] minus 119878119902 (1199052 minus 1199051) 1198681 (119909 (119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

)) minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

) minus int

1199052

0

119878119902 (1199052 minus 119904) times 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) (Γ

119887

0)minus1

times[119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)] minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin (119905119898 119887]

(21)

and define the operator Φ B119887 rarr B119887 by

(Φ119909) (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894)) +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(22)

It should be noted that the control (21) transfers the system(1) from the initial state 120601 to the final state 119909119887 provided thatthe operator Φ has a fixed point In order to prove the exactcontrollability result it is enough to show that the operatorΦhas a fixed point inB119887

Define the function 119910(sdot) (minusinfin 119887] rarr 119883 by

119910 (119905) = 120601 (119905) 119905 isin (minusinfin 0]

0 119905 isin [0 119887] (23)

then 1199100 = 120601 For each 119911 isin 119862(119869 119877) with 119911(0) = 0 let thefunction 119911 be defined by

119911 (119905) = 0 119905 isin (minusinfin 0]

119911 (119905) 119905 isin [0 119887] (24)

If 119909(sdot) satisfies (15) then we can decompose 119909(sdot) as 119909(119905) =119910(119905) + 119911(119905) for 119905 isin 119869 which implies that 119909119905 = 119910119905 + 119911119905 for 119905 isin 119869and the function 119911(sdot) satisfies

6 Abstract and Applied Analysis

119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) times 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119910 (119905minus

1) + 119911 (119905

minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) minus 119892 (119905 119910119905 + 119911119905)]

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(25)

where

119906119910+119911 (119905)

=

119861lowast119878lowast

119902(1199051 minus 119905) (Γ

1199051

0)minus1

times[1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199101199051

+ 1199111199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)) 119889119904)] 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) (Γ

1199052

0)minus1

times [1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)] minus 119878119902 (1199052 minus 1199051) 1198681 (119910 (119905

minus

1) + 119911 (119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)]

+119892 (1199052 1199101199052+ 1199111199052

) minus int

1199052

0

119878119902 (1199052 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) (Γ

119887

0)minus1

times [119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)] minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) + 119892 (119887 119910119887 + 119911119887)

minusint

119887

0

119878119902 (119887 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (119905119898 119887]

(26)

LetB0

119887= 119911 isinB119887 1199110 = 0 isinBℎ For any 119911 isinB

0

119887 we get

119911B0119887

= sup119904isin119869

119911(119904)119883 +100381710038171003817100381711991101003817100381710038171003817Bℎ

= sup119904isin119869

119911(119904)119883 119911 isinB0

119887

(27)

Thus (B0

119887 sdot B0

119887

) is a Banach space Define the operatorΠ B0

119887rarr B0

119887by

Abstract and Applied Analysis 7

Π119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119911 (119905minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119911 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

)]

minus119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(28)

It can be easily seen that the operator Φ has a fixed point ifand only ifΠ has a fixed point Now we will prove thatΠ hasa unique fixed point In order to prove this we show thatΠ isa contraction mapping Let 119911 119911lowast isin B0

119887 for all 119905 isin [0 1199051] we

have

1003817100381710038171003817(Π119911) (119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(1199051 minus 120578) (Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [10038171003817100381710038171003817119892 (1199051 1199101199051

+ 1199111199051) minus 119892 (1199051 1199101199051

+ 119911lowast

1199051)10038171003817100381710038171003817119883

+ int

1199051

0

10038171003817100381710038171003817119878119902 (1199051 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus 119891 (119904 119910119904 + 119911lowast

119904

119867 (119910 (119904) + 119911lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le 119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+1

120574119872

2

119861119872

2int

119905

0

[119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

times1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus 119867 (119910 (119904)

minus119911lowast(119904))1003817100381710038171003817119883) 119889119904] 119889120578

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus119867(119910(119904) minus 119911lowast(119904))1003817100381710038171003817119883)119889119904

le (1 +1

120574119872

2

119861119872

2) [119871119892119862

lowast

1+119872119887 (1205831119862

lowast

1+ 1205832119867

lowast)]

times1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(29)Similarly for 119905 isin (119905119896 119905119896+1] 119896 = 1 2 119898 we can obtain1003817100381710038171003817(Π119911) (119905) minus (Π119911

lowast) (119905)

1003817100381710038171003817119883

le

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894 (119911 (119905minus

119894)) minus 119868119894 (119911 (119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [

10038171003817100381710038171003817100381710038171003817

119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))

10038171003817100381710038171003817100381710038171003817119883

8 Abstract and Applied Analysis

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(119905119896+1 minus 120578)(Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [

119896

sum

119894=1

10038171003817100381710038171003817119878119902(119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894(119911(119905minus

119894)) minus 119868119894(119911(119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [100381710038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894+ 119868119894 (119910119905minus

119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))100381710038171003817100381710038171003817119883

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+10038171003817100381710038171003817119892 (119905119896+1 119910119905119896+1

+ 119911119905119896+1) minus 119892 (119905119896+1 119910119905119896+1

+ 119911lowast

119905119896+1)10038171003817100381710038171003817119883

+ int

119905119896+1

0

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891(119904 119910119904 + 119911lowast

119904 119867(119910(119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le (1 +1

120574119872

2

119861119872

2) [119896119872120588 + 119896119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(30)

Thus for all 119905 isin [0 119887] we get the estimate

1003817100381710038171003817(Π119911)(119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

le 1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(31)

Thus we have (Π119911)(119905) minus (Π119911lowast)(119905)119883 le 119911 minus 119911lowastB0119887

for all119905 isin [0 119887] Since lt 1 this implies that Π is a contractionmapping and hence Π has a unique fixed point 119911 isin B0

119887

Thus the system (1) is exactly controllable on [0 119887]The proofis complete

However the concept of exact controllability is verylimited for many dynamic control systems and the app-roximate controllability is more appropriate for these con-trol systems instead of exact controllability Taking thisinto account in this paper we will also discuss the app-roximate controllability result of the nonlinear impul-sive fractional control system (1) The control system issaid to be approximately controllable if for every initialdata 120601 and every finite time horizon 119887 gt 0 an admis-sible control process can be found such that the corr-esponding solution is arbitrarily close to a given squareintegrable final condition Further approximate controllablesystems are more prevalent and often approximate con-trollability is completely adequate in applications In rec-ent years for deterministic and stochastic control systemsincluding delay term there are several papers devoted tothe study of approximate controllability [23ndash25] Suka-vanam and Kumar [26] obtained a set of conditions whichensure the approximate controllability of a class of semi-linear fractional delay control systems Recently Sakthivelet al [23] formulated and proved a new set of sufficientconditions for approximate controllability of fractional dif-ferential equations by using the fractional calculus theory andsolutions operators

Definition 9 The fractional control system (1) is said to beapproximately controllable on [0 119887] if the closure of thereachable setR(119887 120601) is dense in119883 that isR(119887 120601) = 119883

Remark 10 Assume that the linear fractional control system

119863119902

119905119909 (119905) = 119860119909 (119905) + (119861119906) (119905) 119905 isin [0 119887]

119909 (0) = 120601

(32)

is approximate controllable Let us now introduce the oper-ators associated with (32) as Γ119887

0= int

119887

0S119902(119887 minus 119904)119861119861

lowastSlowast

119902(119887 minus

119904)119889119904 119877(120572 Γ1198870) = (120572119868+Γ

119887

0)minus1 for 120572 gt 0 It should be mentioned

that the approximate controllability of (32) is equivalent tothe convergence of function 120572119877(120572 Γ119887

0) to zero as 120572 rarr 0

+

in the strong operator topology (see [23 27] and referencestherein)

Theorem 11 Assume that conditions (H1)ndash(H4) hold and thatthe family 119878119902(119905) 119905 gt 0 is compact In addition assumethat the function 119891 is uniformly bounded and the linear systemassociated with the system (1) is approximately controllablethen the nonlinear fractional control system with infinite delay(1) is approximately controllable on [0 b]

Proof For each 120572 gt 0 define the operatorΨ B119887 rarr B119887 byΨ119909(119905) = 119911(119905) where

Abstract and Applied Analysis 9

119911 (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909 (sdot)) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

11990521199051) 119901 (119909 (sdot)) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

119905119898) 119901 (119909 (sdot)) 119905 isin (119905119898 119887]

(33)

where

119901 (119909 (sdot))

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

)

minusint

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909 (119905minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(34)

One can easily show that for all 120572 gt 0 the operator Ψ hasa fixed point by employing the technique used in Theorem 8with some changes

Let 119909120572(sdot) be a fixed point of Ψ Further any fixed point ofΨ is a mild solution of (1) under the control

120572(119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(35)

and satisfies

119909120572(1199051) = 1199091199051

minus 120572119877 (120572 Γ1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119909120572(1199052) = 1199091199052

minus 120572119877 (120572 Γ1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119909120572(119887) = 119909119887 minus 120572119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(36)

Moreover by the assumption that 119891 is uniformly boundedthere exists119873 gt 0 such that

int

119887

0

1003817100381710038171003817119891(119904 119909120572

119904 119867119909

120572(119904))1003817100381710038171003817

2119889119904 le 119887119873

2 (37)

and consequently the sequence 119891(119904 119909120572119904 119867119909

120572(119904)) is bounded

in 1198712(119869 119883) Then there is a subsequence still denoted

10 Abstract and Applied Analysis

by 119891(119904 119909120572119904 119867119909

120572(119904)) that converges weakly to say 119891(119904) in

1198712(119869 119883) Now we define

119908

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)]

+119892 (1199051 119909120572

1199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909120572(119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 119909120572

1199051+ 1198681 (119909

120572

119905minus

1

))

minus119892 (1199051 119909120572

1199051)] + 119892 (1199052 119909

120572

1199052)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909120572(119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909120572

119905119894+ 119868119894 (119909

120572

119905minus

119894

))

minus119892 (119905119894 119909120572

119905119894)] + 119892 (119887 119909

120572

119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904) 119889119904 119905 isin (119905119898 119887]

(38)

Now for 119905 isin [0 1199051] we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199051

0

119878119902 (1199051 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin[01199051]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(39)

Also for 119905 isin [119905119894 119905119894+1] 119894 = 1 119898 we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

119905119894+1

0

119878119902 (119905119894+1 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin(119905119894 119905119894+1]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(40)

By using infinite-dimensional version of the Ascoli-Arzelatheorem it is easy to show that an operator 119897(sdot) rarr int

sdot

0119878119902(sdot minus

119904)119897(119904)119889119904 1198711(119869 119883) rarr 119862(119869119883) is compact Hence for all

119905 isin [0 119887] we obtain that 119901(119909120572) minus 119908 rarr 0 as 120572 rarr 0+

Moreover from (36) we get for 119905 isin [0 1199051]10038171003817100381710038171003817119909120572(1199051) minus 1199091199051

10038171003817100381710038171003817le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817

+10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0)10038171003817100381710038171003817

1003817100381710038171003817119901 (119909120572) minus 119908

1003817100381710038171003817

le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817+1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

(41)

It follows from Remark 10 and (39) that 119909120572(1199051) minus 1199091199051 rarr 0

as 120572 rarr 0+ Similarly in the view of (40) for 119905 isin (119905119894 119905119894+1]

119894 = 1 11989810038171003817100381710038171003817119909120572(119905119894+1) minus 119909119905119894+1

10038171003817100381710038171003817997888rarr 0 as 120572 997888rarr 0

+ (42)

Thus for all 119905 isin [0 119887] we get 119909120572(119887) minus 119909119887 rarr 0 as 120572 rarr 0+

This proves the approximate controllability of (1) The proofis completed

Example 12 Now we present an example to illustrate theabstract results of this paper which do not aim at generalitybut indicate how our theorem can be applied to concreteproblems Let 119883 = 119871

2[0 120587] Define 119860 119883 rarr 119883

by 119860119911 = 11991110158401015840 with domain 119863(119860) = 119911 isin 119883

119911 1199111015840 are absolutely continuous 11991110158401015840 isin 119883 119911(0) = 119911(120587) = 0

Then 119860 generates an analytic semigroup 119879(119905) 119905 gt 0 in 119883and it is given by [14]

119879 (119905) 119911 =

infin

sum

119899=1

119890minus1198992119905(119911 119890119899) 119890119899 119911 isin 119883 (43)

where 119890119899(119910) = radic2120587 sin 119899119910 119899 = 1 2 3 is the orthogonalset of eigenvectors of 119860 Also define an infinite dimensionalcontrol space 119880 by 119880 = 119906 | 119906 = sum

infin

119899=2119906119899119890119899 with sum

infin

119899=21199062

119899lt

infin with norm defined by 119906119880 = (suminfin

119899=21199062

119899)12 Define a

continuous linear map 119861 from 119880 to 119883 as 119861119906 = 211990621198901 +

suminfin

119899=2119906119899119890119899 for 119906 = sum

infin

119899=2119906119899119890119899 isin 119880

Note that the subordination principle of solution operatorimplies that 119860 is the infinitesimal generator of a solutionoperator 119878119902(119905)119905ge0 Since 119878119902(119905) is strongly continuous on[0infin) by the uniformly bounded theorem there exists aconstant119872 gt 0 such that 119878119902(119905)119871(119883)

le 119872 for 119905 isin [0 119887] [14]Consider the following fractional partial integrodifferen-

tial equation with infinite delay and control in the followingform

119888119863

119902

119905[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

=1205972

1205971199102[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

+1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902

times [120583 (119904 119910) + int

119904

minusinfin

119870 (119904 119909 120585 minus 119904)

times1198762 (119909 (120585 119910)) 119889120585

+ int

119904

0

119892 (120585 119904) 119890minus119909(120585119910)

119889120585] 119889119904

119905 isin 119869 = [0 1] 119910 isin [0 120587] 119905 = 119905119896

(44)

with 119909(119905 0) = 119909(119905 120587) = 0 119909(119905 119910) = 120601(119905 119910) 119905 isin (minusinfin 0] 119910 isin[0 120587] Δ119909(119905119894)(119910) = int

119905119894

minusinfin119902119894(119905119894 minus 119904)119909(119904 119910)119889119904 and 119910 isin [0 120587]

Abstract and Applied Analysis 11

where 120597119902119905is the Caputo fractional partial derivative of order

0 lt 119902 lt 1 120583 [0 1] times [0 120587] rarr [0 120587] is continuous and in119905 119902119894 119877 rarr 119877 are continuous 0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899 lt 119887 areprefixed numbers and 120601 isinBℎ

Let ℎ(119904) = 1198902119904 119904 lt 0 then 119897 = int

0

infinℎ(119904)119889119904 = 12 Let

Bℎ be the phase space endowed with the norm 120593Bℎ=

int0

minusinfinℎ(119904)sup

120579isin[1199040]120593(120579)

1198712119889119904 Let 119909(119905)(119910) = 119909(119905 119910) and define

the bounded linear operator 119861 119880 rarr 119883 by (119861119906)(119905)(119910) =120583(119905 119910) 0 le 119910 le 120587 119892(119905 120601)(119910) = int0

minusinfin119886(119905 119910 120579)1198761(120601(120579)(119910))119889120579

119868119896(119909(119905minus

119894))(119910) = int

119905119894

minusinfin119902119894(119905119894minus119904)119909(119904 119910)119889119904 and119891(119905 120601119867119909(119905))(119910) =

int0

minusinfin119870(119905 119910 120579)1198762(120601(120579)(119910))119889120579 + 119867119909(119905)(119910) where 119867119909(119905)(119910) =

int119905

0119892(119904 119905)119890

minus119909(119904119910)119889119904 and 120601(120579)(119910) = 120601(120579 119910) (120579 119910) isin (minusinfin 0] times

[0 120587] Moreover the linear fractional control system corre-sponding to (44) is exactly controllable Further if we imposesuitable conditions on 119886 119870 1198761 1198762 119892 119902119894 and 119861 to verifyassumptions onTheorem 8 then the system (44) can be writ-ten in the abstract form of (1)Therefore all the conditions ofTheorem 8 are satisfied and hence the nonlinear fractionalcontrol system (44) is exactly controllable on [0 119887]

Acknowledgment

The work of Yong Ren is supported by the National NaturalScience Foundation of China (no 11371029)

References

[1] P Muthukumar and P Balasubramaniam ldquoApproximate con-trollability of mixed stochastic Volterra-Fredholm type inte-grodifferential systems in Hilbert spacerdquo Journal of the FranklinInstitute vol 348 no 10 pp 2911ndash2922 2011

[2] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009

[3] J Klamka ldquoConstrained controllability of semilinear systemswith delaysrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 169ndash177 2009

[4] J Klamka ldquoStochastic controllability of systems with multipledelays in controlrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 39ndash47 2009

[5] J Klamka ldquoLocal controllability of fractional discrete-timesemilinear systemsrdquo Acta Mechanica et Automatica vol 15 pp55ndash58 2011

[6] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[7] J Dabas A Chauhan and M Kumar ldquoExistence of the mildsolutions for impulsive fractional equations with infinite delayrdquoInternational Journal of Differential Equations vol 2011 ArticleID 793023 20 pages 2011

[8] J Wang M Feckan and Y Zhou ldquoRelaxed controls fornonlinear fractional impulsive evolution equationsrdquo Journal ofOptimization Theory and Applications vol 156 no 1 pp 13ndash322013

[9] M Haase The Functional Calculus for Sectorial Operators vol169 of Operator Theory Advances and Applications BirkhauserBasel Switzerland 2006

[10] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[11] R P Agarwal B de Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[12] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functional evo-lution equations with infinite delayrdquo Applied Mathematics andComputation vol 216 no 1 pp 61ndash69 2010

[13] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications A vol 74 no 5pp 2003ndash2011 2011

[14] J Dabas and A Chauhan ldquoExistence and uniqueness of mildsolution for an impulsive neutral fractional integro-differentialequation with infinite delayrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 754ndash763 2013

[15] R Sakthivel and Y Ren ldquoComplete controllability of stochasticevolution equations with jumpsrdquo Reports on MathematicalPhysics vol 68 no 2 pp 163ndash174 2011

[16] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters of Rapid Publication vol 26 no 4 pp 418ndash4242013

[17] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[18] S Kumar and N Sukavanam ldquoApproximate controllability offractional order semilinear systems with bounded delayrdquo Jour-nal of Differential Equations vol 251 no 11 pp 6163ndash6174 2012

[19] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[20] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends inNanotechnology and Fractional Calculus D Baleanu Z BGuvenc and J A T Machado Eds pp 503ndash509 Springer NewYork NY USA 2010

[21] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[22] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[23] R Sakthivel S Suganya and S M Anthoni ldquoApproximate con-trollability of fractional stochastic evolution equationsrdquo Com-puters amp Mathematics with Applications vol 63 no 3 pp 660ndash668 2012

[24] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[25] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[26] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

The Mittag-Leffler type function in two arguments isdefined by the series expansion

119864119902119901 (119911) =

infin

sum

119896=0

119911119896

Γ (119902119896 + 119901)=1

2120587119894int119862

120583119902minus119901119890120583

120583119902 minus 119911119889120583

119902 119901 gt 0 119911 isin C

(5)

where 119862 is a contour which starts and ends at minusinfin andencircles the disc 120583 le |119911|12 counter clockwise The Laplacetransform of the Mittag-Leffler function is given as follows

int

infin

0

119890minus120582119905119905119901minus1119864119902119901 (120596119905

119902) 119889119905 =

120582119902minus119901

120582119902 minus 120596

Re 120582 gt 1205961119902 120596 gt 0

(6)

and for more details (see [14])

Definition 2 (see [9]) A closed and linear operator 119860 issaid to be sectorial if there are constants 120596 isin 119877 120579 isin

[1205872 120587] and 119872 gt 0 such that the following two conditionsare satisfied

(i) 120588(119860) sub sum(120579120596)

= 120582 isin 119862 120582 = 120596 |arg(120582 minus 120596)| lt 120579

(ii) 119877(120582 119860)119871(119883) le 119872|120582 minus 120596| 120582 isin sum(120579120596)

Definition 3 (see [11]) Let 119860 be a linear closed operator withdomain119863(119860) defined on119883 One can call119860 the generator of asolution operator if there exist120596 ge 0 and strongly continuousfunctions 119878119902 R

+rarr 119871(119909) such that 120582119902 Re 120582 gt 120596 sub 120588(119860)

and

120582119902minus1(120582

119902119868 minus 119860)

minus1119909 = int

infin

0

119890minus120582119905119878119902 (119905) 119909 119889119905

Re 120582 gt 120596 119909 isin 119883

(7)

In this case 119878119902 is called the solution operator generated by119860

Consider the space

B119887 = 119909 (minusinfin 119887] 997888rarr 119883 such that 119909|119869119896 isin 119862 (119869119896 119883)

and there exist 119909 (119905minus119896) and 119909 (119905+

119896)

with 119909 (119905119896) = 119909 (119905minus

119896) 1199090 = 120601 isinBℎ

119896 = 0 1 2 119898

(8)

where 119909|119869119896 is the restriction of 119909 to 119869119896 = (119905119896 119905119896+1] 119896 =

