Research ArticleOptimal Control Problem Investigation forLinear Time-Invariant Systems of Fractional Order withLumped Parameters Described by Equations withRiemann-Liouville Derivative

V A Kubyshkin and S S Postnov

V A Trapeznikov Institute of Control Sciences of Russian Academy of Sciences Profsoyuznaya Street 65 Moscow 117997 Russia

Correspondence should be addressed to S S Postnov postnovsergeyinboxru

Received 29 November 2015 Accepted 3 May 2016

Academic Editor Francisco Gordillo

Copyright copy 2016 V A Kubyshkin and S S Postnov This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper studies two optimal control problems for linear time-invariant systems of fractional orderwith lumpedparameterswhosedynamics is described by equations which contain Riemann-Liouville derivative The first problem is to find control with minimalnorm and the second one is to find control withminimal control time at given restriction for control normTheproblem settingwithnonlocal initial conditions is considered which differs from other known settings for integer-order systems and fractional-ordersystems described in terms of equations with Caputo derivative Admissible controls are allowed to belong to the class of functionswhich are 119901-integrable on half segmentThe basic investigation approach is the moment methodThe correctness and solvability ofmoment problem are validated for considered problem setting for the system of arbitrary dimension It is shown that correspondingconditions are analogous to those derived for systems which are described in terms of equations with Caputo derivative For severalparticular cases of one- and two-dimensional systems the posed problems are solved explicitly The dependencies of basic valuesfrom derivative index and control time are analyzed The comparison is performed of obtained results with known results foranalogous integer-order systems and fractional-order systems which are described by equations with Caputo derivative

1 Introduction

Affairs of dynamics and control for fractional-order systemsattract sufficient attention of modern research communityThis field develops impetuously and is characterized by bothof significant theoretical and very actual applied results Andthis field is much wider than integer-order dynamics andcontains some open problems concerning the foundationsof fractional calculus (eg the problem of unified definitionof fractional derivative and its interpretation) It is knownthat Caputo derivative and Riemann-Liouville derivative arethe most popular definitions of fractional-order derivativeFirst of them is recognized by many researches as moreldquophysicalrdquo more realistic and similar to ordinary derivativeBut this definition imposes rather essential requirements(differentiability) on function from which the fractionalderivative is calculated The Riemann-Liouville definition is

used frequently in theoretical investigations although it hasa physical sense but less similar to ordinary derivative Par-ticularly this derivative is nonzero for constant function andinitial and boundary problems for equations with derivativeof this kind require posing nonlocal conditions On the otherhand for existence of Riemann-Liouville derivative from anyfunction only summability of the function is required

Today optimal control problems for fractional-orderdynamical systems actively develop Many interesting resultsare already obtained in this area for different types offractional derivative [1] So up to recent times there areno theorems analogous to Pontryagin maximum principleIn 2014 an analogue of this principle was formulated andproved in [2] for dynamical system described by equationswith Riemann-Liouville derivative Later the formulation andproof of Pontryagin-like maximum principle were proposedin [3] for linear systems described by equations with Caputo

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016 Article ID 4873083 12 pageshttpdxdoiorg10115520164873083

2 Journal of Control Science and Engineering

derivative Another effective and quite universal approachto the search of optimal control is moment method [4]Based on these methods in [5ndash8] the approach was devel-oped to investigation of optimal control problems for lineardynamic systems of fractional order with lumped parametersdescribed by equations with Caputo derivative In [6 8]this approach was generalized on systems with distributedparameters described by diffusion-like equationwithCaputotime-derivative

In this paper the moment method is applied to investi-gation of optimal control problems for dynamical systemsof fractional order with lumped parameters described byequations with Riemann-Liouville derivative The distinctivefeature of problems considered is the posing of nonlocalinitial conditions The optimal control problem for multidi-mensional linear time-invariant system is considered in gen-eral form Possibility of this problem reduction to momentproblem is demonstrated Then correctness and solvabilityare analyzed for the last problem Further one-dimensionalsystem of general view and single and double integrators isinvestigated in detail Explicit solutions of optimal controlproblem are obtained and analyzed including comparisonwith analogous integer-order systems and systems describedby equations with Caputo derivative