0 1 2 119898 Let sdot B119887 be a seminorm inB119887 defined by

119909B119887= sup

119904isin119869

119909 (119904) +10038171003817100381710038171206011003817100381710038171003817Bℎ 119909 isinB119887 (9)

Lemma 4 (see [14]) If the functions 119892 119869 times Bℎ rarr 119883119891 119869 timesBℎ times 119883 rarr 119883 satisfy the uniform Holder conditionwith the exponent 120573 isin (0 1] and 119860 is a sectorial operator then

a piecewise continuously differentiable function 119909 isin B0

119887is a

mild solution of

119863119902

119905[119909 (119905) + 119892 (119905 119909119905)] = 119860 [119909 (119905) + 119892 (119905 119909119905)]

+ 1198691minus119902

119905119891 (119905 119909119905 119867119909 (119905))

119905 isin 119869 = [0 119887] 119905 = 119905119896

Δ119909 (119905119896) = 119868119896 (119909 (119905minus

119896)) 119896 = 1 2 119898

1199090 = 120601 isinBℎ

(10)

if 119909 is a solution of the following fractional integral equation119909 (119905)

=

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1))

+119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus119892 (1199051 1199091199051)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus 119892 (119905119894 119909119905119894)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(11)where 119878119902(119905) is the solution operator generated by 119860 given by

119878119902 (119905) = 1198641199021 (119860119905119902) =

1

2120587119894int119861119903

119890120582119905 120582

119902minus1

120582119902 minus 119860119889120582 (12)

where 119861119903 denotes the Bromwich path

Let 119909119887(120601 119906) be the state value of system (1) at terminaltime 119887 corresponding to the control 119906 and the initial value120601 isin Bℎ Introduce the set R(119887 120601) = 119909119887(120601 119906)(0) 119906(sdot) isin1198712(119869 119880) which is called the reachable set of system (1) at

terminal time 119887

Definition 5 The fractional control system (1) is said to beexactly controllable on the interval 119869 ifR(119887 120601) = 119883

Assume that the linear fractional differential controlsystem

119863119902

119905119909 (119905) = 119860119909 (119905) + (119861119906) (119905) 119905 isin [0 119887]

119909 (0) = 120601 (0)

(13)

4 Abstract and Applied Analysis

is exactly controllable It is convenient at this point tointroduce the controllability operator associated with (13) as

Γ119887

0= int

119887

0

119878119902 (119887 minus 119904) 119861119861lowast119878lowast

119902(119887 minus 119904) 119889119904 (14)

where 119861lowast denotes the adjoint of 119861 and 119878lowast119902(119905) is the adjoint

of 119878119902(119905) It is straightforward that the operator Γ1198870is a linear

bounded operator [19]

Lemma 6 If the linear fractional system (13) is exactlycontrollable if and only then for some 120574 gt 0 such that ⟨Γ119887

0119909 119909⟩ ge

1205741199092 for all 119909 isin 119883 and consequently (Γ119887

0)minus1 le 1120574

In order to define the concept ofmild solution for the con-trol problem (1) by comparison with the impulsive neutralfractional equations given in [14] we associate problem (1) tothe integral equation

119909 (119905)

=

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1))

+119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus119892 (1199051 1199091199051)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(15)

Definition 7 A function 119909 (minusinfin 119887] rarr 119883 is said to bea mild solution for the system (1) if for each 119906 isin 1198712(119869 119880)1199090 = 120601 isin Bℎ on (minusinfin 0] Δ119909|119905=119905119896 = 119868119896(119909(119905

minus

119896)) 119896 = 1 119898

the restriction of 119909(sdot) to the interval [0 119887) 1199051 119905119898 iscontinuous and the integral equation (15) is satisfied

3 Controllability Results

In this section we formulate and prove a set of sufficientconditions for the exact controllability of impulsive neutralfractional control differential system (1) by using the solutionoperator theory fractional calculations and fixed pointargument To prove the controllability result we need thefollowing hypotheses

(H1) There exists a constant119872 gt 0 such that

10038171003817100381710038171003817119878119902 (119905)

10038171003817100381710038171003817119871(119883)le 119872 forall119905 isin [0 119887] (16)

(H2) The function 119892 119869 times Bℎ rarr 119883 is continuous andthere exists a constant 119871119892 gt 0 such that

1003817100381710038171003817119892 (1199051 1205951) minus 119892 (1199052 1205952)1003817100381710038171003817119883le 119871119892 (

10038161003816100381610038161199051 minus 11990521003816100381610038161003816 +10038171003817100381710038171205951 minus 1205952

1003817100381710038171003817Bℎ)

119905119894 isin 119869 120595119894 isinBℎ 119894 = 1 2

(17)

(H3) There exist constants 1205831 gt 0 and 1205832 gt 0 such that

1003817100381710038171003817119891 (119905 120593 119909) minus 119891 (119905 120595 119910)1003817100381710038171003817119883le 1205831

1003817100381710038171003817120593 minus 1205951003817100381710038171003817Bℎ

+ 12058321003817100381710038171003817119909 minus 119910

1003817100381710038171003817119883

119905 isin 119869 120593 120595 isinBℎ 119909 119910 isin 119883

(18)

(H4) 119868119896 isin 119862(119883119883) and there exist constants 120588 gt 0 suchthat

1003817100381710038171003817119868119896 (119909) minus 119868119896 (119910)1003817100381710038171003817119883le 1205881003817100381710038171003817119909 minus 119910

1003817100381710038171003817119883 119909 119910 isin 119883

for each 119896 = 1 119898(19)

(H5) The linear fractional system (13) is exactly control-lable

Theorem 8 Assume that the hypotheses (H1)ndash(H5) are sat-isfied then the fractional impulsive system (1) is exactlycontrollable on 119869 provided that

= (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588)

+119871119892119862lowast

1+119872119887 (1205831119862

lowast

1+ 1205832119867

lowast) ] lt 1

(20)

where 119862lowast

1= sup

0lt120591lt1198871198621(120591) and 119867lowast

= sup119905isin[0119887]

int119905

0119866(119905 119904)119889119904 lt

infin

Proof For an arbitrary function 119909(sdot) choose the feedbackcontrol function as follows

Abstract and Applied Analysis 5

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) (Γ

1199051

0)minus1

times[1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

) minus int

1199051

0

119878119902 (1199051 minus 119904) times 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) (Γ

1199052

0)minus1

times [1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)] minus 119878119902 (1199052 minus 1199051) 1198681 (119909 (119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

)) minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

) minus int

1199052

0

119878119902 (1199052 minus 119904) times 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) (Γ

119887

0)minus1

times[119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)] minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin (119905119898 119887]

(21)

and define the operator Φ B119887 rarr B119887 by

(Φ119909) (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894)) +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(22)

It should be noted that the control (21) transfers the system(1) from the initial state 120601 to the final state 119909119887 provided thatthe operator Φ has a fixed point In order to prove the exactcontrollability result it is enough to show that the operatorΦhas a fixed point inB119887

Define the function 119910(sdot) (minusinfin 119887] rarr 119883 by

119910 (119905) = 120601 (119905) 119905 isin (minusinfin 0]

0 119905 isin [0 119887] (23)

then 1199100 = 120601 For each 119911 isin 119862(119869 119877) with 119911(0) = 0 let thefunction 119911 be defined by

119911 (119905) = 0 119905 isin (minusinfin 0]

119911 (119905) 119905 isin [0 119887] (24)

If 119909(sdot) satisfies (15) then we can decompose 119909(sdot) as 119909(119905) =119910(119905) + 119911(119905) for 119905 isin 119869 which implies that 119909119905 = 119910119905 + 119911119905 for 119905 isin 119869and the function 119911(sdot) satisfies

6 Abstract and Applied Analysis

119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) times 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119910 (119905minus

1) + 119911 (119905

minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) minus 119892 (119905 119910119905 + 119911119905)]

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(25)

where

119906119910+119911 (119905)

=

119861lowast119878lowast

119902(1199051 minus 119905) (Γ

1199051

0)minus1

times[1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199101199051

+ 1199111199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)) 119889119904)] 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) (Γ

1199052

0)minus1

times [1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)] minus 119878119902 (1199052 minus 1199051) 1198681 (119910 (119905

minus

1) + 119911 (119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)]

+119892 (1199052 1199101199052+ 1199111199052

) minus int

1199052

0

119878119902 (1199052 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) (Γ

119887

0)minus1

times [119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)] minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) + 119892 (119887 119910119887 + 119911119887)

minusint

119887

0

119878119902 (119887 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (119905119898 119887]

(26)

LetB0

119887= 119911 isinB119887 1199110 = 0 isinBℎ For any 119911 isinB

0

119887 we get

119911B0119887

= sup119904isin119869

119911(119904)119883 +100381710038171003817100381711991101003817100381710038171003817Bℎ

= sup119904isin119869

119911(119904)119883 119911 isinB0

119887

(27)

Thus (B0

119887 sdot B0

119887

) is a Banach space Define the operatorΠ B0

119887rarr B0

119887by

Abstract and Applied Analysis 7

Π119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119911 (119905minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119911 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

)]

minus119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(28)

It can be easily seen that the operator Φ has a fixed point ifand only ifΠ has a fixed point Now we will prove thatΠ hasa unique fixed point In order to prove this we show thatΠ isa contraction mapping Let 119911 119911lowast isin B0

119887 for all 119905 isin [0 1199051] we

have

1003817100381710038171003817(Π119911) (119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(1199051 minus 120578) (Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [10038171003817100381710038171003817119892 (1199051 1199101199051

+ 1199111199051) minus 119892 (1199051 1199101199051

+ 119911lowast

1199051)10038171003817100381710038171003817119883

+ int

1199051

0

10038171003817100381710038171003817119878119902 (1199051 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus 119891 (119904 119910119904 + 119911lowast

119904

119867 (119910 (119904) + 119911lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le 119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+1

120574119872

2

119861119872

2int

119905

0

[119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

times1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus 119867 (119910 (119904)

minus119911lowast(119904))1003817100381710038171003817119883) 119889119904] 119889120578

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus119867(119910(119904) minus 119911lowast(119904))1003817100381710038171003817119883)119889119904

le (1 +1

120574119872

2

119861119872

2) [119871119892119862

lowast

1+119872119887 (1205831119862

lowast

1+ 1205832119867

lowast)]

times1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(29)Similarly for 119905 isin (119905119896 119905119896+1] 119896 = 1 2 119898 we can obtain1003817100381710038171003817(Π119911) (119905) minus (Π119911

lowast) (119905)

1003817100381710038171003817119883

le

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894 (119911 (119905minus

119894)) minus 119868119894 (119911 (119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [

10038171003817100381710038171003817100381710038171003817

119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))

10038171003817100381710038171003817100381710038171003817119883

8 Abstract and Applied Analysis

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(119905119896+1 minus 120578)(Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [

119896

sum

119894=1

10038171003817100381710038171003817119878119902(119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894(119911(119905minus

119894)) minus 119868119894(119911(119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [100381710038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894+ 119868119894 (119910119905minus

119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))100381710038171003817100381710038171003817119883

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+10038171003817100381710038171003817119892 (119905119896+1 119910119905119896+1

+ 119911119905119896+1) minus 119892 (119905119896+1 119910119905119896+1

+ 119911lowast

119905119896+1)10038171003817100381710038171003817119883

+ int

119905119896+1

0

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891(119904 119910119904 + 119911lowast

119904 119867(119910(119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le (1 +1

120574119872

2

119861119872

2) [119896119872120588 + 119896119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(30)

Thus for all 119905 isin [0 119887] we get the estimate

1003817100381710038171003817(Π119911)(119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

le 1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(31)

Thus we have (Π119911)(119905) minus (Π119911lowast)(119905)119883 le 119911 minus 119911lowastB0119887

for all119905 isin [0 119887] Since lt 1 this implies that Π is a contractionmapping and hence Π has a unique fixed point 119911 isin B0

119887

Thus the system (1) is exactly controllable on [0 119887]The proofis complete

However the concept of exact controllability is verylimited for many dynamic control systems and the app-roximate controllability is more appropriate for these con-trol systems instead of exact controllability Taking thisinto account in this paper we will also discuss the app-roximate controllability result of the nonlinear impul-sive fractional control system (1) The control system issaid to be approximately controllable if for every initialdata 120601 and every finite time horizon 119887 gt 0 an admis-sible control process can be found such that the corr-esponding solution is arbitrarily close to a given squareintegrable final condition Further approximate controllablesystems are more prevalent and often approximate con-trollability is completely adequate in applications In rec-ent years for deterministic and stochastic control systemsincluding delay term there are several papers devoted tothe study of approximate controllability [23ndash25] Suka-vanam and Kumar [26] obtained a set of conditions whichensure the approximate controllability of a class of semi-linear fractional delay control systems Recently Sakthivelet al [23] formulated and proved a new set of sufficientconditions for approximate controllability of fractional dif-ferential equations by using the fractional calculus theory andsolutions operators

Definition 9 The fractional control system (1) is said to beapproximately controllable on [0 119887] if the closure of thereachable setR(119887 120601) is dense in119883 that isR(119887 120601) = 119883

Remark 10 Assume that the linear fractional control system

119863119902

119905119909 (119905) = 119860119909 (119905) + (119861119906) (119905) 119905 isin [0 119887]

119909 (0) = 120601

(32)

is approximate controllable Let us now introduce the oper-ators associated with (32) as Γ119887

0= int

119887

0S119902(119887 minus 119904)119861119861

lowastSlowast

119902(119887 minus

119904)119889119904 119877(120572 Γ1198870) = (120572119868+Γ

119887

0)minus1 for 120572 gt 0 It should be mentioned

that the approximate controllability of (32) is equivalent tothe convergence of function 120572119877(120572 Γ119887

0) to zero as 120572 rarr 0

+

in the strong operator topology (see [23 27] and referencestherein)

Theorem 11 Assume that conditions (H1)ndash(H4) hold and thatthe family 119878119902(119905) 119905 gt 0 is compact In addition assumethat the function 119891 is uniformly bounded and the linear systemassociated with the system (1) is approximately controllablethen the nonlinear fractional control system with infinite delay(1) is approximately controllable on [0 b]

Proof For each 120572 gt 0 define the operatorΨ B119887 rarr B119887 byΨ119909(119905) = 119911(119905) where

Abstract and Applied Analysis 9

119911 (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909 (sdot)) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

11990521199051) 119901 (119909 (sdot)) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

119905119898) 119901 (119909 (sdot)) 119905 isin (119905119898 119887]

(33)

where

119901 (119909 (sdot))

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

)

minusint

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909 (119905minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(34)

One can easily show that for all 120572 gt 0 the operator Ψ hasa fixed point by employing the technique used in Theorem 8with some changes

Let 119909120572(sdot) be a fixed point of Ψ Further any fixed point ofΨ is a mild solution of (1) under the control

120572(119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(35)

and satisfies

119909120572(1199051) = 1199091199051

minus 120572119877 (120572 Γ1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119909120572(1199052) = 1199091199052

minus 120572119877 (120572 Γ1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119909120572(119887) = 119909119887 minus 120572119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(36)

Moreover by the assumption that 119891 is uniformly boundedthere exists119873 gt 0 such that

int

119887

0

1003817100381710038171003817119891(119904 119909120572

119904 119867119909

120572(119904))1003817100381710038171003817

2119889119904 le 119887119873

2 (37)

and consequently the sequence 119891(119904 119909120572119904 119867119909

120572(119904)) is bounded

in 1198712(119869 119883) Then there is a subsequence still denoted

10 Abstract and Applied Analysis

by 119891(119904 119909120572119904 119867119909

120572(119904)) that converges weakly to say 119891(119904) in

1198712(119869 119883) Now we define

119908

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)]

+119892 (1199051 119909120572

1199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909120572(119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 119909120572

1199051+ 1198681 (119909

120572

119905minus

1

))

minus119892 (1199051 119909120572

1199051)] + 119892 (1199052 119909

120572

1199052)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909120572(119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909120572

119905119894+ 119868119894 (119909

120572

119905minus

119894

))

minus119892 (119905119894 119909120572

119905119894)] + 119892 (119887 119909

120572

119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904) 119889119904 119905 isin (119905119898 119887]

(38)

Now for 119905 isin [0 1199051] we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199051

0

119878119902 (1199051 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin[01199051]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(39)

Also for 119905 isin [119905119894 119905119894+1] 119894 = 1 119898 we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

119905119894+1

0

119878119902 (119905119894+1 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin(119905119894 119905119894+1]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(40)

By using infinite-dimensional version of the Ascoli-Arzelatheorem it is easy to show that an operator 119897(sdot) rarr int

sdot

0119878119902(sdot minus

119904)119897(119904)119889119904 1198711(119869 119883) rarr 119862(119869119883) is compact Hence for all

119905 isin [0 119887] we obtain that 119901(119909120572) minus 119908 rarr 0 as 120572 rarr 0+

Moreover from (36) we get for 119905 isin [0 1199051]10038171003817100381710038171003817119909120572(1199051) minus 1199091199051

10038171003817100381710038171003817le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817

+10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0)10038171003817100381710038171003817

1003817100381710038171003817119901 (119909120572) minus 119908

1003817100381710038171003817

le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817+1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

(41)

It follows from Remark 10 and (39) that 119909120572(1199051) minus 1199091199051 rarr 0

as 120572 rarr 0+ Similarly in the view of (40) for 119905 isin (119905119894 119905119894+1]

119894 = 1 11989810038171003817100381710038171003817119909120572(119905119894+1) minus 119909119905119894+1

10038171003817100381710038171003817997888rarr 0 as 120572 997888rarr 0

+ (42)

Thus for all 119905 isin [0 119887] we get 119909120572(119887) minus 119909119887 rarr 0 as 120572 rarr 0+

This proves the approximate controllability of (1) The proofis completed

Example 12 Now we present an example to illustrate theabstract results of this paper which do not aim at generalitybut indicate how our theorem can be applied to concreteproblems Let 119883 = 119871

2[0 120587] Define 119860 119883 rarr 119883

by 119860119911 = 11991110158401015840 with domain 119863(119860) = 119911 isin 119883

119911 1199111015840 are absolutely continuous 11991110158401015840 isin 119883 119911(0) = 119911(120587) = 0

Then 119860 generates an analytic semigroup 119879(119905) 119905 gt 0 in 119883and it is given by [14]

119879 (119905) 119911 =

infin

sum

119899=1

119890minus1198992119905(119911 119890119899) 119890119899 119911 isin 119883 (43)

where 119890119899(119910) = radic2120587 sin 119899119910 119899 = 1 2 3 is the orthogonalset of eigenvectors of 119860 Also define an infinite dimensionalcontrol space 119880 by 119880 = 119906 | 119906 = sum

infin

119899=2119906119899119890119899 with sum

infin

119899=21199062

119899lt

infin with norm defined by 119906119880 = (suminfin

119899=21199062

119899)12 Define a

continuous linear map 119861 from 119880 to 119883 as 119861119906 = 211990621198901 +

suminfin

119899=2119906119899119890119899 for 119906 = sum

infin

119899=2119906119899119890119899 isin 119880

Note that the subordination principle of solution operatorimplies that 119860 is the infinitesimal generator of a solutionoperator 119878119902(119905)119905ge0 Since 119878119902(119905) is strongly continuous on[0infin) by the uniformly bounded theorem there exists aconstant119872 gt 0 such that 119878119902(119905)119871(119883)

le 119872 for 119905 isin [0 119887] [14]Consider the following fractional partial integrodifferen-

tial equation with infinite delay and control in the followingform

119888119863

119902

119905[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

=1205972

1205971199102[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

+1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902

times [120583 (119904 119910) + int

119904

minusinfin

119870 (119904 119909 120585 minus 119904)

times1198762 (119909 (120585 119910)) 119889120585

+ int

119904

0

119892 (120585 119904) 119890minus119909(120585119910)

119889120585] 119889119904

119905 isin 119869 = [0 1] 119910 isin [0 120587] 119905 = 119905119896

(44)

with 119909(119905 0) = 119909(119905 120587) = 0 119909(119905 119910) = 120601(119905 119910) 119905 isin (minusinfin 0] 119910 isin[0 120587] Δ119909(119905119894)(119910) = int

119905119894

minusinfin119902119894(119905119894 minus 119904)119909(119904 119910)119889119904 and 119910 isin [0 120587]

Abstract and Applied Analysis 11

where 120597119902119905is the Caputo fractional partial derivative of order

0 lt 119902 lt 1 120583 [0 1] times [0 120587] rarr [0 120587] is continuous and in119905 119902119894 119877 rarr 119877 are continuous 0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899 lt 119887 areprefixed numbers and 120601 isinBℎ