2 Problem Statement

Let the system state and control be defined on a half segment(0 119879]119879 gt 0 by vector-functions 119902(119905) = (119902

1(119905) 119902

119873(119905)) and

119906(119905) = (1199061(119905) 119906

119873(119905)) correspondingly

We will consider dynamical systems described by thefollowing equation

0119863120572119894

119905119902119894(119905) = 119886

119894119895119902119895(119905) + 119887

119894119895119906119895(119905) + 119891

119894(119905) (1)

where 119891119894(119905) are perturbations (known) and 119886

119894119895are known

coefficients 119894 119895 = 1119873 The repeated indices suppose sum-mationThe fractional derivative of arbitrary order 120572

119894isin (0 1)

from function 119902119894(119905) is comprehended in our investigation as

left-side Riemann-Liouville derivative [9]

RL0119863120572119894

119905119902119894(119905) =

1

Γ (1 minus 120572119894)

119889

119889119905int

119905

0

119902119894(120591) 119889120591

(119905 minus 120591)120572119894 (2)

Also we will compare obtained results with results describedin [5 6 8] for system (1) with Caputo left-side derivative [9]

119862

0119863120572119894

119905119902119894(119905) =

1

Γ (1 minus 120572119894)int

119905

0

119889119902119894(120591)

119889120591

119889120591

(119905 minus 120591)120572119894 (3)

Definition 1 System (1) with fractional derivative operatorin sense of Riemann-Liouville (Caputo) will be namedas Riemann-Liouville (Caputo) system or RL-system (C-system)

The initial conditions for RL-system are determined innonlocal form [9]

[01198681minus120572119894

119905119902119894(119905)]10038161003816100381610038161003816119905=0+

= 1199040

119894 119894 = 1119873 (4)

where

0119868120572119894

119905119902119894(119905) =

1

Γ (120572119894)int

119905

0

119902119894(120591) 119889120591

(119905 minus 120591)1minus120572119894

(5)

is a left-side Riemann-Liouville integral of order 120572119894[9] The

substitution 119905 = 0+ is comprehended in sense of limitof the expression in square brackets at 119905 rarr 0+ Suchinitial conditions differ from ordinary (local) form used inoptimal control problems Inherently in this case no initialstate is determined (defined by phase coordinates values ininitial time point) but the value of some integral functionalfrom phase coordinates One of the first papers in whichthe optimal control problem with nonlocal initial conditionsis considered was [10] Later such statement was studiedby many authors particularly in [2] Initial conditions oftype (4) can have a quite definite physical interpretation[11 12] For example in some problems of viscoelasticity thestress and strain appear to be coupled by integrodifferentialoperator of fractional orderThen the initial condition of type(4) for one of these functions can be rewritten in local formfor other functions [11]

The final state of system (1) is determined in ordinaryform

119902119894(119879) = 119902

119879

119894 119894 = 1119873 (6)

Let the control belong to the space 119871119901(0 119879] 1 lt 119901 lt infinwith norm [4 13]

119906 (119905) = (int

119879

0

119873

sum

119894=1

1003816100381610038161003816119906119894 (119905)1003816100381610038161003816119901

119889119905)

1119901

(7)

We will consider also the limit case when 119901 rarr infin and thecontrol norm is determined as follows [4 13]

119906 (119905) = vrai max119905isin(0119879]

max119894

1003816100381610038161003816119906119894 (119905)1003816100381610038161003816 (8)

Let us believe that functions 119902119894(119905) 119894 = 1119873 possessing

all properties required for existence of solutions of equationsstudied further on are in particular summable

We will study the two following statements of optimalcontrol problem

Problem 2 (OCP A) Find a control 119906(119905) 119905 isin (0 119879] suchthat system (1) transfers in given final state (6) from the statedetermined by (4) and norm of control will be minimal withthe assigned control time 119879

Problem 3 (OCP B) Find a control 119906(119905) 119905 isin (0 119879] suchthat system (1) transfers in given final state (6) from thestate determined by (4) and control time 119879 will be minimalprovided 119906 le 119897 119897 gt 0 where 119897 is the assigned constant

3 The Moment Problem

31 Preliminaries The classical 119897-problem ofmoments can beformulated in the following way

Journal of Control Science and Engineering 3

Problem 4 Let us have a systemof functions119892119894(119905) isin 119871

1199011015840

(0 119879]119894 = 1119873 1 lt 1199011015840 lt infin Let us also have the assigned numbers119888119894(called moments) 119894 = 1119873 and 119897 gt 0 We should find

function 119906(119905) isin 119871119901(0 119879] 1 lt 119901 lt infin that satisfies thefollowing conditions

int

119879

0

119892119894(120591) 119906 (120591) 119889120591 = 119888

119894(119879) (9)

119906 (119905) le 119897 (10)

1

119901+1

1199011015840= 1 (11)

In order for the moment problem (9) and (10) to besolvable the existence of number 120582

119873gt 0 and numbers

120585lowast

1 120585

lowast

119873that give a solution to the following equivalent

conditional minimum problem is necessary and sufficient[4 14 15]

Problem 5 Find

min1205851 120585119873

(int

119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840

119889119905)

11199011015840

= (int

119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585lowast

119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840

119889119905)

11199011015840

=1

120582119873

(12)

with additional condition119873

sum

119894=1

120585119894119888119894= 1 (13)

In case problem (12) and (13) is solvable the optimalcontrol for OCP A will be given by the following expression[4]

119906 (119905) = 1205821199011015840

119873

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585lowast

119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840minus1

sign[119873

sum

119894=1

120585lowast

119894119892119894(119905)]

119905 isin (0 119879]

(14)

OCP B may be resolved by the following formula

119906 (119905) = 1198971199011015840

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585lowast

119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840minus1

sign[119873

sum

119894=1

120585lowast

119894119892119894(119905)]

119905 isin (0 119879lowast]

(15)

where 119879lowast is the minimal nonnegative real root of equation

120582119873(119879lowast) = 119897 (16)

For solvability of Problem 4 it is necessary and sufficientto satisfy one of two equivalent conditions [4] (1) 120582

119873gt 0

and (2) functions 119892119894(119905) are linearly independent The single

question is the possibility of correct posing of Problem 4which depends on the existence of norm of functions 119892

119894(119905)

in space 1198711199011015840

(0 119879] and the existence of at least one nonzerocomponent in number set 119888

119894

Definition 6 The moment problem (9) and (10) is calledcorrect if the norm of functions 119892

119894(119905) is determined in space

1198711199011015840

(0 119879] 1 lt 1199011015840 lt infin and there exists at least one nonzerocomponent in number set 119888

119894

It is known that optimal control problems in form ofProblems 2 and 3 can be reduced to Problem 4 in case ofinteger-order systems with lumped parameters [4] It wasshown in [5ndash8] that the same is valid for Caputo systemsBelow we will demonstrate that for Riemann-Liouville sys-tems Problems 2 and 3 also can be reduced to Problem 4

32 The Correctness and Solvability of Moment Problemfor Multidimensional Fractional-Order System The generalsolution of (1) in case of RL-system at 120572

1= sdot sdot sdot = 120572

119873= 120572

can be represented by the following expression [9 sect742]

119902 (119905) = 119890A1199051205721199040+ int

119905

0

119890A(119905minus120591)120572

119891 (120591) 119889120591 + int

119905

0

119890A(119905minus120591)120572

119906 (120591) 119889120591 (17)

where 119890A119905120572= 119911120572minus1suminfin

119896=0A119896(119905120572119896Γ((119896 + 1)120572)) is matrix 120572-

exponent [9] A is a matrix of coefficients 119886119894119895 and 1199040 is the

vector of values 1199040119894 defined by initial conditions (4) It is also

known that general solution of (1) in case of Caputo systemcan be written as follows [9 sect742]

119902 (119905) = 119902 (0) + int

119905

0

119890A(119905minus120591)120572

[A119902 (0) + 119891 (120591)] 119889120591

+ int

119905

0

119890A(119905minus120591)120572

119906 (120591) 119889120591

(18)