Let ℎ(119904) = 1198902119904 119904 lt 0 then 119897 = int

0

infinℎ(119904)119889119904 = 12 Let

Bℎ be the phase space endowed with the norm 120593Bℎ=

int0

minusinfinℎ(119904)sup

120579isin[1199040]120593(120579)

1198712119889119904 Let 119909(119905)(119910) = 119909(119905 119910) and define

the bounded linear operator 119861 119880 rarr 119883 by (119861119906)(119905)(119910) =120583(119905 119910) 0 le 119910 le 120587 119892(119905 120601)(119910) = int0

minusinfin119886(119905 119910 120579)1198761(120601(120579)(119910))119889120579

119868119896(119909(119905minus

119894))(119910) = int

119905119894

minusinfin119902119894(119905119894minus119904)119909(119904 119910)119889119904 and119891(119905 120601119867119909(119905))(119910) =

int0

minusinfin119870(119905 119910 120579)1198762(120601(120579)(119910))119889120579 + 119867119909(119905)(119910) where 119867119909(119905)(119910) =

int119905

0119892(119904 119905)119890

minus119909(119904119910)119889119904 and 120601(120579)(119910) = 120601(120579 119910) (120579 119910) isin (minusinfin 0] times

[0 120587] Moreover the linear fractional control system corre-sponding to (44) is exactly controllable Further if we imposesuitable conditions on 119886 119870 1198761 1198762 119892 119902119894 and 119861 to verifyassumptions onTheorem 8 then the system (44) can be writ-ten in the abstract form of (1)Therefore all the conditions ofTheorem 8 are satisfied and hence the nonlinear fractionalcontrol system (44) is exactly controllable on [0 119887]

Acknowledgment

The work of Yong Ren is supported by the National NaturalScience Foundation of China (no 11371029)

References

[1] P Muthukumar and P Balasubramaniam ldquoApproximate con-trollability of mixed stochastic Volterra-Fredholm type inte-grodifferential systems in Hilbert spacerdquo Journal of the FranklinInstitute vol 348 no 10 pp 2911ndash2922 2011

[2] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009

[3] J Klamka ldquoConstrained controllability of semilinear systemswith delaysrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 169ndash177 2009

[4] J Klamka ldquoStochastic controllability of systems with multipledelays in controlrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 39ndash47 2009

[5] J Klamka ldquoLocal controllability of fractional discrete-timesemilinear systemsrdquo Acta Mechanica et Automatica vol 15 pp55ndash58 2011

[6] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[7] J Dabas A Chauhan and M Kumar ldquoExistence of the mildsolutions for impulsive fractional equations with infinite delayrdquoInternational Journal of Differential Equations vol 2011 ArticleID 793023 20 pages 2011

[8] J Wang M Feckan and Y Zhou ldquoRelaxed controls fornonlinear fractional impulsive evolution equationsrdquo Journal ofOptimization Theory and Applications vol 156 no 1 pp 13ndash322013

[9] M Haase The Functional Calculus for Sectorial Operators vol169 of Operator Theory Advances and Applications BirkhauserBasel Switzerland 2006

[10] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[11] R P Agarwal B de Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[12] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functional evo-lution equations with infinite delayrdquo Applied Mathematics andComputation vol 216 no 1 pp 61ndash69 2010

[13] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications A vol 74 no 5pp 2003ndash2011 2011

[14] J Dabas and A Chauhan ldquoExistence and uniqueness of mildsolution for an impulsive neutral fractional integro-differentialequation with infinite delayrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 754ndash763 2013

[15] R Sakthivel and Y Ren ldquoComplete controllability of stochasticevolution equations with jumpsrdquo Reports on MathematicalPhysics vol 68 no 2 pp 163ndash174 2011

[16] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters of Rapid Publication vol 26 no 4 pp 418ndash4242013

[17] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[18] S Kumar and N Sukavanam ldquoApproximate controllability offractional order semilinear systems with bounded delayrdquo Jour-nal of Differential Equations vol 251 no 11 pp 6163ndash6174 2012

[19] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[20] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends inNanotechnology and Fractional Calculus D Baleanu Z BGuvenc and J A T Machado Eds pp 503ndash509 Springer NewYork NY USA 2010

[21] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[22] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[23] R Sakthivel S Suganya and S M Anthoni ldquoApproximate con-trollability of fractional stochastic evolution equationsrdquo Com-puters amp Mathematics with Applications vol 63 no 3 pp 660ndash668 2012

[24] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[25] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[26] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

is exactly controllable It is convenient at this point tointroduce the controllability operator associated with (13) as

Γ119887

0= int

119887

0

119878119902 (119887 minus 119904) 119861119861lowast119878lowast

119902(119887 minus 119904) 119889119904 (14)

where 119861lowast denotes the adjoint of 119861 and 119878lowast119902(119905) is the adjoint

of 119878119902(119905) It is straightforward that the operator Γ1198870is a linear

bounded operator [19]

Lemma 6 If the linear fractional system (13) is exactlycontrollable if and only then for some 120574 gt 0 such that ⟨Γ119887

0119909 119909⟩ ge

1205741199092 for all 119909 isin 119883 and consequently (Γ119887

0)minus1 le 1120574

In order to define the concept ofmild solution for the con-trol problem (1) by comparison with the impulsive neutralfractional equations given in [14] we associate problem (1) tothe integral equation

119909 (119905)

=

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1))

+119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus119892 (1199051 1199091199051)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)]

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] minus 119892 (119905 119909119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(15)

Definition 7 A function 119909 (minusinfin 119887] rarr 119883 is said to bea mild solution for the system (1) if for each 119906 isin 1198712(119869 119880)1199090 = 120601 isin Bℎ on (minusinfin 0] Δ119909|119905=119905119896 = 119868119896(119909(119905

minus

119896)) 119896 = 1 119898

the restriction of 119909(sdot) to the interval [0 119887) 1199051 119905119898 iscontinuous and the integral equation (15) is satisfied

3 Controllability Results

In this section we formulate and prove a set of sufficientconditions for the exact controllability of impulsive neutralfractional control differential system (1) by using the solutionoperator theory fractional calculations and fixed pointargument To prove the controllability result we need thefollowing hypotheses

(H1) There exists a constant119872 gt 0 such that

10038171003817100381710038171003817119878119902 (119905)

10038171003817100381710038171003817119871(119883)le 119872 forall119905 isin [0 119887] (16)

(H2) The function 119892 119869 times Bℎ rarr 119883 is continuous andthere exists a constant 119871119892 gt 0 such that

1003817100381710038171003817119892 (1199051 1205951) minus 119892 (1199052 1205952)1003817100381710038171003817119883le 119871119892 (

10038161003816100381610038161199051 minus 11990521003816100381610038161003816 +10038171003817100381710038171205951 minus 1205952

1003817100381710038171003817Bℎ)

119905119894 isin 119869 120595119894 isinBℎ 119894 = 1 2

(17)

(H3) There exist constants 1205831 gt 0 and 1205832 gt 0 such that

1003817100381710038171003817119891 (119905 120593 119909) minus 119891 (119905 120595 119910)1003817100381710038171003817119883le 1205831

1003817100381710038171003817120593 minus 1205951003817100381710038171003817Bℎ

+ 12058321003817100381710038171003817119909 minus 119910

1003817100381710038171003817119883

119905 isin 119869 120593 120595 isinBℎ 119909 119910 isin 119883

(18)

(H4) 119868119896 isin 119862(119883119883) and there exist constants 120588 gt 0 suchthat

1003817100381710038171003817119868119896 (119909) minus 119868119896 (119910)1003817100381710038171003817119883le 1205881003817100381710038171003817119909 minus 119910

1003817100381710038171003817119883 119909 119910 isin 119883

for each 119896 = 1 119898(19)

(H5) The linear fractional system (13) is exactly control-lable

Theorem 8 Assume that the hypotheses (H1)ndash(H5) are sat-isfied then the fractional impulsive system (1) is exactlycontrollable on 119869 provided that

= (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588)

+119871119892119862lowast

1+119872119887 (1205831119862

lowast

1+ 1205832119867

lowast) ] lt 1

(20)

where 119862lowast

1= sup

0lt120591lt1198871198621(120591) and 119867lowast

= sup119905isin[0119887]

int119905

0119866(119905 119904)119889119904 lt

infin

Proof For an arbitrary function 119909(sdot) choose the feedbackcontrol function as follows

Abstract and Applied Analysis 5

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) (Γ

1199051

0)minus1

times[1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

) minus int

1199051

0

119878119902 (1199051 minus 119904) times 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) (Γ

1199052

0)minus1

times [1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)] minus 119878119902 (1199052 minus 1199051) 1198681 (119909 (119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

)) minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

) minus int

1199052

0

119878119902 (1199052 minus 119904) times 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) (Γ

119887

0)minus1

times[119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)] minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin (119905119898 119887]

(21)

and define the operator Φ B119887 rarr B119887 by

(Φ119909) (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894)) +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(22)

It should be noted that the control (21) transfers the system(1) from the initial state 120601 to the final state 119909119887 provided thatthe operator Φ has a fixed point In order to prove the exactcontrollability result it is enough to show that the operatorΦhas a fixed point inB119887

Define the function 119910(sdot) (minusinfin 119887] rarr 119883 by

119910 (119905) = 120601 (119905) 119905 isin (minusinfin 0]

0 119905 isin [0 119887] (23)

then 1199100 = 120601 For each 119911 isin 119862(119869 119877) with 119911(0) = 0 let thefunction 119911 be defined by

119911 (119905) = 0 119905 isin (minusinfin 0]

119911 (119905) 119905 isin [0 119887] (24)

If 119909(sdot) satisfies (15) then we can decompose 119909(sdot) as 119909(119905) =119910(119905) + 119911(119905) for 119905 isin 119869 which implies that 119909119905 = 119910119905 + 119911119905 for 119905 isin 119869and the function 119911(sdot) satisfies

6 Abstract and Applied Analysis

119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) times 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119910 (119905minus

1) + 119911 (119905

minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) minus 119892 (119905 119910119905 + 119911119905)]

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(25)

where

119906119910+119911 (119905)

=

119861lowast119878lowast

119902(1199051 minus 119905) (Γ

1199051

0)minus1

times[1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199101199051

+ 1199111199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)) 119889119904)] 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) (Γ

1199052

0)minus1

times [1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)] minus 119878119902 (1199052 minus 1199051) 1198681 (119910 (119905

minus

1) + 119911 (119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)]

+119892 (1199052 1199101199052+ 1199111199052

) minus int

1199052

0

119878119902 (1199052 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) (Γ

119887

0)minus1

times [119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)] minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) + 119892 (119887 119910119887 + 119911119887)

minusint

119887

0

119878119902 (119887 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (119905119898 119887]

(26)

LetB0

119887= 119911 isinB119887 1199110 = 0 isinBℎ For any 119911 isinB

0

119887 we get

119911B0119887

= sup119904isin119869

119911(119904)119883 +100381710038171003817100381711991101003817100381710038171003817Bℎ

= sup119904isin119869

119911(119904)119883 119911 isinB0

119887

(27)

Thus (B0

119887 sdot B0

119887

) is a Banach space Define the operatorΠ B0

119887rarr B0

119887by

Abstract and Applied Analysis 7

Π119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119911 (119905minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119911 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

)]

minus119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(28)

It can be easily seen that the operator Φ has a fixed point ifand only ifΠ has a fixed point Now we will prove thatΠ hasa unique fixed point In order to prove this we show thatΠ isa contraction mapping Let 119911 119911lowast isin B0

119887 for all 119905 isin [0 1199051] we

have

1003817100381710038171003817(Π119911) (119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(1199051 minus 120578) (Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [10038171003817100381710038171003817119892 (1199051 1199101199051

+ 1199111199051) minus 119892 (1199051 1199101199051

+ 119911lowast

1199051)10038171003817100381710038171003817119883

+ int

1199051

0

10038171003817100381710038171003817119878119902 (1199051 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus 119891 (119904 119910119904 + 119911lowast

119904

119867 (119910 (119904) + 119911lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le 119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+1

120574119872

2

119861119872

2int

119905

0

[119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

times1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus 119867 (119910 (119904)

minus119911lowast(119904))1003817100381710038171003817119883) 119889119904] 119889120578

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus119867(119910(119904) minus 119911lowast(119904))1003817100381710038171003817119883)119889119904

le (1 +1

120574119872

2

119861119872

2) [119871119892119862

lowast

1+119872119887 (1205831119862

lowast

1+ 1205832119867

lowast)]

times1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(29)Similarly for 119905 isin (119905119896 119905119896+1] 119896 = 1 2 119898 we can obtain1003817100381710038171003817(Π119911) (119905) minus (Π119911

lowast) (119905)

1003817100381710038171003817119883

le

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894 (119911 (119905minus

119894)) minus 119868119894 (119911 (119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [

10038171003817100381710038171003817100381710038171003817

119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))

10038171003817100381710038171003817100381710038171003817119883

8 Abstract and Applied Analysis

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(119905119896+1 minus 120578)(Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [

119896

sum

119894=1

10038171003817100381710038171003817119878119902(119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894(119911(119905minus

119894)) minus 119868119894(119911(119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [100381710038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894+ 119868119894 (119910119905minus

119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))100381710038171003817100381710038171003817119883

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+10038171003817100381710038171003817119892 (119905119896+1 119910119905119896+1

+ 119911119905119896+1) minus 119892 (119905119896+1 119910119905119896+1

+ 119911lowast

119905119896+1)10038171003817100381710038171003817119883

+ int

119905119896+1

0

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891(119904 119910119904 + 119911lowast

119904 119867(119910(119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le (1 +1

120574119872

2

119861119872

2) [119896119872120588 + 119896119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(30)

Thus for all 119905 isin [0 119887] we get the estimate

1003817100381710038171003817(Π119911)(119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

le 1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(31)

Thus we have (Π119911)(119905) minus (Π119911lowast)(119905)119883 le 119911 minus 119911lowastB0119887

for all119905 isin [0 119887] Since lt 1 this implies that Π is a contractionmapping and hence Π has a unique fixed point 119911 isin B0

119887

Thus the system (1) is exactly controllable on [0 119887]The proofis complete

However the concept of exact controllability is verylimited for many dynamic control systems and the app-roximate controllability is more appropriate for these con-trol systems instead of exact controllability Taking thisinto account in this paper we will also discuss the app-roximate controllability result of the nonlinear impul-sive fractional control system (1) The control system issaid to be approximately controllable if for every initialdata 120601 and every finite time horizon 119887 gt 0 an admis-sible control process can be found such that the corr-esponding solution is arbitrarily close to a given squareintegrable final condition Further approximate controllablesystems are more prevalent and often approximate con-trollability is completely adequate in applications In rec-ent years for deterministic and stochastic control systemsincluding delay term there are several papers devoted tothe study of approximate controllability [23ndash25] Suka-vanam and Kumar [26] obtained a set of conditions whichensure the approximate controllability of a class of semi-linear fractional delay control systems Recently Sakthivelet al [23] formulated and proved a new set of sufficientconditions for approximate controllability of fractional dif-ferential equations by using the fractional calculus theory andsolutions operators

Definition 9 The fractional control system (1) is said to beapproximately controllable on [0 119887] if the closure of thereachable setR(119887 120601) is dense in119883 that isR(119887 120601) = 119883

Remark 10 Assume that the linear fractional control system

119863119902

119905119909 (119905) = 119860119909 (119905) + (119861119906) (119905) 119905 isin [0 119887]

119909 (0) = 120601

(32)

is approximate controllable Let us now introduce the oper-ators associated with (32) as Γ119887

0= int

119887

0S119902(119887 minus 119904)119861119861

lowastSlowast

119902(119887 minus

119904)119889119904 119877(120572 Γ1198870) = (120572119868+Γ

119887

0)minus1 for 120572 gt 0 It should be mentioned

that the approximate controllability of (32) is equivalent tothe convergence of function 120572119877(120572 Γ119887

0) to zero as 120572 rarr 0

+

in the strong operator topology (see [23 27] and referencestherein)

Theorem 11 Assume that conditions (H1)ndash(H4) hold and thatthe family 119878119902(119905) 119905 gt 0 is compact In addition assumethat the function 119891 is uniformly bounded and the linear systemassociated with the system (1) is approximately controllablethen the nonlinear fractional control system with infinite delay(1) is approximately controllable on [0 b]

Proof For each 120572 gt 0 define the operatorΨ B119887 rarr B119887 byΨ119909(119905) = 119911(119905) where

Abstract and Applied Analysis 9

119911 (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909 (sdot)) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

11990521199051) 119901 (119909 (sdot)) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

119905119898) 119901 (119909 (sdot)) 119905 isin (119905119898 119887]

(33)

where

119901 (119909 (sdot))

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

)

minusint

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909 (119905minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(34)

One can easily show that for all 120572 gt 0 the operator Ψ hasa fixed point by employing the technique used in Theorem 8with some changes

Let 119909120572(sdot) be a fixed point of Ψ Further any fixed point ofΨ is a mild solution of (1) under the control

120572(119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(35)

and satisfies

119909120572(1199051) = 1199091199051

minus 120572119877 (120572 Γ1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119909120572(1199052) = 1199091199052

minus 120572119877 (120572 Γ1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119909120572(119887) = 119909119887 minus 120572119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(36)

Moreover by the assumption that 119891 is uniformly boundedthere exists119873 gt 0 such that

int

119887

0

1003817100381710038171003817119891(119904 119909120572

119904 119867119909

120572(119904))1003817100381710038171003817

2119889119904 le 119887119873

2 (37)

and consequently the sequence 119891(119904 119909120572119904 119867119909

120572(119904)) is bounded

in 1198712(119869 119883) Then there is a subsequence still denoted

10 Abstract and Applied Analysis

by 119891(119904 119909120572119904 119867119909

120572(119904)) that converges weakly to say 119891(119904) in

1198712(119869 119883) Now we define

119908

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)]

+119892 (1199051 119909120572

1199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909120572(119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 119909120572

1199051+ 1198681 (119909

120572

119905minus

1

))

minus119892 (1199051 119909120572

1199051)] + 119892 (1199052 119909

120572

1199052)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909120572(119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909120572

119905119894+ 119868119894 (119909

120572

119905minus

119894

))

minus119892 (119905119894 119909120572

119905119894)] + 119892 (119887 119909

120572

119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904) 119889119904 119905 isin (119905119898 119887]

(38)

Now for 119905 isin [0 1199051] we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199051

0

119878119902 (1199051 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin[01199051]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(39)

Also for 119905 isin [119905119894 119905119894+1] 119894 = 1 119898 we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

119905119894+1

0

119878119902 (119905119894+1 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin(119905119894 119905119894+1]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(40)

By using infinite-dimensional version of the Ascoli-Arzelatheorem it is easy to show that an operator 119897(sdot) rarr int

sdot

0119878119902(sdot minus

119904)119897(119904)119889119904 1198711(119869 119883) rarr 119862(119869119883) is compact Hence for all

119905 isin [0 119887] we obtain that 119901(119909120572) minus 119908 rarr 0 as 120572 rarr 0+

Moreover from (36) we get for 119905 isin [0 1199051]10038171003817100381710038171003817119909120572(1199051) minus 1199091199051

10038171003817100381710038171003817le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817

+10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0)10038171003817100381710038171003817

1003817100381710038171003817119901 (119909120572) minus 119908

1003817100381710038171003817

le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817+1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

(41)

It follows from Remark 10 and (39) that 119909120572(1199051) minus 1199091199051 rarr 0

as 120572 rarr 0+ Similarly in the view of (40) for 119905 isin (119905119894 119905119894+1]

119894 = 1 11989810038171003817100381710038171003817119909120572(119905119894+1) minus 119909119905119894+1

10038171003817100381710038171003817997888rarr 0 as 120572 997888rarr 0

+ (42)

Thus for all 119905 isin [0 119887] we get 119909120572(119887) minus 119909119887 rarr 0 as 120572 rarr 0+

This proves the approximate controllability of (1) The proofis completed

Example 12 Now we present an example to illustrate theabstract results of this paper which do not aim at generalitybut indicate how our theorem can be applied to concreteproblems Let 119883 = 119871

2[0 120587] Define 119860 119883 rarr 119883

by 119860119911 = 11991110158401015840 with domain 119863(119860) = 119911 isin 119883

119911 1199111015840 are absolutely continuous 11991110158401015840 isin 119883 119911(0) = 119911(120587) = 0

Then 119860 generates an analytic semigroup 119879(119905) 119905 gt 0 in 119883and it is given by [14]

119879 (119905) 119911 =

infin

sum

119899=1

119890minus1198992119905(119911 119890119899) 119890119899 119911 isin 119883 (43)

where 119890119899(119910) = radic2120587 sin 119899119910 119899 = 1 2 3 is the orthogonalset of eigenvectors of 119860 Also define an infinite dimensionalcontrol space 119880 by 119880 = 119906 | 119906 = sum

infin

119899=2119906119899119890119899 with sum

infin

119899=21199062

119899lt

infin with norm defined by 119906119880 = (suminfin

119899=21199062

119899)12 Define a

continuous linear map 119861 from 119880 to 119883 as 119861119906 = 211990621198901 +

suminfin

119899=2119906119899119890119899 for 119906 = sum

infin

119899=2119906119899119890119899 isin 119880

Note that the subordination principle of solution operatorimplies that 119860 is the infinitesimal generator of a solutionoperator 119878119902(119905)119905ge0 Since 119878119902(119905) is strongly continuous on[0infin) by the uniformly bounded theorem there exists aconstant119872 gt 0 such that 119878119902(119905)119871(119883)

le 119872 for 119905 isin [0 119887] [14]Consider the following fractional partial integrodifferen-

tial equation with infinite delay and control in the followingform

119888119863

119902

119905[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

=1205972

1205971199102[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

+1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902

times [120583 (119904 119910) + int

119904

minusinfin

119870 (119904 119909 120585 minus 119904)

times1198762 (119909 (120585 119910)) 119889120585

+ int

119904

0

119892 (120585 119904) 119890minus119909(120585119910)

119889120585] 119889119904

119905 isin 119869 = [0 1] 119910 isin [0 120587] 119905 = 119905119896

(44)

with 119909(119905 0) = 119909(119905 120587) = 0 119909(119905 119910) = 120601(119905 119910) 119905 isin (minusinfin 0] 119910 isin[0 120587] Δ119909(119905119894)(119910) = int

119905119894

minusinfin119902119894(119905119894 minus 119904)119909(119904 119910)119889119904 and 119910 isin [0 120587]

Abstract and Applied Analysis 11

where 120597119902119905is the Caputo fractional partial derivative of order

0 lt 119902 lt 1 120583 [0 1] times [0 120587] rarr [0 120587] is continuous and in119905 119902119894 119877 rarr 119877 are continuous 0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899 lt 119887 areprefixed numbers and 120601 isinBℎ

Let ℎ(119904) = 1198902119904 119904 lt 0 then 119897 = int

0

infinℎ(119904)119889119904 = 12 Let

Bℎ be the phase space endowed with the norm 120593Bℎ=

int0

minusinfinℎ(119904)sup

120579isin[1199040]120593(120579)

1198712119889119904 Let 119909(119905)(119910) = 119909(119905 119910) and define

the bounded linear operator 119861 119880 rarr 119883 by (119861119906)(119905)(119910) =120583(119905 119910) 0 le 119910 le 120587 119892(119905 120601)(119910) = int0

minusinfin119886(119905 119910 120579)1198761(120601(120579)(119910))119889120579

119868119896(119909(119905minus

119894))(119910) = int

119905119894

minusinfin119902119894(119905119894minus119904)119909(119904 119910)119889119904 and119891(119905 120601119867119909(119905))(119910) =

int0

minusinfin119870(119905 119910 120579)1198762(120601(120579)(119910))119889120579 + 119867119909(119905)(119910) where 119867119909(119905)(119910) =

int119905

0119892(119904 119905)119890

minus119909(119904119910)119889119904 and 120601(120579)(119910) = 120601(120579 119910) (120579 119910) isin (minusinfin 0] times

[0 120587] Moreover the linear fractional control system corre-sponding to (44) is exactly controllable Further if we imposesuitable conditions on 119886 119870 1198761 1198762 119892 119902119894 and 119861 to verifyassumptions onTheorem 8 then the system (44) can be writ-ten in the abstract form of (1)Therefore all the conditions ofTheorem 8 are satisfied and hence the nonlinear fractionalcontrol system (44) is exactly controllable on [0 119887]

Acknowledgment

The work of Yong Ren is supported by the National NaturalScience Foundation of China (no 11371029)

References

[1] P Muthukumar and P Balasubramaniam ldquoApproximate con-trollability of mixed stochastic Volterra-Fredholm type inte-grodifferential systems in Hilbert spacerdquo Journal of the FranklinInstitute vol 348 no 10 pp 2911ndash2922 2011

[2] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009

[3] J Klamka ldquoConstrained controllability of semilinear systemswith delaysrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 169ndash177 2009

[4] J Klamka ldquoStochastic controllability of systems with multipledelays in controlrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 39ndash47 2009

[5] J Klamka ldquoLocal controllability of fractional discrete-timesemilinear systemsrdquo Acta Mechanica et Automatica vol 15 pp55ndash58 2011

[6] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[7] J Dabas A Chauhan and M Kumar ldquoExistence of the mildsolutions for impulsive fractional equations with infinite delayrdquoInternational Journal of Differential Equations vol 2011 ArticleID 793023 20 pages 2011

[8] J Wang M Feckan and Y Zhou ldquoRelaxed controls fornonlinear fractional impulsive evolution equationsrdquo Journal ofOptimization Theory and Applications vol 156 no 1 pp 13ndash322013

[9] M Haase The Functional Calculus for Sectorial Operators vol169 of Operator Theory Advances and Applications BirkhauserBasel Switzerland 2006

[10] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[11] R P Agarwal B de Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[12] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functional evo-lution equations with infinite delayrdquo Applied Mathematics andComputation vol 216 no 1 pp 61ndash69 2010

[13] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications A vol 74 no 5pp 2003ndash2011 2011

[14] J Dabas and A Chauhan ldquoExistence and uniqueness of mildsolution for an impulsive neutral fractional integro-differentialequation with infinite delayrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 754ndash763 2013

[15] R Sakthivel and Y Ren ldquoComplete controllability of stochasticevolution equations with jumpsrdquo Reports on MathematicalPhysics vol 68 no 2 pp 163ndash174 2011

[16] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters of Rapid Publication vol 26 no 4 pp 418ndash4242013

[17] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[18] S Kumar and N Sukavanam ldquoApproximate controllability offractional order semilinear systems with bounded delayrdquo Jour-nal of Differential Equations vol 251 no 11 pp 6163ndash6174 2012

[19] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[20] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends inNanotechnology and Fractional Calculus D Baleanu Z BGuvenc and J A T Machado Eds pp 503ndash509 Springer NewYork NY USA 2010

[21] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[22] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[23] R Sakthivel S Suganya and S M Anthoni ldquoApproximate con-trollability of fractional stochastic evolution equationsrdquo Com-puters amp Mathematics with Applications vol 63 no 3 pp 660ndash668 2012

[24] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[25] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[26] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) (Γ

1199051

0)minus1

times[1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

) minus int

1199051

0

119878119902 (1199051 minus 119904) times 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) (Γ

1199052

0)minus1

times [1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)] minus 119878119902 (1199052 minus 1199051) 1198681 (119909 (119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

)) minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

) minus int

1199052

0

119878119902 (1199052 minus 119904) times 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) (Γ

119887

0)minus1

times[119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)] minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904] 119905 isin (119905119898 119887]

(21)

and define the operator Φ B119887 rarr B119887 by

(Φ119909) (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894)) +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(22)

It should be noted that the control (21) transfers the system(1) from the initial state 120601 to the final state 119909119887 provided thatthe operator Φ has a fixed point In order to prove the exactcontrollability result it is enough to show that the operatorΦhas a fixed point inB119887

Define the function 119910(sdot) (minusinfin 119887] rarr 119883 by

119910 (119905) = 120601 (119905) 119905 isin (minusinfin 0]

0 119905 isin [0 119887] (23)

then 1199100 = 120601 For each 119911 isin 119862(119869 119877) with 119911(0) = 0 let thefunction 119911 be defined by

119911 (119905) = 0 119905 isin (minusinfin 0]

119911 (119905) 119905 isin [0 119887] (24)

If 119909(sdot) satisfies (15) then we can decompose 119909(sdot) as 119909(119905) =119910(119905) + 119911(119905) for 119905 isin 119869 which implies that 119909119905 = 119910119905 + 119911119905 for 119905 isin 119869and the function 119911(sdot) satisfies

6 Abstract and Applied Analysis

119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) times 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119910 (119905minus

1) + 119911 (119905

minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) minus 119892 (119905 119910119905 + 119911119905)]

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(25)

where

119906119910+119911 (119905)

=

119861lowast119878lowast

119902(1199051 minus 119905) (Γ

1199051

0)minus1

times[1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199101199051

+ 1199111199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)) 119889119904)] 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) (Γ

1199052

0)minus1

times [1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)] minus 119878119902 (1199052 minus 1199051) 1198681 (119910 (119905

minus

1) + 119911 (119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)]

+119892 (1199052 1199101199052+ 1199111199052

) minus int

1199052

0

119878119902 (1199052 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) (Γ

119887

0)minus1

times [119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)] minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) + 119892 (119887 119910119887 + 119911119887)

minusint

119887

0

119878119902 (119887 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (119905119898 119887]

(26)

LetB0

119887= 119911 isinB119887 1199110 = 0 isinBℎ For any 119911 isinB

0

119887 we get

119911B0119887

= sup119904isin119869

119911(119904)119883 +100381710038171003817100381711991101003817100381710038171003817Bℎ

= sup119904isin119869

119911(119904)119883 119911 isinB0

119887

(27)

Thus (B0

119887 sdot B0

119887

) is a Banach space Define the operatorΠ B0

119887rarr B0

119887by

Abstract and Applied Analysis 7

Π119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119911 (119905minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119911 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

)]

minus119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(28)

It can be easily seen that the operator Φ has a fixed point ifand only ifΠ has a fixed point Now we will prove thatΠ hasa unique fixed point In order to prove this we show thatΠ isa contraction mapping Let 119911 119911lowast isin B0

119887 for all 119905 isin [0 1199051] we

have

1003817100381710038171003817(Π119911) (119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(1199051 minus 120578) (Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [10038171003817100381710038171003817119892 (1199051 1199101199051

+ 1199111199051) minus 119892 (1199051 1199101199051

+ 119911lowast

1199051)10038171003817100381710038171003817119883

+ int

1199051

0

10038171003817100381710038171003817119878119902 (1199051 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus 119891 (119904 119910119904 + 119911lowast

119904

119867 (119910 (119904) + 119911lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le 119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+1

120574119872

2

119861119872

2int

119905

0

[119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

times1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus 119867 (119910 (119904)

minus119911lowast(119904))1003817100381710038171003817119883) 119889119904] 119889120578

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus119867(119910(119904) minus 119911lowast(119904))1003817100381710038171003817119883)119889119904

le (1 +1

120574119872

2

119861119872

2) [119871119892119862

lowast

1+119872119887 (1205831119862

lowast

1+ 1205832119867

lowast)]

times1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(29)Similarly for 119905 isin (119905119896 119905119896+1] 119896 = 1 2 119898 we can obtain1003817100381710038171003817(Π119911) (119905) minus (Π119911

lowast) (119905)

1003817100381710038171003817119883

le

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894 (119911 (119905minus

119894)) minus 119868119894 (119911 (119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [

10038171003817100381710038171003817100381710038171003817

119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))

10038171003817100381710038171003817100381710038171003817119883

8 Abstract and Applied Analysis

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(119905119896+1 minus 120578)(Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [

119896

sum

119894=1

10038171003817100381710038171003817119878119902(119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894(119911(119905minus

119894)) minus 119868119894(119911(119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [100381710038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894+ 119868119894 (119910119905minus

119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))100381710038171003817100381710038171003817119883

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+10038171003817100381710038171003817119892 (119905119896+1 119910119905119896+1

+ 119911119905119896+1) minus 119892 (119905119896+1 119910119905119896+1

+ 119911lowast

119905119896+1)10038171003817100381710038171003817119883

+ int

119905119896+1

0

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891(119904 119910119904 + 119911lowast

119904 119867(119910(119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le (1 +1

120574119872

2

119861119872

2) [119896119872120588 + 119896119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(30)

Thus for all 119905 isin [0 119887] we get the estimate

1003817100381710038171003817(Π119911)(119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

le 1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(31)

Thus we have (Π119911)(119905) minus (Π119911lowast)(119905)119883 le 119911 minus 119911lowastB0119887

for all119905 isin [0 119887] Since lt 1 this implies that Π is a contractionmapping and hence Π has a unique fixed point 119911 isin B0

119887

Thus the system (1) is exactly controllable on [0 119887]The proofis complete

However the concept of exact controllability is verylimited for many dynamic control systems and the app-roximate controllability is more appropriate for these con-trol systems instead of exact controllability Taking thisinto account in this paper we will also discuss the app-roximate controllability result of the nonlinear impul-sive fractional control system (1) The control system issaid to be approximately controllable if for every initialdata 120601 and every finite time horizon 119887 gt 0 an admis-sible control process can be found such that the corr-esponding solution is arbitrarily close to a given squareintegrable final condition Further approximate controllablesystems are more prevalent and often approximate con-trollability is completely adequate in applications In rec-ent years for deterministic and stochastic control systemsincluding delay term there are several papers devoted tothe study of approximate controllability [23ndash25] Suka-vanam and Kumar [26] obtained a set of conditions whichensure the approximate controllability of a class of semi-linear fractional delay control systems Recently Sakthivelet al [23] formulated and proved a new set of sufficientconditions for approximate controllability of fractional dif-ferential equations by using the fractional calculus theory andsolutions operators

Definition 9 The fractional control system (1) is said to beapproximately controllable on [0 119887] if the closure of thereachable setR(119887 120601) is dense in119883 that isR(119887 120601) = 119883

Remark 10 Assume that the linear fractional control system

119863119902

119905119909 (119905) = 119860119909 (119905) + (119861119906) (119905) 119905 isin [0 119887]

119909 (0) = 120601

(32)

is approximate controllable Let us now introduce the oper-ators associated with (32) as Γ119887

0= int

119887

0S119902(119887 minus 119904)119861119861

lowastSlowast

119902(119887 minus

119904)119889119904 119877(120572 Γ1198870) = (120572119868+Γ

119887

0)minus1 for 120572 gt 0 It should be mentioned

that the approximate controllability of (32) is equivalent tothe convergence of function 120572119877(120572 Γ119887

0) to zero as 120572 rarr 0

+

in the strong operator topology (see [23 27] and referencestherein)

Theorem 11 Assume that conditions (H1)ndash(H4) hold and thatthe family 119878119902(119905) 119905 gt 0 is compact In addition assumethat the function 119891 is uniformly bounded and the linear systemassociated with the system (1) is approximately controllablethen the nonlinear fractional control system with infinite delay(1) is approximately controllable on [0 b]

Proof For each 120572 gt 0 define the operatorΨ B119887 rarr B119887 byΨ119909(119905) = 119911(119905) where

Abstract and Applied Analysis 9

119911 (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909 (sdot)) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

11990521199051) 119901 (119909 (sdot)) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

119905119898) 119901 (119909 (sdot)) 119905 isin (119905119898 119887]

(33)

where

119901 (119909 (sdot))

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

)

minusint

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909 (119905minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(34)

One can easily show that for all 120572 gt 0 the operator Ψ hasa fixed point by employing the technique used in Theorem 8with some changes

Let 119909120572(sdot) be a fixed point of Ψ Further any fixed point ofΨ is a mild solution of (1) under the control

120572(119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(35)

and satisfies

119909120572(1199051) = 1199091199051

minus 120572119877 (120572 Γ1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119909120572(1199052) = 1199091199052

minus 120572119877 (120572 Γ1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119909120572(119887) = 119909119887 minus 120572119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(36)

Moreover by the assumption that 119891 is uniformly boundedthere exists119873 gt 0 such that

int

119887

0

1003817100381710038171003817119891(119904 119909120572

119904 119867119909

120572(119904))1003817100381710038171003817

2119889119904 le 119887119873

2 (37)

and consequently the sequence 119891(119904 119909120572119904 119867119909

120572(119904)) is bounded

in 1198712(119869 119883) Then there is a subsequence still denoted

10 Abstract and Applied Analysis

by 119891(119904 119909120572119904 119867119909

120572(119904)) that converges weakly to say 119891(119904) in

1198712(119869 119883) Now we define

119908

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)]

+119892 (1199051 119909120572

1199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909120572(119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 119909120572

1199051+ 1198681 (119909

120572

119905minus

1

))

minus119892 (1199051 119909120572

1199051)] + 119892 (1199052 119909

120572

1199052)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909120572(119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909120572

119905119894+ 119868119894 (119909

120572

119905minus

119894

))

minus119892 (119905119894 119909120572

119905119894)] + 119892 (119887 119909

120572

119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904) 119889119904 119905 isin (119905119898 119887]

(38)

Now for 119905 isin [0 1199051] we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199051

0

119878119902 (1199051 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin[01199051]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(39)

Also for 119905 isin [119905119894 119905119894+1] 119894 = 1 119898 we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

119905119894+1

0

119878119902 (119905119894+1 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin(119905119894 119905119894+1]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(40)

By using infinite-dimensional version of the Ascoli-Arzelatheorem it is easy to show that an operator 119897(sdot) rarr int

sdot

0119878119902(sdot minus

119904)119897(119904)119889119904 1198711(119869 119883) rarr 119862(119869119883) is compact Hence for all

119905 isin [0 119887] we obtain that 119901(119909120572) minus 119908 rarr 0 as 120572 rarr 0+

Moreover from (36) we get for 119905 isin [0 1199051]10038171003817100381710038171003817119909120572(1199051) minus 1199091199051

10038171003817100381710038171003817le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817

+10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0)10038171003817100381710038171003817

1003817100381710038171003817119901 (119909120572) minus 119908

1003817100381710038171003817

le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817+1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

(41)

It follows from Remark 10 and (39) that 119909120572(1199051) minus 1199091199051 rarr 0

as 120572 rarr 0+ Similarly in the view of (40) for 119905 isin (119905119894 119905119894+1]

119894 = 1 11989810038171003817100381710038171003817119909120572(119905119894+1) minus 119909119905119894+1

10038171003817100381710038171003817997888rarr 0 as 120572 997888rarr 0

+ (42)

Thus for all 119905 isin [0 119887] we get 119909120572(119887) minus 119909119887 rarr 0 as 120572 rarr 0+

This proves the approximate controllability of (1) The proofis completed

Example 12 Now we present an example to illustrate theabstract results of this paper which do not aim at generalitybut indicate how our theorem can be applied to concreteproblems Let 119883 = 119871

2[0 120587] Define 119860 119883 rarr 119883

by 119860119911 = 11991110158401015840 with domain 119863(119860) = 119911 isin 119883

119911 1199111015840 are absolutely continuous 11991110158401015840 isin 119883 119911(0) = 119911(120587) = 0

Then 119860 generates an analytic semigroup 119879(119905) 119905 gt 0 in 119883and it is given by [14]

119879 (119905) 119911 =

infin

sum

119899=1

119890minus1198992119905(119911 119890119899) 119890119899 119911 isin 119883 (43)

where 119890119899(119910) = radic2120587 sin 119899119910 119899 = 1 2 3 is the orthogonalset of eigenvectors of 119860 Also define an infinite dimensionalcontrol space 119880 by 119880 = 119906 | 119906 = sum

infin

119899=2119906119899119890119899 with sum

infin

119899=21199062

119899lt

infin with norm defined by 119906119880 = (suminfin

119899=21199062

119899)12 Define a

continuous linear map 119861 from 119880 to 119883 as 119861119906 = 211990621198901 +

suminfin

119899=2119906119899119890119899 for 119906 = sum

infin

119899=2119906119899119890119899 isin 119880

Note that the subordination principle of solution operatorimplies that 119860 is the infinitesimal generator of a solutionoperator 119878119902(119905)119905ge0 Since 119878119902(119905) is strongly continuous on[0infin) by the uniformly bounded theorem there exists aconstant119872 gt 0 such that 119878119902(119905)119871(119883)

le 119872 for 119905 isin [0 119887] [14]Consider the following fractional partial integrodifferen-

tial equation with infinite delay and control in the followingform

119888119863

119902

119905[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

=1205972

1205971199102[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

+1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902

times [120583 (119904 119910) + int

119904

minusinfin

119870 (119904 119909 120585 minus 119904)

times1198762 (119909 (120585 119910)) 119889120585

+ int

119904

0

119892 (120585 119904) 119890minus119909(120585119910)

119889120585] 119889119904

119905 isin 119869 = [0 1] 119910 isin [0 120587] 119905 = 119905119896

(44)

with 119909(119905 0) = 119909(119905 120587) = 0 119909(119905 119910) = 120601(119905 119910) 119905 isin (minusinfin 0] 119910 isin[0 120587] Δ119909(119905119894)(119910) = int

119905119894

minusinfin119902119894(119905119894 minus 119904)119909(119904 119910)119889119904 and 119910 isin [0 120587]

Abstract and Applied Analysis 11

where 120597119902119905is the Caputo fractional partial derivative of order

0 lt 119902 lt 1 120583 [0 1] times [0 120587] rarr [0 120587] is continuous and in119905 119902119894 119877 rarr 119877 are continuous 0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899 lt 119887 areprefixed numbers and 120601 isinBℎ