As in case of integer-order systems solutions (17) and(18) at 119905 = 119879 can be written in the form of expression (9)and consequently can be represented in the form of momentproblem (in case of Caputo system it is demonstrated indetail in [5ndash8]) And in both of these solutions the com-ponents of matrix 120572-exponent act as functions 119892

119894(119905) Other

terms in expressions form the moments Consequently themoment problem for Riemann-Liouville system differs fromthe problem for analogousCaputo systemonly by expressions(and values) for the moments So the theorems concerningcorrectness and solvability of the moment problem provedfor Caputo systems [5ndash8] will be valid also for analogousRiemann-Liouville systems Thus if there exists at least onenonzero moment in the set 119888

119894 119894 = 1119873 we have the following

conditions for multidimensional Riemann-Liouville system

(1) In case of equal differentiation indices 1205721= sdot sdot sdot =

120572119873= 120572 themoment problem of type (9) derived from

(17) at 119905 = 119879will be correct and solvable for all120572whichsatisfy the following condition

120572 gt1199011015840minus 1

1199011015840 (19)

(2) In case of system (1) at 119886119894119895= 120575119894119895+1

119906(119905) = (0 0119906119873(119905)) and119891

119894(119905) = 0 119894 119895 = 1119873 themoment problem

of type (9) derived from (17) at 119905 = 119879 will be correct

4 Journal of Control Science and Engineering

and solvable for every 1205721 120572

119873minus1and 120572

119873which

satisfy the following condition

120572119873gt1199011015840minus 1

1199011015840 (20)

33 The Moment Problem for One-Dimensional System ofFractional-Order Problem Setting and Study In case of119873 =1 the solution of (1) with initial condition (4) can be writtenas follows [9 sect411] (the subscripts are omitted)

119902 (119905) = 1199040119905120572minus1119864120572120572(119886119905120572)

+ int

119905

0

119906 (120591) + 119891 (120591)

(119905 minus 120591)1minus120572119864120572120572[119886 (119905 minus 120591)

120572] 119889120591

(21)

where 119864120572120572(119905) is two-parameter Mittag-Leffler function [9

sect18] By direct calculation one can obtain that at 119905 = 119879expression (21) with regard to (6) may be written over as (9)with the following symbols

119892 (120591) =119864120572120572[119886 (119879 minus 120591)

120572]

(119879 minus 120591)1minus120572

(22)

119888 (119879) = 119902119879minus 1199040119879120572minus1119864120572120572(119886119879120572)

minus int

119879

0

119864120572120572[119886 (119879 minus 120591)

120572]

(119879 minus 120591)1minus120572

119891 (120591) 119889120591

(23)

Expression (22) is identical to analogous expression forone-dimensional Caputo system (formula (17) at 119887 = 1 in[5]) So expressions that result from (12) and (13) will matchwith analogous expressions obtained for the Caputo system[5 6 8] (the difference will appear only after substitution ofexpression (23) for the moment) Consequently if we keepin mind the noted difference we can use the solutions forProblem 5 obtained for Caputo system As shown in [5 6 8]the analytical solution of Problem 4 for one-dimensionalsystem can be obtained for arbitrary 1199011015840 gt 1 (if condition (19)is satisfied)

Let 119891(119905) = 0 Using (13) we can reduce Problem 5 tosimple integral calculation which can be carried out similarlyto case of Caputo system

Consider the case 119906(119905) isin 119871infin(0 119879] Direct calculationby formula (12) with regard to (13) and (22) leads us to thefollowing expression (which matches with analogous resultfor Caputo system [5 6 8])

120582 =|119888 (119879)| 119886

119864120572(119886119879120572) minus 1

(24)

where 119864120572(119886119879120572) = 119864

1205721(119886119879120572) Using (24) we can obtain

from (14) and (15) the solutions of OCP A and OCP Bcorrespondingly

119906 (119905) =|119888 (119879)| 119886

119864120572(119886119879120572) minus 1

119905 isin (0 119879]