Let ℎ(119904) = 1198902119904 119904 lt 0 then 119897 = int

0

infinℎ(119904)119889119904 = 12 Let

Bℎ be the phase space endowed with the norm 120593Bℎ=

int0

minusinfinℎ(119904)sup

120579isin[1199040]120593(120579)

1198712119889119904 Let 119909(119905)(119910) = 119909(119905 119910) and define

the bounded linear operator 119861 119880 rarr 119883 by (119861119906)(119905)(119910) =120583(119905 119910) 0 le 119910 le 120587 119892(119905 120601)(119910) = int0

minusinfin119886(119905 119910 120579)1198761(120601(120579)(119910))119889120579

119868119896(119909(119905minus

119894))(119910) = int

119905119894

minusinfin119902119894(119905119894minus119904)119909(119904 119910)119889119904 and119891(119905 120601119867119909(119905))(119910) =

int0

minusinfin119870(119905 119910 120579)1198762(120601(120579)(119910))119889120579 + 119867119909(119905)(119910) where 119867119909(119905)(119910) =

int119905

0119892(119904 119905)119890

minus119909(119904119910)119889119904 and 120601(120579)(119910) = 120601(120579 119910) (120579 119910) isin (minusinfin 0] times

[0 120587] Moreover the linear fractional control system corre-sponding to (44) is exactly controllable Further if we imposesuitable conditions on 119886 119870 1198761 1198762 119892 119902119894 and 119861 to verifyassumptions onTheorem 8 then the system (44) can be writ-ten in the abstract form of (1)Therefore all the conditions ofTheorem 8 are satisfied and hence the nonlinear fractionalcontrol system (44) is exactly controllable on [0 119887]

Acknowledgment

The work of Yong Ren is supported by the National NaturalScience Foundation of China (no 11371029)

References

[1] P Muthukumar and P Balasubramaniam ldquoApproximate con-trollability of mixed stochastic Volterra-Fredholm type inte-grodifferential systems in Hilbert spacerdquo Journal of the FranklinInstitute vol 348 no 10 pp 2911ndash2922 2011

[2] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009

[3] J Klamka ldquoConstrained controllability of semilinear systemswith delaysrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 169ndash177 2009

[4] J Klamka ldquoStochastic controllability of systems with multipledelays in controlrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 39ndash47 2009

[5] J Klamka ldquoLocal controllability of fractional discrete-timesemilinear systemsrdquo Acta Mechanica et Automatica vol 15 pp55ndash58 2011

[6] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[7] J Dabas A Chauhan and M Kumar ldquoExistence of the mildsolutions for impulsive fractional equations with infinite delayrdquoInternational Journal of Differential Equations vol 2011 ArticleID 793023 20 pages 2011

[8] J Wang M Feckan and Y Zhou ldquoRelaxed controls fornonlinear fractional impulsive evolution equationsrdquo Journal ofOptimization Theory and Applications vol 156 no 1 pp 13ndash322013

[9] M Haase The Functional Calculus for Sectorial Operators vol169 of Operator Theory Advances and Applications BirkhauserBasel Switzerland 2006

[10] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[11] R P Agarwal B de Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[12] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functional evo-lution equations with infinite delayrdquo Applied Mathematics andComputation vol 216 no 1 pp 61ndash69 2010

[13] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications A vol 74 no 5pp 2003ndash2011 2011

[14] J Dabas and A Chauhan ldquoExistence and uniqueness of mildsolution for an impulsive neutral fractional integro-differentialequation with infinite delayrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 754ndash763 2013

[15] R Sakthivel and Y Ren ldquoComplete controllability of stochasticevolution equations with jumpsrdquo Reports on MathematicalPhysics vol 68 no 2 pp 163ndash174 2011

[16] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters of Rapid Publication vol 26 no 4 pp 418ndash4242013

[17] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[18] S Kumar and N Sukavanam ldquoApproximate controllability offractional order semilinear systems with bounded delayrdquo Jour-nal of Differential Equations vol 251 no 11 pp 6163ndash6174 2012

[19] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[20] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends inNanotechnology and Fractional Calculus D Baleanu Z BGuvenc and J A T Machado Eds pp 503ndash509 Springer NewYork NY USA 2010

[21] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[22] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[23] R Sakthivel S Suganya and S M Anthoni ldquoApproximate con-trollability of fractional stochastic evolution equationsrdquo Com-puters amp Mathematics with Applications vol 63 no 3 pp 660ndash668 2012

[24] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[25] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[26] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) times 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119910 (119905minus

1) + 119911 (119905

minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) minus 119892 (119905 119910119905 + 119911119905)]

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(25)

where

119906119910+119911 (119905)

=

119861lowast119878lowast

119902(1199051 minus 119905) (Γ

1199051

0)minus1

times[1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199101199051

+ 1199111199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)) 119889119904)] 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) (Γ

1199052

0)minus1

times [1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)] minus 119878119902 (1199052 minus 1199051) 1198681 (119910 (119905

minus

1) + 119911 (119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)]

+119892 (1199052 1199101199052+ 1199111199052

) minus int

1199052

0

119878119902 (1199052 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) (Γ

119887

0)minus1

times [119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)] minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119910 (119905minus

119894) + 119911 (119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

) + 119892 (119887 119910119887 + 119911119887)

minusint

119887

0

119878119902 (119887 minus 119904) times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904] 119905 isin (119905119898 119887]

(26)

LetB0

119887= 119911 isinB119887 1199110 = 0 isinBℎ For any 119911 isinB

0

119887 we get

119911B0119887

= sup119904isin119869

119911(119904)119883 +100381710038171003817100381711991101003817100381710038171003817Bℎ

= sup119904isin119869

119911(119904)119883 119911 isinB0

119887

(27)

Thus (B0

119887 sdot B0

119887

) is a Banach space Define the operatorΠ B0

119887rarr B0

119887by

Abstract and Applied Analysis 7

Π119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119911 (119905minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119911 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

)]

minus119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(28)

It can be easily seen that the operator Φ has a fixed point ifand only ifΠ has a fixed point Now we will prove thatΠ hasa unique fixed point In order to prove this we show thatΠ isa contraction mapping Let 119911 119911lowast isin B0

119887 for all 119905 isin [0 1199051] we

have

1003817100381710038171003817(Π119911) (119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(1199051 minus 120578) (Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [10038171003817100381710038171003817119892 (1199051 1199101199051

+ 1199111199051) minus 119892 (1199051 1199101199051

+ 119911lowast

1199051)10038171003817100381710038171003817119883

+ int

1199051

0

10038171003817100381710038171003817119878119902 (1199051 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus 119891 (119904 119910119904 + 119911lowast

119904

119867 (119910 (119904) + 119911lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le 119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+1

120574119872

2

119861119872

2int

119905

0

[119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

times1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus 119867 (119910 (119904)

minus119911lowast(119904))1003817100381710038171003817119883) 119889119904] 119889120578

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus119867(119910(119904) minus 119911lowast(119904))1003817100381710038171003817119883)119889119904

le (1 +1

120574119872

2

119861119872

2) [119871119892119862

lowast

1+119872119887 (1205831119862

lowast

1+ 1205832119867

lowast)]

times1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(29)Similarly for 119905 isin (119905119896 119905119896+1] 119896 = 1 2 119898 we can obtain1003817100381710038171003817(Π119911) (119905) minus (Π119911

lowast) (119905)

1003817100381710038171003817119883

le

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894 (119911 (119905minus

119894)) minus 119868119894 (119911 (119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [

10038171003817100381710038171003817100381710038171003817

119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))

10038171003817100381710038171003817100381710038171003817119883

8 Abstract and Applied Analysis

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(119905119896+1 minus 120578)(Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [

119896

sum

119894=1

10038171003817100381710038171003817119878119902(119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894(119911(119905minus

119894)) minus 119868119894(119911(119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [100381710038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894+ 119868119894 (119910119905minus

119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))100381710038171003817100381710038171003817119883

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+10038171003817100381710038171003817119892 (119905119896+1 119910119905119896+1

+ 119911119905119896+1) minus 119892 (119905119896+1 119910119905119896+1

+ 119911lowast

119905119896+1)10038171003817100381710038171003817119883

+ int

119905119896+1

0

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891(119904 119910119904 + 119911lowast

119904 119867(119910(119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le (1 +1

120574119872

2

119861119872

2) [119896119872120588 + 119896119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(30)

Thus for all 119905 isin [0 119887] we get the estimate

1003817100381710038171003817(Π119911)(119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

le 1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(31)

Thus we have (Π119911)(119905) minus (Π119911lowast)(119905)119883 le 119911 minus 119911lowastB0119887

for all119905 isin [0 119887] Since lt 1 this implies that Π is a contractionmapping and hence Π has a unique fixed point 119911 isin B0

119887

Thus the system (1) is exactly controllable on [0 119887]The proofis complete

However the concept of exact controllability is verylimited for many dynamic control systems and the app-roximate controllability is more appropriate for these con-trol systems instead of exact controllability Taking thisinto account in this paper we will also discuss the app-roximate controllability result of the nonlinear impul-sive fractional control system (1) The control system issaid to be approximately controllable if for every initialdata 120601 and every finite time horizon 119887 gt 0 an admis-sible control process can be found such that the corr-esponding solution is arbitrarily close to a given squareintegrable final condition Further approximate controllablesystems are more prevalent and often approximate con-trollability is completely adequate in applications In rec-ent years for deterministic and stochastic control systemsincluding delay term there are several papers devoted tothe study of approximate controllability [23ndash25] Suka-vanam and Kumar [26] obtained a set of conditions whichensure the approximate controllability of a class of semi-linear fractional delay control systems Recently Sakthivelet al [23] formulated and proved a new set of sufficientconditions for approximate controllability of fractional dif-ferential equations by using the fractional calculus theory andsolutions operators

Definition 9 The fractional control system (1) is said to beapproximately controllable on [0 119887] if the closure of thereachable setR(119887 120601) is dense in119883 that isR(119887 120601) = 119883

Remark 10 Assume that the linear fractional control system

119863119902

119905119909 (119905) = 119860119909 (119905) + (119861119906) (119905) 119905 isin [0 119887]

119909 (0) = 120601

(32)

is approximate controllable Let us now introduce the oper-ators associated with (32) as Γ119887

0= int

119887

0S119902(119887 minus 119904)119861119861

lowastSlowast

119902(119887 minus

119904)119889119904 119877(120572 Γ1198870) = (120572119868+Γ

119887

0)minus1 for 120572 gt 0 It should be mentioned

that the approximate controllability of (32) is equivalent tothe convergence of function 120572119877(120572 Γ119887

0) to zero as 120572 rarr 0

+

in the strong operator topology (see [23 27] and referencestherein)

Theorem 11 Assume that conditions (H1)ndash(H4) hold and thatthe family 119878119902(119905) 119905 gt 0 is compact In addition assumethat the function 119891 is uniformly bounded and the linear systemassociated with the system (1) is approximately controllablethen the nonlinear fractional control system with infinite delay(1) is approximately controllable on [0 b]

Proof For each 120572 gt 0 define the operatorΨ B119887 rarr B119887 byΨ119909(119905) = 119911(119905) where

Abstract and Applied Analysis 9

119911 (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909 (sdot)) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

11990521199051) 119901 (119909 (sdot)) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

119905119898) 119901 (119909 (sdot)) 119905 isin (119905119898 119887]

(33)

where

119901 (119909 (sdot))

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

)

minusint

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909 (119905minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(34)

One can easily show that for all 120572 gt 0 the operator Ψ hasa fixed point by employing the technique used in Theorem 8with some changes

Let 119909120572(sdot) be a fixed point of Ψ Further any fixed point ofΨ is a mild solution of (1) under the control

120572(119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(35)

and satisfies

119909120572(1199051) = 1199091199051

minus 120572119877 (120572 Γ1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119909120572(1199052) = 1199091199052

minus 120572119877 (120572 Γ1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119909120572(119887) = 119909119887 minus 120572119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(36)

Moreover by the assumption that 119891 is uniformly boundedthere exists119873 gt 0 such that

int

119887

0

1003817100381710038171003817119891(119904 119909120572

119904 119867119909

120572(119904))1003817100381710038171003817

2119889119904 le 119887119873

2 (37)

and consequently the sequence 119891(119904 119909120572119904 119867119909

120572(119904)) is bounded

in 1198712(119869 119883) Then there is a subsequence still denoted

10 Abstract and Applied Analysis

by 119891(119904 119909120572119904 119867119909

120572(119904)) that converges weakly to say 119891(119904) in

1198712(119869 119883) Now we define

119908

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)]

+119892 (1199051 119909120572

1199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909120572(119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 119909120572

1199051+ 1198681 (119909

120572

119905minus

1

))

minus119892 (1199051 119909120572

1199051)] + 119892 (1199052 119909

120572

1199052)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909120572(119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909120572

119905119894+ 119868119894 (119909

120572

119905minus

119894

))

minus119892 (119905119894 119909120572

119905119894)] + 119892 (119887 119909

120572

119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904) 119889119904 119905 isin (119905119898 119887]

(38)

Now for 119905 isin [0 1199051] we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199051

0

119878119902 (1199051 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin[01199051]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(39)

Also for 119905 isin [119905119894 119905119894+1] 119894 = 1 119898 we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

119905119894+1

0

119878119902 (119905119894+1 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin(119905119894 119905119894+1]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(40)

By using infinite-dimensional version of the Ascoli-Arzelatheorem it is easy to show that an operator 119897(sdot) rarr int

sdot

0119878119902(sdot minus

119904)119897(119904)119889119904 1198711(119869 119883) rarr 119862(119869119883) is compact Hence for all

119905 isin [0 119887] we obtain that 119901(119909120572) minus 119908 rarr 0 as 120572 rarr 0+

Moreover from (36) we get for 119905 isin [0 1199051]10038171003817100381710038171003817119909120572(1199051) minus 1199091199051

10038171003817100381710038171003817le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817

+10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0)10038171003817100381710038171003817

1003817100381710038171003817119901 (119909120572) minus 119908

1003817100381710038171003817

le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817+1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

(41)

It follows from Remark 10 and (39) that 119909120572(1199051) minus 1199091199051 rarr 0

as 120572 rarr 0+ Similarly in the view of (40) for 119905 isin (119905119894 119905119894+1]

119894 = 1 11989810038171003817100381710038171003817119909120572(119905119894+1) minus 119909119905119894+1

10038171003817100381710038171003817997888rarr 0 as 120572 997888rarr 0

+ (42)

Thus for all 119905 isin [0 119887] we get 119909120572(119887) minus 119909119887 rarr 0 as 120572 rarr 0+

This proves the approximate controllability of (1) The proofis completed

Example 12 Now we present an example to illustrate theabstract results of this paper which do not aim at generalitybut indicate how our theorem can be applied to concreteproblems Let 119883 = 119871

2[0 120587] Define 119860 119883 rarr 119883

by 119860119911 = 11991110158401015840 with domain 119863(119860) = 119911 isin 119883

119911 1199111015840 are absolutely continuous 11991110158401015840 isin 119883 119911(0) = 119911(120587) = 0

Then 119860 generates an analytic semigroup 119879(119905) 119905 gt 0 in 119883and it is given by [14]

119879 (119905) 119911 =

infin

sum

119899=1

119890minus1198992119905(119911 119890119899) 119890119899 119911 isin 119883 (43)

where 119890119899(119910) = radic2120587 sin 119899119910 119899 = 1 2 3 is the orthogonalset of eigenvectors of 119860 Also define an infinite dimensionalcontrol space 119880 by 119880 = 119906 | 119906 = sum

infin

119899=2119906119899119890119899 with sum

infin

119899=21199062

119899lt

infin with norm defined by 119906119880 = (suminfin

119899=21199062

119899)12 Define a

continuous linear map 119861 from 119880 to 119883 as 119861119906 = 211990621198901 +

suminfin

119899=2119906119899119890119899 for 119906 = sum

infin

119899=2119906119899119890119899 isin 119880

Note that the subordination principle of solution operatorimplies that 119860 is the infinitesimal generator of a solutionoperator 119878119902(119905)119905ge0 Since 119878119902(119905) is strongly continuous on[0infin) by the uniformly bounded theorem there exists aconstant119872 gt 0 such that 119878119902(119905)119871(119883)

le 119872 for 119905 isin [0 119887] [14]Consider the following fractional partial integrodifferen-

tial equation with infinite delay and control in the followingform

119888119863

119902

119905[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

=1205972

1205971199102[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

+1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902

times [120583 (119904 119910) + int

119904

minusinfin

119870 (119904 119909 120585 minus 119904)

times1198762 (119909 (120585 119910)) 119889120585

+ int

119904

0

119892 (120585 119904) 119890minus119909(120585119910)

119889120585] 119889119904

119905 isin 119869 = [0 1] 119910 isin [0 120587] 119905 = 119905119896

(44)

with 119909(119905 0) = 119909(119905 120587) = 0 119909(119905 119910) = 120601(119905 119910) 119905 isin (minusinfin 0] 119910 isin[0 120587] Δ119909(119905119894)(119910) = int

119905119894

minusinfin119902119894(119905119894 minus 119904)119909(119904 119910)119889119904 and 119910 isin [0 120587]

Abstract and Applied Analysis 11

where 120597119902119905is the Caputo fractional partial derivative of order

0 lt 119902 lt 1 120583 [0 1] times [0 120587] rarr [0 120587] is continuous and in119905 119902119894 119877 rarr 119877 are continuous 0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899 lt 119887 areprefixed numbers and 120601 isinBℎ

Let ℎ(119904) = 1198902119904 119904 lt 0 then 119897 = int

0

infinℎ(119904)119889119904 = 12 Let

Bℎ be the phase space endowed with the norm 120593Bℎ=

int0

minusinfinℎ(119904)sup

120579isin[1199040]120593(120579)

1198712119889119904 Let 119909(119905)(119910) = 119909(119905 119910) and define

the bounded linear operator 119861 119880 rarr 119883 by (119861119906)(119905)(119910) =120583(119905 119910) 0 le 119910 le 120587 119892(119905 120601)(119910) = int0

minusinfin119886(119905 119910 120579)1198761(120601(120579)(119910))119889120579

119868119896(119909(119905minus

119894))(119910) = int

119905119894

minusinfin119902119894(119905119894minus119904)119909(119904 119910)119889119904 and119891(119905 120601119867119909(119905))(119910) =

int0

minusinfin119870(119905 119910 120579)1198762(120601(120579)(119910))119889120579 + 119867119909(119905)(119910) where 119867119909(119905)(119910) =

int119905

0119892(119904 119905)119890

minus119909(119904119910)119889119904 and 120601(120579)(119910) = 120601(120579 119910) (120579 119910) isin (minusinfin 0] times

[0 120587] Moreover the linear fractional control system corre-sponding to (44) is exactly controllable Further if we imposesuitable conditions on 119886 119870 1198761 1198762 119892 119902119894 and 119861 to verifyassumptions onTheorem 8 then the system (44) can be writ-ten in the abstract form of (1)Therefore all the conditions ofTheorem 8 are satisfied and hence the nonlinear fractionalcontrol system (44) is exactly controllable on [0 119887]

Acknowledgment

The work of Yong Ren is supported by the National NaturalScience Foundation of China (no 11371029)

References

[1] P Muthukumar and P Balasubramaniam ldquoApproximate con-trollability of mixed stochastic Volterra-Fredholm type inte-grodifferential systems in Hilbert spacerdquo Journal of the FranklinInstitute vol 348 no 10 pp 2911ndash2922 2011

[2] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009

[3] J Klamka ldquoConstrained controllability of semilinear systemswith delaysrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 169ndash177 2009

[4] J Klamka ldquoStochastic controllability of systems with multipledelays in controlrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 39ndash47 2009

[5] J Klamka ldquoLocal controllability of fractional discrete-timesemilinear systemsrdquo Acta Mechanica et Automatica vol 15 pp55ndash58 2011

[6] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[7] J Dabas A Chauhan and M Kumar ldquoExistence of the mildsolutions for impulsive fractional equations with infinite delayrdquoInternational Journal of Differential Equations vol 2011 ArticleID 793023 20 pages 2011

[8] J Wang M Feckan and Y Zhou ldquoRelaxed controls fornonlinear fractional impulsive evolution equationsrdquo Journal ofOptimization Theory and Applications vol 156 no 1 pp 13ndash322013

[9] M Haase The Functional Calculus for Sectorial Operators vol169 of Operator Theory Advances and Applications BirkhauserBasel Switzerland 2006

[10] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[11] R P Agarwal B de Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[12] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functional evo-lution equations with infinite delayrdquo Applied Mathematics andComputation vol 216 no 1 pp 61ndash69 2010

[13] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications A vol 74 no 5pp 2003ndash2011 2011

[14] J Dabas and A Chauhan ldquoExistence and uniqueness of mildsolution for an impulsive neutral fractional integro-differentialequation with infinite delayrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 754ndash763 2013

[15] R Sakthivel and Y Ren ldquoComplete controllability of stochasticevolution equations with jumpsrdquo Reports on MathematicalPhysics vol 68 no 2 pp 163ndash174 2011

[16] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters of Rapid Publication vol 26 no 4 pp 418ndash4242013

[17] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[18] S Kumar and N Sukavanam ldquoApproximate controllability offractional order semilinear systems with bounded delayrdquo Jour-nal of Differential Equations vol 251 no 11 pp 6163ndash6174 2012

[19] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[20] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends inNanotechnology and Fractional Calculus D Baleanu Z BGuvenc and J A T Machado Eds pp 503ndash509 Springer NewYork NY USA 2010

[21] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[22] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[23] R Sakthivel S Suganya and S M Anthoni ldquoApproximate con-trollability of fractional stochastic evolution equationsrdquo Com-puters amp Mathematics with Applications vol 63 no 3 pp 660ndash668 2012

[24] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[25] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[26] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Π119911 (119905) =

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119911 (119905minus

1)) + 119878119902 (119905 minus 1199051)

times [119892 (1199051 1199101199051+ 1199111199051

+ 1198681 (119910119905minus1

+ 119911119905minus1

)) minus 119892 (1199051 1199101199051+ 1199111199051

)] minus 119892 (119905 119910119905 + 119911119905)

+int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119911 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

)) minus 119892 (119905119894 119910119905119894+ 119911119905119894

)]

minus119892 (119905 119910119905 + 119911119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119910+119911 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904)

times119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904))) 119889119904 119905 isin (119905119898 119887]

(28)

It can be easily seen that the operator Φ has a fixed point ifand only ifΠ has a fixed point Now we will prove thatΠ hasa unique fixed point In order to prove this we show thatΠ isa contraction mapping Let 119911 119911lowast isin B0

119887 for all 119905 isin [0 1199051] we

have

1003817100381710038171003817(Π119911) (119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(1199051 minus 120578) (Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [10038171003817100381710038171003817119892 (1199051 1199101199051

+ 1199111199051) minus 119892 (1199051 1199101199051

+ 119911lowast

1199051)10038171003817100381710038171003817119883

+ int

1199051

0

10038171003817100381710038171003817119878119902 (1199051 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus 119891 (119904 119910119904 + 119911lowast

119904

119867 (119910 (119904) + 119911lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le 119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+1

120574119872

2

119861119872

2int

119905

0

[119871119892

1003817100381710038171003817119911119905 minus 119911lowast

119905

1003817100381710038171003817Bℎ

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

times1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus 119867 (119910 (119904)

minus119911lowast(119904))1003817100381710038171003817119883) 119889119904] 119889120578

+119872int

1199051

0

(12058311003817100381710038171003817119911119904 minus 119911

lowast

119904

1003817100381710038171003817Bℎ+ 1205832

1003817100381710038171003817119867 (119910 (119904) minus 119911 (119904))

minus119867(119910(119904) minus 119911lowast(119904))1003817100381710038171003817119883)119889119904

le (1 +1

120574119872

2

119861119872

2) [119871119892119862

lowast

1+119872119887 (1205831119862

lowast

1+ 1205832119867

lowast)]

times1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(29)Similarly for 119905 isin (119905119896 119905119896+1] 119896 = 1 2 119898 we can obtain1003817100381710038171003817(Π119911) (119905) minus (Π119911

lowast) (119905)

1003817100381710038171003817119883

le

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894 (119911 (119905minus

119894)) minus 119868119894 (119911 (119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [

10038171003817100381710038171003817100381710038171003817

119892 (119905119894 119910119905119894+ 119911119905119894

+ 119868119894 (119910119905minus119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))

10038171003817100381710038171003817100381710038171003817119883

8 Abstract and Applied Analysis

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(119905119896+1 minus 120578)(Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [

119896

sum

119894=1

10038171003817100381710038171003817119878119902(119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894(119911(119905minus

119894)) minus 119868119894(119911(119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [100381710038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894+ 119868119894 (119910119905minus

119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))100381710038171003817100381710038171003817119883

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+10038171003817100381710038171003817119892 (119905119896+1 119910119905119896+1

+ 119911119905119896+1) minus 119892 (119905119896+1 119910119905119896+1

+ 119911lowast

119905119896+1)10038171003817100381710038171003817119883

+ int

119905119896+1

0

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891(119904 119910119904 + 119911lowast

119904 119867(119910(119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le (1 +1

120574119872

2

119861119872

2) [119896119872120588 + 119896119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(30)

Thus for all 119905 isin [0 119887] we get the estimate

1003817100381710038171003817(Π119911)(119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

le 1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(31)

Thus we have (Π119911)(119905) minus (Π119911lowast)(119905)119883 le 119911 minus 119911lowastB0119887

for all119905 isin [0 119887] Since lt 1 this implies that Π is a contractionmapping and hence Π has a unique fixed point 119911 isin B0

119887

Thus the system (1) is exactly controllable on [0 119887]The proofis complete

However the concept of exact controllability is verylimited for many dynamic control systems and the app-roximate controllability is more appropriate for these con-trol systems instead of exact controllability Taking thisinto account in this paper we will also discuss the app-roximate controllability result of the nonlinear impul-sive fractional control system (1) The control system issaid to be approximately controllable if for every initialdata 120601 and every finite time horizon 119887 gt 0 an admis-sible control process can be found such that the corr-esponding solution is arbitrarily close to a given squareintegrable final condition Further approximate controllablesystems are more prevalent and often approximate con-trollability is completely adequate in applications In rec-ent years for deterministic and stochastic control systemsincluding delay term there are several papers devoted tothe study of approximate controllability [23ndash25] Suka-vanam and Kumar [26] obtained a set of conditions whichensure the approximate controllability of a class of semi-linear fractional delay control systems Recently Sakthivelet al [23] formulated and proved a new set of sufficientconditions for approximate controllability of fractional dif-ferential equations by using the fractional calculus theory andsolutions operators

Definition 9 The fractional control system (1) is said to beapproximately controllable on [0 119887] if the closure of thereachable setR(119887 120601) is dense in119883 that isR(119887 120601) = 119883

Remark 10 Assume that the linear fractional control system

119863119902

119905119909 (119905) = 119860119909 (119905) + (119861119906) (119905) 119905 isin [0 119887]

119909 (0) = 120601

(32)

is approximate controllable Let us now introduce the oper-ators associated with (32) as Γ119887

0= int

119887

0S119902(119887 minus 119904)119861119861

lowastSlowast

119902(119887 minus

119904)119889119904 119877(120572 Γ1198870) = (120572119868+Γ

119887

0)minus1 for 120572 gt 0 It should be mentioned

that the approximate controllability of (32) is equivalent tothe convergence of function 120572119877(120572 Γ119887

0) to zero as 120572 rarr 0

+

in the strong operator topology (see [23 27] and referencestherein)

Theorem 11 Assume that conditions (H1)ndash(H4) hold and thatthe family 119878119902(119905) 119905 gt 0 is compact In addition assumethat the function 119891 is uniformly bounded and the linear systemassociated with the system (1) is approximately controllablethen the nonlinear fractional control system with infinite delay(1) is approximately controllable on [0 b]

Proof For each 120572 gt 0 define the operatorΨ B119887 rarr B119887 byΨ119909(119905) = 119911(119905) where

Abstract and Applied Analysis 9

119911 (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909 (sdot)) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

11990521199051) 119901 (119909 (sdot)) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

119905119898) 119901 (119909 (sdot)) 119905 isin (119905119898 119887]

(33)

where

119901 (119909 (sdot))

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

)

minusint

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909 (119905minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(34)

One can easily show that for all 120572 gt 0 the operator Ψ hasa fixed point by employing the technique used in Theorem 8with some changes

Let 119909120572(sdot) be a fixed point of Ψ Further any fixed point ofΨ is a mild solution of (1) under the control

120572(119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(35)

and satisfies

119909120572(1199051) = 1199091199051

minus 120572119877 (120572 Γ1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119909120572(1199052) = 1199091199052

minus 120572119877 (120572 Γ1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119909120572(119887) = 119909119887 minus 120572119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(36)

Moreover by the assumption that 119891 is uniformly boundedthere exists119873 gt 0 such that

int

119887

0

1003817100381710038171003817119891(119904 119909120572

119904 119867119909

120572(119904))1003817100381710038171003817

2119889119904 le 119887119873

2 (37)

and consequently the sequence 119891(119904 119909120572119904 119867119909

120572(119904)) is bounded

in 1198712(119869 119883) Then there is a subsequence still denoted

10 Abstract and Applied Analysis

by 119891(119904 119909120572119904 119867119909

120572(119904)) that converges weakly to say 119891(119904) in

1198712(119869 119883) Now we define

119908

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)]

+119892 (1199051 119909120572

1199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909120572(119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 119909120572

1199051+ 1198681 (119909

120572

119905minus

1

))

minus119892 (1199051 119909120572

1199051)] + 119892 (1199052 119909

120572

1199052)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909120572(119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909120572

119905119894+ 119868119894 (119909

120572

119905minus

119894

))

minus119892 (119905119894 119909120572

119905119894)] + 119892 (119887 119909

120572

119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904) 119889119904 119905 isin (119905119898 119887]

(38)

Now for 119905 isin [0 1199051] we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199051

0

119878119902 (1199051 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin[01199051]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(39)

Also for 119905 isin [119905119894 119905119894+1] 119894 = 1 119898 we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

119905119894+1

0

119878119902 (119905119894+1 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin(119905119894 119905119894+1]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(40)

By using infinite-dimensional version of the Ascoli-Arzelatheorem it is easy to show that an operator 119897(sdot) rarr int

sdot

0119878119902(sdot minus

119904)119897(119904)119889119904 1198711(119869 119883) rarr 119862(119869119883) is compact Hence for all

119905 isin [0 119887] we obtain that 119901(119909120572) minus 119908 rarr 0 as 120572 rarr 0+

Moreover from (36) we get for 119905 isin [0 1199051]10038171003817100381710038171003817119909120572(1199051) minus 1199091199051

10038171003817100381710038171003817le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817

+10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0)10038171003817100381710038171003817

1003817100381710038171003817119901 (119909120572) minus 119908

1003817100381710038171003817

le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817+1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

(41)

It follows from Remark 10 and (39) that 119909120572(1199051) minus 1199091199051 rarr 0

as 120572 rarr 0+ Similarly in the view of (40) for 119905 isin (119905119894 119905119894+1]

119894 = 1 11989810038171003817100381710038171003817119909120572(119905119894+1) minus 119909119905119894+1

10038171003817100381710038171003817997888rarr 0 as 120572 997888rarr 0

+ (42)

Thus for all 119905 isin [0 119887] we get 119909120572(119887) minus 119909119887 rarr 0 as 120572 rarr 0+

This proves the approximate controllability of (1) The proofis completed

Example 12 Now we present an example to illustrate theabstract results of this paper which do not aim at generalitybut indicate how our theorem can be applied to concreteproblems Let 119883 = 119871

2[0 120587] Define 119860 119883 rarr 119883

by 119860119911 = 11991110158401015840 with domain 119863(119860) = 119911 isin 119883

119911 1199111015840 are absolutely continuous 11991110158401015840 isin 119883 119911(0) = 119911(120587) = 0

Then 119860 generates an analytic semigroup 119879(119905) 119905 gt 0 in 119883and it is given by [14]

119879 (119905) 119911 =

infin

sum

119899=1

119890minus1198992119905(119911 119890119899) 119890119899 119911 isin 119883 (43)

where 119890119899(119910) = radic2120587 sin 119899119910 119899 = 1 2 3 is the orthogonalset of eigenvectors of 119860 Also define an infinite dimensionalcontrol space 119880 by 119880 = 119906 | 119906 = sum

infin

119899=2119906119899119890119899 with sum

infin

119899=21199062

119899lt

infin with norm defined by 119906119880 = (suminfin

119899=21199062

119899)12 Define a

continuous linear map 119861 from 119880 to 119883 as 119861119906 = 211990621198901 +

suminfin

119899=2119906119899119890119899 for 119906 = sum

infin

119899=2119906119899119890119899 isin 119880

Note that the subordination principle of solution operatorimplies that 119860 is the infinitesimal generator of a solutionoperator 119878119902(119905)119905ge0 Since 119878119902(119905) is strongly continuous on[0infin) by the uniformly bounded theorem there exists aconstant119872 gt 0 such that 119878119902(119905)119871(119883)

le 119872 for 119905 isin [0 119887] [14]Consider the following fractional partial integrodifferen-

tial equation with infinite delay and control in the followingform

119888119863

119902

119905[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

=1205972

1205971199102[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

+1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902

times [120583 (119904 119910) + int

119904

minusinfin

119870 (119904 119909 120585 minus 119904)

times1198762 (119909 (120585 119910)) 119889120585

+ int

119904

0

119892 (120585 119904) 119890minus119909(120585119910)

119889120585] 119889119904

119905 isin 119869 = [0 1] 119910 isin [0 120587] 119905 = 119905119896

(44)

with 119909(119905 0) = 119909(119905 120587) = 0 119909(119905 119910) = 120601(119905 119910) 119905 isin (minusinfin 0] 119910 isin[0 120587] Δ119909(119905119894)(119910) = int

119905119894

minusinfin119902119894(119905119894 minus 119904)119909(119904 119910)119889119904 and 119910 isin [0 120587]

Abstract and Applied Analysis 11

where 120597119902119905is the Caputo fractional partial derivative of order

0 lt 119902 lt 1 120583 [0 1] times [0 120587] rarr [0 120587] is continuous and in119905 119902119894 119877 rarr 119877 are continuous 0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899 lt 119887 areprefixed numbers and 120601 isinBℎ

Let ℎ(119904) = 1198902119904 119904 lt 0 then 119897 = int

0

infinℎ(119904)119889119904 = 12 Let

Bℎ be the phase space endowed with the norm 120593Bℎ=

int0

minusinfinℎ(119904)sup

120579isin[1199040]120593(120579)

1198712119889119904 Let 119909(119905)(119910) = 119909(119905 119910) and define

the bounded linear operator 119861 119880 rarr 119883 by (119861119906)(119905)(119910) =120583(119905 119910) 0 le 119910 le 120587 119892(119905 120601)(119910) = int0

minusinfin119886(119905 119910 120579)1198761(120601(120579)(119910))119889120579

119868119896(119909(119905minus

119894))(119910) = int

119905119894

minusinfin119902119894(119905119894minus119904)119909(119904 119910)119889119904 and119891(119905 120601119867119909(119905))(119910) =

int0

minusinfin119870(119905 119910 120579)1198762(120601(120579)(119910))119889120579 + 119867119909(119905)(119910) where 119867119909(119905)(119910) =

int119905

0119892(119904 119905)119890

minus119909(119904119910)119889119904 and 120601(120579)(119910) = 120601(120579 119910) (120579 119910) isin (minusinfin 0] times

[0 120587] Moreover the linear fractional control system corre-sponding to (44) is exactly controllable Further if we imposesuitable conditions on 119886 119870 1198761 1198762 119892 119902119894 and 119861 to verifyassumptions onTheorem 8 then the system (44) can be writ-ten in the abstract form of (1)Therefore all the conditions ofTheorem 8 are satisfied and hence the nonlinear fractionalcontrol system (44) is exactly controllable on [0 119887]

Acknowledgment

The work of Yong Ren is supported by the National NaturalScience Foundation of China (no 11371029)

References

[1] P Muthukumar and P Balasubramaniam ldquoApproximate con-trollability of mixed stochastic Volterra-Fredholm type inte-grodifferential systems in Hilbert spacerdquo Journal of the FranklinInstitute vol 348 no 10 pp 2911ndash2922 2011

[2] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009

[3] J Klamka ldquoConstrained controllability of semilinear systemswith delaysrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 169ndash177 2009

[4] J Klamka ldquoStochastic controllability of systems with multipledelays in controlrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 39ndash47 2009

[5] J Klamka ldquoLocal controllability of fractional discrete-timesemilinear systemsrdquo Acta Mechanica et Automatica vol 15 pp55ndash58 2011

[6] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[7] J Dabas A Chauhan and M Kumar ldquoExistence of the mildsolutions for impulsive fractional equations with infinite delayrdquoInternational Journal of Differential Equations vol 2011 ArticleID 793023 20 pages 2011

[8] J Wang M Feckan and Y Zhou ldquoRelaxed controls fornonlinear fractional impulsive evolution equationsrdquo Journal ofOptimization Theory and Applications vol 156 no 1 pp 13ndash322013

[9] M Haase The Functional Calculus for Sectorial Operators vol169 of Operator Theory Advances and Applications BirkhauserBasel Switzerland 2006

[10] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[11] R P Agarwal B de Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[12] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functional evo-lution equations with infinite delayrdquo Applied Mathematics andComputation vol 216 no 1 pp 61ndash69 2010

[13] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications A vol 74 no 5pp 2003ndash2011 2011

[14] J Dabas and A Chauhan ldquoExistence and uniqueness of mildsolution for an impulsive neutral fractional integro-differentialequation with infinite delayrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 754ndash763 2013

[15] R Sakthivel and Y Ren ldquoComplete controllability of stochasticevolution equations with jumpsrdquo Reports on MathematicalPhysics vol 68 no 2 pp 163ndash174 2011

[16] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters of Rapid Publication vol 26 no 4 pp 418ndash4242013

[17] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[18] S Kumar and N Sukavanam ldquoApproximate controllability offractional order semilinear systems with bounded delayrdquo Jour-nal of Differential Equations vol 251 no 11 pp 6163ndash6174 2012

[19] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[20] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends inNanotechnology and Fractional Calculus D Baleanu Z BGuvenc and J A T Machado Eds pp 503ndash509 Springer NewYork NY USA 2010

[21] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[22] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[23] R Sakthivel S Suganya and S M Anthoni ldquoApproximate con-trollability of fractional stochastic evolution equationsrdquo Com-puters amp Mathematics with Applications vol 63 no 3 pp 660ndash668 2012

[24] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[25] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[26] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

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Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+1003817100381710038171003817119892 (119905 119910119905 + 119911119905) minus 119892 (119905 119910119905 + 119911

lowast

119905)1003817100381710038171003817119883

+ int

119905

0

100381710038171003817100381710038171003817119878119902 (119905 minus 120578) 119861119861

lowast119878lowast

119902(119905119896+1 minus 120578)(Γ

1199051

0)minus1100381710038171003817100381710038171003817119883

times [

119896

sum

119894=1

10038171003817100381710038171003817119878119902(119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119909)

1003817100381710038171003817119868119894(119911(119905minus

119894)) minus 119868119894(119911(119905

minus

119894))1003817100381710038171003817119883

+

119896

sum

119894=1

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119905119894)

10038171003817100381710038171003817119871(119883)

times [100381710038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894+ 119868119894 (119910119905minus

119894

+ 119911119905minus119894

))

minus119892 (119905119894 119910119905119894+ 119911

lowast

119905119894+ 119868119894 (119910119905minus

119894

+ 119911lowast

119905minus

119894

))100381710038171003817100381710038171003817119883

+10038171003817100381710038171003817119892 (119905119894 119910119905119894

+ 119911119905119894) minus 119892 (119905119894 119910119905119894

+ 119911lowast

119905119894)10038171003817100381710038171003817119883]

+10038171003817100381710038171003817119892 (119905119896+1 119910119905119896+1

+ 119911119905119896+1) minus 119892 (119905119896+1 119910119905119896+1

+ 119911lowast

119905119896+1)10038171003817100381710038171003817119883

+ int

119905119896+1

0

10038171003817100381710038171003817119878119902 (119905119896+1 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891(119904 119910119904 + 119911lowast

119904 119867(119910(119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904] 119889120578

+ int

119905

0

10038171003817100381710038171003817119878119902 (119905 minus 119904)

10038171003817100381710038171003817119871(119883)

times1003817100381710038171003817119891 (119904 119910119904 + 119911119904 119867 (119910 (119904) + 119911 (119904)))

minus119891 (119904 119910119904 + 119911lowast

119904 119867 (119910 (119904) + 119911

lowast(119904)))

1003817100381710038171003817119883119889119904

le (1 +1

120574119872

2

119861119872

2) [119896119872120588 + 119896119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(30)

Thus for all 119905 isin [0 119887] we get the estimate

1003817100381710038171003817(Π119911)(119905) minus (Π119911lowast) (119905)

1003817100381710038171003817119883

le (1 +1

120574119872

2

119861119872

2) [119898119872120588 + 119898119872119871119892119862

lowast

1(2 + 120588) + 119871119892119862

lowast

1

+119872119887 (1205831119862lowast

1+ 1205832119867

lowast) ]1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

le 1003817100381710038171003817119911 minus 119911

lowast1003817100381710038171003817B0119887

(31)