119906 (119905) = 119897 sign (119888 (119879lowast)) 119905 isin (0 119879lowast] (25)

where 119879lowast may be received from (16) with (24) It is seen thatcontrols (25) have not one switching point It is similar tothe system of order [120572] + 1 behaviour in accordance withFeldbaumrsquos theorem about 119899 intervals [16]

In case of 119906(119905) isin 119871119901(0 119879] 1 lt 119901 lt infin we can obtain thefollowing from (12) and (13) subject to (22)

120582 =|119888 (119879)|

119865119879

(26)

where

119865119879= [

[

int

119879

0

(119864120572120572(119886 (119879 minus 119905)

120572)

(119879 minus 119905)1minus120572

)

1199011015840

119889119905]

]

11199011015840

(27)

Taking into account (14) one can obtain further for OCP Athe following

119906 (119905) =119888 (119879)

1198651199011015840

119879

(119864120572120572(119886 (119879 minus 119905)

120572)

(119879 minus 119905)1minus120572

)

1199011015840minus1

(28)

For OCP B the solution can be expressed by the followingformula

119906 (119905) = 1198971199011015840

(119864120572120572(119886 (119879lowastminus 119905)120572

)

(119879lowast minus 119905)1minus120572

)

1199011015840minus1

sign (119888 (119879lowast)) (29)

where 119879lowast can be calculated from (16) using (26)Let us now consider a single integrator the special case

of one-dimensional system at 119886 = 0 Instead of (22) and(23) we will have more simple expressions written in termsof elementary functions

119892 (120591) =1

Γ (120572)

1

(119879 minus 120591)1minus120572 (30)

119888 (119879) = 119902119879minus1199040119879120572minus1

Γ (120572) (31)

Taking into account formula (30) one can obtain thefollowing from (12) and (13)

120582 =|119888 (119879)| Γ (120572) (119901

1015840(120572 minus 1) + 1)

11199011015840

119879120572minus1+11199011015840

(32)

From formula (32) using (14) we can derive the followingexpression for control

119906 (119905) =119888 (119879) (119901

1015840(120572 minus 1) + 1) Γ (120572)

1198791199011015840(120572minus1)+1

(119879 minus 119905)(1199011015840minus1)(120572minus1)

119905 isin (0 119879]

(33)

Analogously in case of OCP B we can derive the followingusing (15)

119906 (119905) = 1198971199011015840

(1

Γ (120572)

1

(119879lowast minus 119905)1minus120572)

1199011015840minus1

sign (119888 (119879lowast))

119905 isin (0 119879lowast]

(34)

where 119879lowast can be calculated from (16) using (32)

Journal of Control Science and Engineering 5

Note that expressions (32) (33) and (34) provide anexplicit solution of optimal control problem for Riemann-Liouville single integrator at arbitrary 1199011015840 ge 1

34 The Moment Problem for Double Integrator of FractionalOrder Problem Setting and Study Two-dimensional system(1) at 119891

1(119905) = 119891

2(119905) = 0 119886

11= 11988621= 11988622= 0 119886

12= 1 119906

1(119905) =

0 and 1199062(119905) = 119906(119905) represents itself as a double integrator

Solution (17) subject to initial conditions (4) for this systemcan be written in the following form

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)

+1

Γ (1205721+ 1205722)int

119905

0

119906 (120591) 119889120591

(119905 minus 120591)1minus1205721minus1205722

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)+1

Γ (1205722)int

119905

0

119906 (120591) 119889120591

(119905 minus 120591)1minus1205722

(35)

It is clear that solutions (35) at 119905 = 119879 can be written inform of expression (9) with the following functions 119892

119894(120591) and

moments

1198921(120591) =

1

Γ (1205721+ 1205722)

1

(119879 minus 120591)1minus1205721minus1205722

1198922(120591) =

1

Γ (1205722)

1

(119879 minus 120591)1minus1205722

(36)

1198881(119879) = 119902

119879

1minus1199040

21198791205721+1205722minus1

Γ (1205721+ 1205722)minus1199040

11198791205721minus1

Γ (1205721) (37)

1198882(119879) = 119902

119879

2minus1199040

21198791205722minus1

Γ (1205722) (38)

The functions 11989212(120591) defined by (36) are identical to

the analogous functions obtained in [5] for Caputo doubleintegrator On the contrary the moments defined by (37)-(38) differ in general from the moments obtained in [5]for Caputo double integrator Consequently as for one-dimensional system the form of solution for moment prob-lem will match with that for Caputo double integrator So wecan use the solutions of OCP A and OCP B obtained in [5]taking into account the moments (37)

Firstly consider the case when 119906(119905) isin 119871infin(0 119879] It isshown [5] that in this case theminimization problem (12) and(13) reduces to the following algebraic equation

21205721

(1205721+ 1205722) Γ (1205722+ 1)[Γ (1205721+ 1205722)

Γ (1205722)]

12057221205721

sdot (11988821205852minus 1

1198881

)

minus12057221205721minus1 11988821205852minus 1 minus 120572

21205721

1198881

12058512057221205721

2

=1198791205722

Γ (1205722+ 1)minus1198882

1198881

1198791205721+1205722

Γ (1205721+ 1205722+ 1)

(39)

There is no explicit solution of (39) that can be foundat arbitrary 120572

1and 120572

2 In some particular cases the explicit

solution exists [5] for example in case of zero-secondmoment 119888

2(119879) = 0 Then the solutions of (12) and (13) lead

us to the following expression

1205822=212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

1198881

1198791205721+1205722 (40)

The solution of OCP A will be written as

119906 (119905) = minus212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

1198881

1198791205721+1205722

sdot sign[(119879 minus 119905)1205721 minus 2

minus120572112057221198791205721

1198881

] 119905 isin (0 119879]

(41)

The OCP B explicit solution can be found at additionalassumption 119902119879

1= 0

119906 (119905) = 119897 sign[2minus12057211205722 (119879

lowast)1205721minus (119879lowastminus 119905)1205721

11990201

]

119905 isin (0 119879lowast]

(42)

The minimal control time 119879lowast can be calculated based on(16) and (40)

119879lowast= (212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897Γ (1205721))

1(1+1205722)

(43)

In case of 119906(119905) isin 1198712(0 119879] calculations by (12) and (13) givean expression

1205822=radic1205741

120572111987912057422

11988512 (44)

where 1205741= 1205721+ 21205722minus 1 120574

2= 21205721+ 21205722minus 1 120574

3= 21205722minus 1

and 119885 = |120574112057421198882

1Γ2(1205721+ 1205722) minus 2120574

2120574311988811198882Γ(1205721+ 1205722)Γ(1205722)1198791205721 +

120574112057431198882

2Γ2(1205722)11987921205721 |

According to (14) the OCP A solution will be written as

119906 (119905)

=120574112057431198882Γ (1205722) (1205742(119879 minus 119905)

1205721 minus 12057411198791205721)

120572211198791205741 (119879 minus 119905)

1minus1205722[12057421198881Γ (1205721+ 1205722)

12057431198882Γ (1205722) 1198791205721

sdot1205741(119879 minus 119905)

1205721 minus 12057431198791205721

1205742(119879 minus 119905)

1205721 minus 12057411198791205721minus 1]

(45)

TheOCPB solution can be derived from (15) and (16) andwill be represented by the following expression

6 Journal of Control Science and Engineering

119906 (119905) =119897212057431198882Γ (1205722) (119879lowast)1205721(1205742(119879lowastminus 119905)1205721minus 1205741(119879lowast)1205721)

119885 (119879lowast minus 119905)1minus1205722

[12057421198881Γ (1205721+ 1205722)

12057431198882Γ (1205722) (119879lowast)

1205721

1205741(119879lowastminus 119905)1205721minus 1205743(119879lowast)1205721

1205742(119879lowast minus 119905)

1205721 minus 1205741(119879lowast)1205721minus 1] (46)

where 119879lowast can be found as least positive real root of theequation

11988512=1205721119897

radic120574111987912057422 (47)

Note also that for Caputo double integrator the case when1198882(119879) = 0 corresponds to the case when 1199020