Thus we have (Π119911)(119905) minus (Π119911lowast)(119905)119883 le 119911 minus 119911lowastB0119887

for all119905 isin [0 119887] Since lt 1 this implies that Π is a contractionmapping and hence Π has a unique fixed point 119911 isin B0

119887

Thus the system (1) is exactly controllable on [0 119887]The proofis complete

However the concept of exact controllability is verylimited for many dynamic control systems and the app-roximate controllability is more appropriate for these con-trol systems instead of exact controllability Taking thisinto account in this paper we will also discuss the app-roximate controllability result of the nonlinear impul-sive fractional control system (1) The control system issaid to be approximately controllable if for every initialdata 120601 and every finite time horizon 119887 gt 0 an admis-sible control process can be found such that the corr-esponding solution is arbitrarily close to a given squareintegrable final condition Further approximate controllablesystems are more prevalent and often approximate con-trollability is completely adequate in applications In rec-ent years for deterministic and stochastic control systemsincluding delay term there are several papers devoted tothe study of approximate controllability [23ndash25] Suka-vanam and Kumar [26] obtained a set of conditions whichensure the approximate controllability of a class of semi-linear fractional delay control systems Recently Sakthivelet al [23] formulated and proved a new set of sufficientconditions for approximate controllability of fractional dif-ferential equations by using the fractional calculus theory andsolutions operators

Definition 9 The fractional control system (1) is said to beapproximately controllable on [0 119887] if the closure of thereachable setR(119887 120601) is dense in119883 that isR(119887 120601) = 119883

Remark 10 Assume that the linear fractional control system

119863119902

119905119909 (119905) = 119860119909 (119905) + (119861119906) (119905) 119905 isin [0 119887]

119909 (0) = 120601

(32)

is approximate controllable Let us now introduce the oper-ators associated with (32) as Γ119887

0= int

119887

0S119902(119887 minus 119904)119861119861

lowastSlowast

119902(119887 minus

119904)119889119904 119877(120572 Γ1198870) = (120572119868+Γ

119887

0)minus1 for 120572 gt 0 It should be mentioned

that the approximate controllability of (32) is equivalent tothe convergence of function 120572119877(120572 Γ119887

0) to zero as 120572 rarr 0

+

in the strong operator topology (see [23 27] and referencestherein)

Theorem 11 Assume that conditions (H1)ndash(H4) hold and thatthe family 119878119902(119905) 119905 gt 0 is compact In addition assumethat the function 119891 is uniformly bounded and the linear systemassociated with the system (1) is approximately controllablethen the nonlinear fractional control system with infinite delay(1) is approximately controllable on [0 b]

Proof For each 120572 gt 0 define the operatorΨ B119887 rarr B119887 byΨ119909(119905) = 119911(119905) where

Abstract and Applied Analysis 9

119911 (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909 (sdot)) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

11990521199051) 119901 (119909 (sdot)) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

119905119898) 119901 (119909 (sdot)) 119905 isin (119905119898 119887]

(33)

where

119901 (119909 (sdot))

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

)

minusint

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909 (119905minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(34)

One can easily show that for all 120572 gt 0 the operator Ψ hasa fixed point by employing the technique used in Theorem 8with some changes

Let 119909120572(sdot) be a fixed point of Ψ Further any fixed point ofΨ is a mild solution of (1) under the control

120572(119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(35)

and satisfies

119909120572(1199051) = 1199091199051

minus 120572119877 (120572 Γ1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119909120572(1199052) = 1199091199052

minus 120572119877 (120572 Γ1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119909120572(119887) = 119909119887 minus 120572119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(36)

Moreover by the assumption that 119891 is uniformly boundedthere exists119873 gt 0 such that

int

119887

0

1003817100381710038171003817119891(119904 119909120572

119904 119867119909

120572(119904))1003817100381710038171003817

2119889119904 le 119887119873

2 (37)

and consequently the sequence 119891(119904 119909120572119904 119867119909

120572(119904)) is bounded

in 1198712(119869 119883) Then there is a subsequence still denoted

10 Abstract and Applied Analysis

by 119891(119904 119909120572119904 119867119909

120572(119904)) that converges weakly to say 119891(119904) in

1198712(119869 119883) Now we define

119908

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)]

+119892 (1199051 119909120572

1199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909120572(119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 119909120572

1199051+ 1198681 (119909

120572

119905minus

1

))

minus119892 (1199051 119909120572

1199051)] + 119892 (1199052 119909

120572

1199052)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909120572(119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909120572

119905119894+ 119868119894 (119909

120572

119905minus

119894

))

minus119892 (119905119894 119909120572

119905119894)] + 119892 (119887 119909

120572

119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904) 119889119904 119905 isin (119905119898 119887]

(38)

Now for 119905 isin [0 1199051] we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199051

0

119878119902 (1199051 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin[01199051]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(39)

Also for 119905 isin [119905119894 119905119894+1] 119894 = 1 119898 we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

119905119894+1

0

119878119902 (119905119894+1 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin(119905119894 119905119894+1]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(40)

By using infinite-dimensional version of the Ascoli-Arzelatheorem it is easy to show that an operator 119897(sdot) rarr int

sdot

0119878119902(sdot minus

119904)119897(119904)119889119904 1198711(119869 119883) rarr 119862(119869119883) is compact Hence for all

119905 isin [0 119887] we obtain that 119901(119909120572) minus 119908 rarr 0 as 120572 rarr 0+

Moreover from (36) we get for 119905 isin [0 1199051]10038171003817100381710038171003817119909120572(1199051) minus 1199091199051

10038171003817100381710038171003817le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817

+10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0)10038171003817100381710038171003817

1003817100381710038171003817119901 (119909120572) minus 119908

1003817100381710038171003817

le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817+1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

(41)

It follows from Remark 10 and (39) that 119909120572(1199051) minus 1199091199051 rarr 0

as 120572 rarr 0+ Similarly in the view of (40) for 119905 isin (119905119894 119905119894+1]

119894 = 1 11989810038171003817100381710038171003817119909120572(119905119894+1) minus 119909119905119894+1

10038171003817100381710038171003817997888rarr 0 as 120572 997888rarr 0

+ (42)

Thus for all 119905 isin [0 119887] we get 119909120572(119887) minus 119909119887 rarr 0 as 120572 rarr 0+

This proves the approximate controllability of (1) The proofis completed

Example 12 Now we present an example to illustrate theabstract results of this paper which do not aim at generalitybut indicate how our theorem can be applied to concreteproblems Let 119883 = 119871

2[0 120587] Define 119860 119883 rarr 119883

by 119860119911 = 11991110158401015840 with domain 119863(119860) = 119911 isin 119883

119911 1199111015840 are absolutely continuous 11991110158401015840 isin 119883 119911(0) = 119911(120587) = 0

Then 119860 generates an analytic semigroup 119879(119905) 119905 gt 0 in 119883and it is given by [14]

119879 (119905) 119911 =

infin

sum

119899=1

119890minus1198992119905(119911 119890119899) 119890119899 119911 isin 119883 (43)

where 119890119899(119910) = radic2120587 sin 119899119910 119899 = 1 2 3 is the orthogonalset of eigenvectors of 119860 Also define an infinite dimensionalcontrol space 119880 by 119880 = 119906 | 119906 = sum

infin

119899=2119906119899119890119899 with sum

infin

119899=21199062

119899lt

infin with norm defined by 119906119880 = (suminfin

119899=21199062

119899)12 Define a

continuous linear map 119861 from 119880 to 119883 as 119861119906 = 211990621198901 +

suminfin

119899=2119906119899119890119899 for 119906 = sum

infin

119899=2119906119899119890119899 isin 119880

Note that the subordination principle of solution operatorimplies that 119860 is the infinitesimal generator of a solutionoperator 119878119902(119905)119905ge0 Since 119878119902(119905) is strongly continuous on[0infin) by the uniformly bounded theorem there exists aconstant119872 gt 0 such that 119878119902(119905)119871(119883)

le 119872 for 119905 isin [0 119887] [14]Consider the following fractional partial integrodifferen-

tial equation with infinite delay and control in the followingform

119888119863

119902

119905[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

=1205972

1205971199102[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

+1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902

times [120583 (119904 119910) + int

119904

minusinfin

119870 (119904 119909 120585 minus 119904)

times1198762 (119909 (120585 119910)) 119889120585

+ int

119904

0

119892 (120585 119904) 119890minus119909(120585119910)

119889120585] 119889119904

119905 isin 119869 = [0 1] 119910 isin [0 120587] 119905 = 119905119896

(44)

with 119909(119905 0) = 119909(119905 120587) = 0 119909(119905 119910) = 120601(119905 119910) 119905 isin (minusinfin 0] 119910 isin[0 120587] Δ119909(119905119894)(119910) = int

119905119894

minusinfin119902119894(119905119894 minus 119904)119909(119904 119910)119889119904 and 119910 isin [0 120587]

Abstract and Applied Analysis 11

where 120597119902119905is the Caputo fractional partial derivative of order

0 lt 119902 lt 1 120583 [0 1] times [0 120587] rarr [0 120587] is continuous and in119905 119902119894 119877 rarr 119877 are continuous 0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899 lt 119887 areprefixed numbers and 120601 isinBℎ

Let ℎ(119904) = 1198902119904 119904 lt 0 then 119897 = int

0

infinℎ(119904)119889119904 = 12 Let

Bℎ be the phase space endowed with the norm 120593Bℎ=

int0

minusinfinℎ(119904)sup

120579isin[1199040]120593(120579)

1198712119889119904 Let 119909(119905)(119910) = 119909(119905 119910) and define

the bounded linear operator 119861 119880 rarr 119883 by (119861119906)(119905)(119910) =120583(119905 119910) 0 le 119910 le 120587 119892(119905 120601)(119910) = int0

minusinfin119886(119905 119910 120579)1198761(120601(120579)(119910))119889120579

119868119896(119909(119905minus

119894))(119910) = int

119905119894

minusinfin119902119894(119905119894minus119904)119909(119904 119910)119889119904 and119891(119905 120601119867119909(119905))(119910) =

int0

minusinfin119870(119905 119910 120579)1198762(120601(120579)(119910))119889120579 + 119867119909(119905)(119910) where 119867119909(119905)(119910) =

int119905

0119892(119904 119905)119890

minus119909(119904119910)119889119904 and 120601(120579)(119910) = 120601(120579 119910) (120579 119910) isin (minusinfin 0] times

[0 120587] Moreover the linear fractional control system corre-sponding to (44) is exactly controllable Further if we imposesuitable conditions on 119886 119870 1198761 1198762 119892 119902119894 and 119861 to verifyassumptions onTheorem 8 then the system (44) can be writ-ten in the abstract form of (1)Therefore all the conditions ofTheorem 8 are satisfied and hence the nonlinear fractionalcontrol system (44) is exactly controllable on [0 119887]

Acknowledgment

The work of Yong Ren is supported by the National NaturalScience Foundation of China (no 11371029)

References

[1] P Muthukumar and P Balasubramaniam ldquoApproximate con-trollability of mixed stochastic Volterra-Fredholm type inte-grodifferential systems in Hilbert spacerdquo Journal of the FranklinInstitute vol 348 no 10 pp 2911ndash2922 2011

[2] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009

[3] J Klamka ldquoConstrained controllability of semilinear systemswith delaysrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 169ndash177 2009

[4] J Klamka ldquoStochastic controllability of systems with multipledelays in controlrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 39ndash47 2009

[5] J Klamka ldquoLocal controllability of fractional discrete-timesemilinear systemsrdquo Acta Mechanica et Automatica vol 15 pp55ndash58 2011

[6] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[7] J Dabas A Chauhan and M Kumar ldquoExistence of the mildsolutions for impulsive fractional equations with infinite delayrdquoInternational Journal of Differential Equations vol 2011 ArticleID 793023 20 pages 2011

[8] J Wang M Feckan and Y Zhou ldquoRelaxed controls fornonlinear fractional impulsive evolution equationsrdquo Journal ofOptimization Theory and Applications vol 156 no 1 pp 13ndash322013

[9] M Haase The Functional Calculus for Sectorial Operators vol169 of Operator Theory Advances and Applications BirkhauserBasel Switzerland 2006

[10] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[11] R P Agarwal B de Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[12] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functional evo-lution equations with infinite delayrdquo Applied Mathematics andComputation vol 216 no 1 pp 61ndash69 2010

[13] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications A vol 74 no 5pp 2003ndash2011 2011

[14] J Dabas and A Chauhan ldquoExistence and uniqueness of mildsolution for an impulsive neutral fractional integro-differentialequation with infinite delayrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 754ndash763 2013

[15] R Sakthivel and Y Ren ldquoComplete controllability of stochasticevolution equations with jumpsrdquo Reports on MathematicalPhysics vol 68 no 2 pp 163ndash174 2011

[16] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters of Rapid Publication vol 26 no 4 pp 418ndash4242013

[17] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[18] S Kumar and N Sukavanam ldquoApproximate controllability offractional order semilinear systems with bounded delayrdquo Jour-nal of Differential Equations vol 251 no 11 pp 6163ndash6174 2012

[19] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[20] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends inNanotechnology and Fractional Calculus D Baleanu Z BGuvenc and J A T Machado Eds pp 503ndash509 Springer NewYork NY USA 2010

[21] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[22] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[23] R Sakthivel S Suganya and S M Anthoni ldquoApproximate con-trollability of fractional stochastic evolution equationsrdquo Com-puters amp Mathematics with Applications vol 63 no 3 pp 660ndash668 2012

[24] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[25] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[26] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

119911 (119905) =

120601 (119905) 119905 isin (minusinfin 0]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] minus 119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904

+int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] + 119878119902 (119905 minus 1199051) 1198681 (119909 (119905minus

1)) + 119878119902 (119905 minus 1199051) [119892 (1199051 1199091199051

+ 1198681 (119909119905minus1

)) minus 119892 (1199051 1199091199051)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119878119902 (119905) [120601 (0) + 119892 (0 120601)] +

119898

sum

119894=1

119878119902 (119905 minus 119905119894) 119868119894 (119909 (119905minus

119894))

+

119898

sum

119894=1

119878119902 (119905 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

)) minus 119892 (119905119894 119909119905119894)]

minus119892 (119905 119909119905) + int

119905

0

119878119902 (119905 minus 119904) 119861119906119909 (119904) 119889119904 + int

119905

0

119878119902 (119905 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

119906119909 (119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909 (sdot)) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

11990521199051) 119901 (119909 (sdot)) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

119905119898) 119901 (119909 (sdot)) 119905 isin (119905119898 119887]

(33)

where

119901 (119909 (sdot))

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)] + 119892 (1199051 1199091199051

)

minusint

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909 (119905minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 1199091199051+ 1198681 (119909119905minus

1

))

minus 119892 (1199051 1199091199051)] + 119892 (1199052 1199091199052

)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909 (119905minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909119905119894+ 119868119894 (119909119905minus

119894

))

minus119892 (119905119894 119909119905119894)] + 119892 (119887 119909119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904 119909119904 119867119909 (119904)) 119889119904 119905 isin (119905119898 119887]

(34)

One can easily show that for all 120572 gt 0 the operator Ψ hasa fixed point by employing the technique used in Theorem 8with some changes

Let 119909120572(sdot) be a fixed point of Ψ Further any fixed point ofΨ is a mild solution of (1) under the control

120572(119905) =

119861lowast119878lowast

119902(1199051 minus 119905) 119877 (120572 Γ

1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119861lowast119878lowast

119902(1199052 minus 119905) 119877 (120572 Γ

1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119861lowast119878lowast

119902(119887 minus 119905) 119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(35)

and satisfies

119909120572(1199051) = 1199091199051

minus 120572119877 (120572 Γ1199051

0) 119901 (119909

120572) 119905 isin [0 1199051]

119909120572(1199052) = 1199091199052

minus 120572119877 (120572 Γ1199052

0) 119901 (119909

120572) 119905 isin (1199051 1199052]

119909120572(119887) = 119909119887 minus 120572119877 (120572 Γ

119887

0) 119901 (119909

120572) 119905 isin (119905119898 119887]

(36)

Moreover by the assumption that 119891 is uniformly boundedthere exists119873 gt 0 such that

int

119887

0

1003817100381710038171003817119891(119904 119909120572

119904 119867119909

120572(119904))1003817100381710038171003817

2119889119904 le 119887119873

2 (37)

and consequently the sequence 119891(119904 119909120572119904 119867119909

120572(119904)) is bounded

in 1198712(119869 119883) Then there is a subsequence still denoted

10 Abstract and Applied Analysis

by 119891(119904 119909120572119904 119867119909

120572(119904)) that converges weakly to say 119891(119904) in

1198712(119869 119883) Now we define

119908

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)]

+119892 (1199051 119909120572

1199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909120572(119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 119909120572

1199051+ 1198681 (119909

120572

119905minus

1

))

minus119892 (1199051 119909120572

1199051)] + 119892 (1199052 119909

120572

1199052)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909120572(119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909120572

119905119894+ 119868119894 (119909

120572

119905minus

119894

))

minus119892 (119905119894 119909120572

119905119894)] + 119892 (119887 119909

120572

119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904) 119889119904 119905 isin (119905119898 119887]

(38)

Now for 119905 isin [0 1199051] we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199051

0

119878119902 (1199051 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin[01199051]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(39)

Also for 119905 isin [119905119894 119905119894+1] 119894 = 1 119898 we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

119905119894+1

0

119878119902 (119905119894+1 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin(119905119894 119905119894+1]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(40)

By using infinite-dimensional version of the Ascoli-Arzelatheorem it is easy to show that an operator 119897(sdot) rarr int

sdot

0119878119902(sdot minus

119904)119897(119904)119889119904 1198711(119869 119883) rarr 119862(119869119883) is compact Hence for all

119905 isin [0 119887] we obtain that 119901(119909120572) minus 119908 rarr 0 as 120572 rarr 0+

Moreover from (36) we get for 119905 isin [0 1199051]10038171003817100381710038171003817119909120572(1199051) minus 1199091199051

10038171003817100381710038171003817le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817

+10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0)10038171003817100381710038171003817

1003817100381710038171003817119901 (119909120572) minus 119908

1003817100381710038171003817

le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817+1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

(41)

It follows from Remark 10 and (39) that 119909120572(1199051) minus 1199091199051 rarr 0

as 120572 rarr 0+ Similarly in the view of (40) for 119905 isin (119905119894 119905119894+1]

119894 = 1 11989810038171003817100381710038171003817119909120572(119905119894+1) minus 119909119905119894+1

10038171003817100381710038171003817997888rarr 0 as 120572 997888rarr 0

+ (42)

Thus for all 119905 isin [0 119887] we get 119909120572(119887) minus 119909119887 rarr 0 as 120572 rarr 0+

This proves the approximate controllability of (1) The proofis completed

Example 12 Now we present an example to illustrate theabstract results of this paper which do not aim at generalitybut indicate how our theorem can be applied to concreteproblems Let 119883 = 119871

2[0 120587] Define 119860 119883 rarr 119883

by 119860119911 = 11991110158401015840 with domain 119863(119860) = 119911 isin 119883

119911 1199111015840 are absolutely continuous 11991110158401015840 isin 119883 119911(0) = 119911(120587) = 0

Then 119860 generates an analytic semigroup 119879(119905) 119905 gt 0 in 119883and it is given by [14]

119879 (119905) 119911 =

infin

sum

119899=1

119890minus1198992119905(119911 119890119899) 119890119899 119911 isin 119883 (43)

where 119890119899(119910) = radic2120587 sin 119899119910 119899 = 1 2 3 is the orthogonalset of eigenvectors of 119860 Also define an infinite dimensionalcontrol space 119880 by 119880 = 119906 | 119906 = sum

infin

119899=2119906119899119890119899 with sum

infin

119899=21199062

119899lt

infin with norm defined by 119906119880 = (suminfin

119899=21199062

119899)12 Define a

continuous linear map 119861 from 119880 to 119883 as 119861119906 = 211990621198901 +

suminfin

119899=2119906119899119890119899 for 119906 = sum

infin

119899=2119906119899119890119899 isin 119880

Note that the subordination principle of solution operatorimplies that 119860 is the infinitesimal generator of a solutionoperator 119878119902(119905)119905ge0 Since 119878119902(119905) is strongly continuous on[0infin) by the uniformly bounded theorem there exists aconstant119872 gt 0 such that 119878119902(119905)119871(119883)

le 119872 for 119905 isin [0 119887] [14]Consider the following fractional partial integrodifferen-

tial equation with infinite delay and control in the followingform

119888119863

119902

119905[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

=1205972

1205971199102[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

+1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902

times [120583 (119904 119910) + int

119904

minusinfin

119870 (119904 119909 120585 minus 119904)

times1198762 (119909 (120585 119910)) 119889120585

+ int

119904

0

119892 (120585 119904) 119890minus119909(120585119910)

119889120585] 119889119904

119905 isin 119869 = [0 1] 119910 isin [0 120587] 119905 = 119905119896

(44)

with 119909(119905 0) = 119909(119905 120587) = 0 119909(119905 119910) = 120601(119905 119910) 119905 isin (minusinfin 0] 119910 isin[0 120587] Δ119909(119905119894)(119910) = int

119905119894

minusinfin119902119894(119905119894 minus 119904)119909(119904 119910)119889119904 and 119910 isin [0 120587]

Abstract and Applied Analysis 11

where 120597119902119905is the Caputo fractional partial derivative of order

0 lt 119902 lt 1 120583 [0 1] times [0 120587] rarr [0 120587] is continuous and in119905 119902119894 119877 rarr 119877 are continuous 0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899 lt 119887 areprefixed numbers and 120601 isinBℎ

Let ℎ(119904) = 1198902119904 119904 lt 0 then 119897 = int

0

infinℎ(119904)119889119904 = 12 Let

Bℎ be the phase space endowed with the norm 120593Bℎ=

int0

minusinfinℎ(119904)sup

120579isin[1199040]120593(120579)

1198712119889119904 Let 119909(119905)(119910) = 119909(119905 119910) and define

the bounded linear operator 119861 119880 rarr 119883 by (119861119906)(119905)(119910) =120583(119905 119910) 0 le 119910 le 120587 119892(119905 120601)(119910) = int0

minusinfin119886(119905 119910 120579)1198761(120601(120579)(119910))119889120579

119868119896(119909(119905minus

119894))(119910) = int

119905119894

minusinfin119902119894(119905119894minus119904)119909(119904 119910)119889119904 and119891(119905 120601119867119909(119905))(119910) =

int0

minusinfin119870(119905 119910 120579)1198762(120601(120579)(119910))119889120579 + 119867119909(119905)(119910) where 119867119909(119905)(119910) =

int119905

0119892(119904 119905)119890

minus119909(119904119910)119889119904 and 120601(120579)(119910) = 120601(120579 119910) (120579 119910) isin (minusinfin 0] times

[0 120587] Moreover the linear fractional control system corre-sponding to (44) is exactly controllable Further if we imposesuitable conditions on 119886 119870 1198761 1198762 119892 119902119894 and 119861 to verifyassumptions onTheorem 8 then the system (44) can be writ-ten in the abstract form of (1)Therefore all the conditions ofTheorem 8 are satisfied and hence the nonlinear fractionalcontrol system (44) is exactly controllable on [0 119887]

Acknowledgment

The work of Yong Ren is supported by the National NaturalScience Foundation of China (no 11371029)

References

[1] P Muthukumar and P Balasubramaniam ldquoApproximate con-trollability of mixed stochastic Volterra-Fredholm type inte-grodifferential systems in Hilbert spacerdquo Journal of the FranklinInstitute vol 348 no 10 pp 2911ndash2922 2011

[2] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009

[3] J Klamka ldquoConstrained controllability of semilinear systemswith delaysrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 169ndash177 2009

[4] J Klamka ldquoStochastic controllability of systems with multipledelays in controlrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 39ndash47 2009

[5] J Klamka ldquoLocal controllability of fractional discrete-timesemilinear systemsrdquo Acta Mechanica et Automatica vol 15 pp55ndash58 2011

[6] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[7] J Dabas A Chauhan and M Kumar ldquoExistence of the mildsolutions for impulsive fractional equations with infinite delayrdquoInternational Journal of Differential Equations vol 2011 ArticleID 793023 20 pages 2011

[8] J Wang M Feckan and Y Zhou ldquoRelaxed controls fornonlinear fractional impulsive evolution equationsrdquo Journal ofOptimization Theory and Applications vol 156 no 1 pp 13ndash322013

[9] M Haase The Functional Calculus for Sectorial Operators vol169 of Operator Theory Advances and Applications BirkhauserBasel Switzerland 2006

[10] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[11] R P Agarwal B de Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[12] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functional evo-lution equations with infinite delayrdquo Applied Mathematics andComputation vol 216 no 1 pp 61ndash69 2010

[13] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications A vol 74 no 5pp 2003ndash2011 2011

[14] J Dabas and A Chauhan ldquoExistence and uniqueness of mildsolution for an impulsive neutral fractional integro-differentialequation with infinite delayrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 754ndash763 2013

[15] R Sakthivel and Y Ren ldquoComplete controllability of stochasticevolution equations with jumpsrdquo Reports on MathematicalPhysics vol 68 no 2 pp 163ndash174 2011

[16] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters of Rapid Publication vol 26 no 4 pp 418ndash4242013

[17] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[18] S Kumar and N Sukavanam ldquoApproximate controllability offractional order semilinear systems with bounded delayrdquo Jour-nal of Differential Equations vol 251 no 11 pp 6163ndash6174 2012

[19] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[20] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends inNanotechnology and Fractional Calculus D Baleanu Z BGuvenc and J A T Machado Eds pp 503ndash509 Springer NewYork NY USA 2010

[21] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[22] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[23] R Sakthivel S Suganya and S M Anthoni ldquoApproximate con-trollability of fractional stochastic evolution equationsrdquo Com-puters amp Mathematics with Applications vol 63 no 3 pp 660ndash668 2012

[24] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[25] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[26] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

by 119891(119904 119909120572119904 119867119909

120572(119904)) that converges weakly to say 119891(119904) in

1198712(119869 119883) Now we define

119908

=

1199091199051minus 119878119902 (1199051) [120601 (0) + 119892 (0 120601)]

+119892 (1199051 119909120572

1199051) minus int

1199051

0

119878119902 (1199051 minus 119904) 119891 (119904) 119889119904 119905 isin [0 1199051]

1199091199051minus 119878119902 (1199052) [120601 (0) + 119892 (0 120601)]

minus119878119902 (1199052 minus 1199051) 1198681 (119909120572(119905

minus

1))

minus119878119902 (1199052 minus 1199051) [119892 (1199051 119909120572

1199051+ 1198681 (119909

120572

119905minus

1

))

minus119892 (1199051 119909120572

1199051)] + 119892 (1199052 119909

120572

1199052)

minusint

1199052

0

119878119902 (1199052 minus 119904) 119891 (119904) 119889119904 119905 isin (1199051 1199052]

119909119887 minus 119878119902 (119887) [120601 (0) + 119892 (0 120601)]

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) 119868119894 (119909120572(119905

minus

119894))

minus

119898

sum

119894=1

119878119902 (119887 minus 119905119894) [119892 (119905119894 119909120572

119905119894+ 119868119894 (119909

120572

119905minus

119894

))

minus119892 (119905119894 119909120572

119905119894)] + 119892 (119887 119909

120572

119887)

minusint

119887

0

119878119902 (119887 minus 119904) 119891 (119904) 119889119904 119905 isin (119905119898 119887]

(38)

Now for 119905 isin [0 1199051] we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199051

0

119878119902 (1199051 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin[01199051]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(39)

Also for 119905 isin [119905119894 119905119894+1] 119894 = 1 119898 we have1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

119905119894+1

0

119878119902 (119905119894+1 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le sup119905isin(119905119894 119905119894+1]

10038171003817100381710038171003817100381710038171003817

int

119905

0

119878119902 (119905 minus 119904) [119891 (119904 119909120572

119904 119867119909

120572(119904)) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

(40)

By using infinite-dimensional version of the Ascoli-Arzelatheorem it is easy to show that an operator 119897(sdot) rarr int

sdot

0119878119902(sdot minus

119904)119897(119904)119889119904 1198711(119869 119883) rarr 119862(119869119883) is compact Hence for all

119905 isin [0 119887] we obtain that 119901(119909120572) minus 119908 rarr 0 as 120572 rarr 0+

Moreover from (36) we get for 119905 isin [0 1199051]10038171003817100381710038171003817119909120572(1199051) minus 1199091199051

10038171003817100381710038171003817le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817

+10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0)10038171003817100381710038171003817

1003817100381710038171003817119901 (119909120572) minus 119908

1003817100381710038171003817

le10038171003817100381710038171003817120572119877 (120572 Γ

1199051

0) (119908)

10038171003817100381710038171003817+1003817100381710038171003817119901 (119909

120572) minus 119908

1003817100381710038171003817

(41)

It follows from Remark 10 and (39) that 119909120572(1199051) minus 1199091199051 rarr 0

as 120572 rarr 0+ Similarly in the view of (40) for 119905 isin (119905119894 119905119894+1]

119894 = 1 11989810038171003817100381710038171003817119909120572(119905119894+1) minus 119909119905119894+1

10038171003817100381710038171003817997888rarr 0 as 120572 997888rarr 0

+ (42)

Thus for all 119905 isin [0 119887] we get 119909120572(119887) minus 119909119887 rarr 0 as 120572 rarr 0+

This proves the approximate controllability of (1) The proofis completed

Example 12 Now we present an example to illustrate theabstract results of this paper which do not aim at generalitybut indicate how our theorem can be applied to concreteproblems Let 119883 = 119871

2[0 120587] Define 119860 119883 rarr 119883

by 119860119911 = 11991110158401015840 with domain 119863(119860) = 119911 isin 119883

119911 1199111015840 are absolutely continuous 11991110158401015840 isin 119883 119911(0) = 119911(120587) = 0

Then 119860 generates an analytic semigroup 119879(119905) 119905 gt 0 in 119883and it is given by [14]

119879 (119905) 119911 =

infin

sum

119899=1

119890minus1198992119905(119911 119890119899) 119890119899 119911 isin 119883 (43)

where 119890119899(119910) = radic2120587 sin 119899119910 119899 = 1 2 3 is the orthogonalset of eigenvectors of 119860 Also define an infinite dimensionalcontrol space 119880 by 119880 = 119906 | 119906 = sum

infin

119899=2119906119899119890119899 with sum

infin

119899=21199062

119899lt

infin with norm defined by 119906119880 = (suminfin

119899=21199062

119899)12 Define a

continuous linear map 119861 from 119880 to 119883 as 119861119906 = 211990621198901 +

suminfin

119899=2119906119899119890119899 for 119906 = sum

infin

119899=2119906119899119890119899 isin 119880

Note that the subordination principle of solution operatorimplies that 119860 is the infinitesimal generator of a solutionoperator 119878119902(119905)119905ge0 Since 119878119902(119905) is strongly continuous on[0infin) by the uniformly bounded theorem there exists aconstant119872 gt 0 such that 119878119902(119905)119871(119883)

le 119872 for 119905 isin [0 119887] [14]Consider the following fractional partial integrodifferen-

tial equation with infinite delay and control in the followingform

119888119863

119902

119905[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

=1205972

1205971199102[119909 (119905 119910) + int

119905

minusinfin

119886 (119905 119910 119904 minus 119905) 1198761 (119909 (119904 119910)) 119889119904]

+1

Γ (1 minus 119902)int

119905

0

(119905 minus 119904)minus119902

times [120583 (119904 119910) + int

119904

minusinfin

119870 (119904 119909 120585 minus 119904)

times1198762 (119909 (120585 119910)) 119889120585

+ int

119904

0

119892 (120585 119904) 119890minus119909(120585119910)

119889120585] 119889119904

119905 isin 119869 = [0 1] 119910 isin [0 120587] 119905 = 119905119896

(44)

with 119909(119905 0) = 119909(119905 120587) = 0 119909(119905 119910) = 120601(119905 119910) 119905 isin (minusinfin 0] 119910 isin[0 120587] Δ119909(119905119894)(119910) = int

119905119894

minusinfin119902119894(119905119894 minus 119904)119909(119904 119910)119889119904 and 119910 isin [0 120587]

Abstract and Applied Analysis 11

where 120597119902119905is the Caputo fractional partial derivative of order

0 lt 119902 lt 1 120583 [0 1] times [0 120587] rarr [0 120587] is continuous and in119905 119902119894 119877 rarr 119877 are continuous 0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899 lt 119887 areprefixed numbers and 120601 isinBℎ

Let ℎ(119904) = 1198902119904 119904 lt 0 then 119897 = int

0

infinℎ(119904)119889119904 = 12 Let

Bℎ be the phase space endowed with the norm 120593Bℎ=

int0

minusinfinℎ(119904)sup

120579isin[1199040]120593(120579)

1198712119889119904 Let 119909(119905)(119910) = 119909(119905 119910) and define

the bounded linear operator 119861 119880 rarr 119883 by (119861119906)(119905)(119910) =120583(119905 119910) 0 le 119910 le 120587 119892(119905 120601)(119910) = int0

minusinfin119886(119905 119910 120579)1198761(120601(120579)(119910))119889120579

119868119896(119909(119905minus

119894))(119910) = int

119905119894

minusinfin119902119894(119905119894minus119904)119909(119904 119910)119889119904 and119891(119905 120601119867119909(119905))(119910) =

int0

minusinfin119870(119905 119910 120579)1198762(120601(120579)(119910))119889120579 + 119867119909(119905)(119910) where 119867119909(119905)(119910) =

int119905

0119892(119904 119905)119890

minus119909(119904119910)119889119904 and 120601(120579)(119910) = 120601(120579 119910) (120579 119910) isin (minusinfin 0] times

[0 120587] Moreover the linear fractional control system corre-sponding to (44) is exactly controllable Further if we imposesuitable conditions on 119886 119870 1198761 1198762 119892 119902119894 and 119861 to verifyassumptions onTheorem 8 then the system (44) can be writ-ten in the abstract form of (1)Therefore all the conditions ofTheorem 8 are satisfied and hence the nonlinear fractionalcontrol system (44) is exactly controllable on [0 119887]

Acknowledgment

The work of Yong Ren is supported by the National NaturalScience Foundation of China (no 11371029)

References

[1] P Muthukumar and P Balasubramaniam ldquoApproximate con-trollability of mixed stochastic Volterra-Fredholm type inte-grodifferential systems in Hilbert spacerdquo Journal of the FranklinInstitute vol 348 no 10 pp 2911ndash2922 2011

[2] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009

[3] J Klamka ldquoConstrained controllability of semilinear systemswith delaysrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 169ndash177 2009

[4] J Klamka ldquoStochastic controllability of systems with multipledelays in controlrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 39ndash47 2009

[5] J Klamka ldquoLocal controllability of fractional discrete-timesemilinear systemsrdquo Acta Mechanica et Automatica vol 15 pp55ndash58 2011

[6] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[7] J Dabas A Chauhan and M Kumar ldquoExistence of the mildsolutions for impulsive fractional equations with infinite delayrdquoInternational Journal of Differential Equations vol 2011 ArticleID 793023 20 pages 2011

[8] J Wang M Feckan and Y Zhou ldquoRelaxed controls fornonlinear fractional impulsive evolution equationsrdquo Journal ofOptimization Theory and Applications vol 156 no 1 pp 13ndash322013

[9] M Haase The Functional Calculus for Sectorial Operators vol169 of Operator Theory Advances and Applications BirkhauserBasel Switzerland 2006

[10] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[11] R P Agarwal B de Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[12] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functional evo-lution equations with infinite delayrdquo Applied Mathematics andComputation vol 216 no 1 pp 61ndash69 2010

[13] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications A vol 74 no 5pp 2003ndash2011 2011

[14] J Dabas and A Chauhan ldquoExistence and uniqueness of mildsolution for an impulsive neutral fractional integro-differentialequation with infinite delayrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 754ndash763 2013

[15] R Sakthivel and Y Ren ldquoComplete controllability of stochasticevolution equations with jumpsrdquo Reports on MathematicalPhysics vol 68 no 2 pp 163ndash174 2011

[16] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters of Rapid Publication vol 26 no 4 pp 418ndash4242013

[17] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[18] S Kumar and N Sukavanam ldquoApproximate controllability offractional order semilinear systems with bounded delayrdquo Jour-nal of Differential Equations vol 251 no 11 pp 6163ndash6174 2012

[19] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[20] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends inNanotechnology and Fractional Calculus D Baleanu Z BGuvenc and J A T Machado Eds pp 503ndash509 Springer NewYork NY USA 2010

[21] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[22] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[23] R Sakthivel S Suganya and S M Anthoni ldquoApproximate con-trollability of fractional stochastic evolution equationsrdquo Com-puters amp Mathematics with Applications vol 63 no 3 pp 660ndash668 2012

[24] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[25] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[26] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

where 120597119902119905is the Caputo fractional partial derivative of order

0 lt 119902 lt 1 120583 [0 1] times [0 120587] rarr [0 120587] is continuous and in119905 119902119894 119877 rarr 119877 are continuous 0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899 lt 119887 areprefixed numbers and 120601 isinBℎ

Let ℎ(119904) = 1198902119904 119904 lt 0 then 119897 = int

0

infinℎ(119904)119889119904 = 12 Let

Bℎ be the phase space endowed with the norm 120593Bℎ=

int0

minusinfinℎ(119904)sup

120579isin[1199040]120593(120579)

1198712119889119904 Let 119909(119905)(119910) = 119909(119905 119910) and define

the bounded linear operator 119861 119880 rarr 119883 by (119861119906)(119905)(119910) =120583(119905 119910) 0 le 119910 le 120587 119892(119905 120601)(119910) = int0

minusinfin119886(119905 119910 120579)1198761(120601(120579)(119910))119889120579

119868119896(119909(119905minus

119894))(119910) = int

119905119894

minusinfin119902119894(119905119894minus119904)119909(119904 119910)119889119904 and119891(119905 120601119867119909(119905))(119910) =

int0

minusinfin119870(119905 119910 120579)1198762(120601(120579)(119910))119889120579 + 119867119909(119905)(119910) where 119867119909(119905)(119910) =

int119905

0119892(119904 119905)119890

minus119909(119904119910)119889119904 and 120601(120579)(119910) = 120601(120579 119910) (120579 119910) isin (minusinfin 0] times

[0 120587] Moreover the linear fractional control system corre-sponding to (44) is exactly controllable Further if we imposesuitable conditions on 119886 119870 1198761 1198762 119892 119902119894 and 119861 to verifyassumptions onTheorem 8 then the system (44) can be writ-ten in the abstract form of (1)Therefore all the conditions ofTheorem 8 are satisfied and hence the nonlinear fractionalcontrol system (44) is exactly controllable on [0 119887]

Acknowledgment

The work of Yong Ren is supported by the National NaturalScience Foundation of China (no 11371029)

References

[1] P Muthukumar and P Balasubramaniam ldquoApproximate con-trollability of mixed stochastic Volterra-Fredholm type inte-grodifferential systems in Hilbert spacerdquo Journal of the FranklinInstitute vol 348 no 10 pp 2911ndash2922 2011

[2] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009

[3] J Klamka ldquoConstrained controllability of semilinear systemswith delaysrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 169ndash177 2009

[4] J Klamka ldquoStochastic controllability of systems with multipledelays in controlrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 39ndash47 2009

[5] J Klamka ldquoLocal controllability of fractional discrete-timesemilinear systemsrdquo Acta Mechanica et Automatica vol 15 pp55ndash58 2011

[6] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[7] J Dabas A Chauhan and M Kumar ldquoExistence of the mildsolutions for impulsive fractional equations with infinite delayrdquoInternational Journal of Differential Equations vol 2011 ArticleID 793023 20 pages 2011

[8] J Wang M Feckan and Y Zhou ldquoRelaxed controls fornonlinear fractional impulsive evolution equationsrdquo Journal ofOptimization Theory and Applications vol 156 no 1 pp 13ndash322013

[9] M Haase The Functional Calculus for Sectorial Operators vol169 of Operator Theory Advances and Applications BirkhauserBasel Switzerland 2006

[10] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[11] R P Agarwal B de Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[12] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functional evo-lution equations with infinite delayrdquo Applied Mathematics andComputation vol 216 no 1 pp 61ndash69 2010

[13] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications A vol 74 no 5pp 2003ndash2011 2011

[14] J Dabas and A Chauhan ldquoExistence and uniqueness of mildsolution for an impulsive neutral fractional integro-differentialequation with infinite delayrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 754ndash763 2013

[15] R Sakthivel and Y Ren ldquoComplete controllability of stochasticevolution equations with jumpsrdquo Reports on MathematicalPhysics vol 68 no 2 pp 163ndash174 2011

[16] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters of Rapid Publication vol 26 no 4 pp 418ndash4242013

[17] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[18] S Kumar and N Sukavanam ldquoApproximate controllability offractional order semilinear systems with bounded delayrdquo Jour-nal of Differential Equations vol 251 no 11 pp 6163ndash6174 2012

[19] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[20] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends inNanotechnology and Fractional Calculus D Baleanu Z BGuvenc and J A T Machado Eds pp 503ndash509 Springer NewYork NY USA 2010

[21] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[22] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[23] R Sakthivel S Suganya and S M Anthoni ldquoApproximate con-trollability of fractional stochastic evolution equationsrdquo Com-puters amp Mathematics with Applications vol 63 no 3 pp 660ndash668 2012

[24] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[25] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[26] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Abstract and Applied Analysis

[27] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of