2= 119902119879

2[5] From

formula (38) one can see that the condition 1198882(119879) = 0 is not

so clear

4 Qualitative Analysis of Results

In this chapter the obtained solutions of OCP are analyzedIt will be shown that at integer (equal to 1) values ofdifferentiation indices these solutions reduce to the knownresults for corresponding integer-order systems For doubleintegrator analytical expressions derived for system phasetrajectories in different modes

41 Optimal Control Behaviour at Integer-Order Differentia-tion Indices For one-dimensional system of first order onecan obtain the solution of moment problem and explicitexpressions for optimal controls For example in case ofsingle integrator at 119906(119905) isin 119871infin(0 119879] the OCP A solution isexpressed by the following formula

1199061(119905) =

119902119879minus 1199020

119879 119905 isin (0 119879] (48)

The OCP B solution gives the following [4]

1199061(119905) = 119897 sign (119902119879 minus 1199020) 119905 isin (0 119879lowast]

119879lowast

1=

10038161003816100381610038161003816119902119879minus 119902010038161003816100381610038161003816

119897

(49)

It is easy to see that solution of OCP A defined by (33)(taking into account (31)) at 120572 = 1 is identical to expression(48) The same is true for OCP B solution expressions (34)and (16) subject to (32) at 120572 = 1 give formulas (49)

For double integrator of first order at arbitrary initial andfinal conditions the solutions of OCP A and OCP B lead toquadratic equation for 120585

2[4] Equation (39) reduces to that

equation at 1205721= 1205722= 1 The solution of OCP A for double

integrator of first order at 11990202= 119902119879

2= 0 for 119906(119905) isin 119871infin(0 119879] is

given by the following formula [4]

1199062(119905) =

4 (119902119879

1minus 1199020

1)

1198792sign(1198792 minus 119905

1199021198791minus 11990201

) (50)

By direct calculation one can obtain that formula (41) (ieOCP A solution for Riemann-Liouville double integrator)

subject to (37) at 11990202= 119902119879

2= 0 and 120572

1= 1205722= 1 reduces to

(50)Consider OCP B in case of 119902119879

1= 0 Then the solution of

the problem for double integrator of first order will be givenby the following formulas

119879lowast

2= 2radic

1199020

1

119897

1199062(119905) = 119897 sign( 119905

11990201

minus1

radic11989711990201

) 119905 isin (0 119879lowast

2]

(51)

Expressions (42) and (43) at 1205721= 1205722= 1 transform

into formulas (51) which can be proved by correspondingsubstitution

Thus as for Caputo systems [5 6 8] all the resultsobtained for single and double integrators of fractional orderreduce to corresponding formulas for integer-order systemswhen differentiation indices are equated to 1

42 Investigation of Qualitative Dynamics for Double Integra-tor For two-dimensional systems the analysis of qualitativedynamics is interesting itself We will calculate below theboundary trajectories for double integrator and its trajecto-ries corresponding to optimal control mode Consider 119906(119905) isin119871infin(0 119879]

Definition 7 The boundary trajectories of some system arethe phase trajectories corresponding to boundary values ofcontrol 119906(119905) = plusmn119897

In case of differentiation indices equal to 1 the boundarytrajectories of some system represent the boundaries ofintegral vortex for differential inclusion corresponding to thissystem [17] This manifold bounds the phase space regionwhich contains all admissible trajectories of the systemIn case of fractional-order systems (Caputo and Riemann-Liouville) this property is also valid caused by comparisontheorem [18 19]

Substituting the boundary values of control 119906(119905) =plusmn119897 to solutions (35) one can obtain the following explicitexpressions for boundary trajectories

119902plusmn119897

1(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)plusmn

1198971199051205721+1205722

Γ (1205721+ 1205722+ 1)

119902plusmn119897

2(119905) =

1199040

21199051205722minus1

Γ (1205722)plusmn

1198971199051205722

Γ (1205722+ 1)

(52)

It is seen that expressions (52) at arbitrary1205721and1205722donot

allow eliminating time and obtaining the explicit dependency

Journal of Control Science and Engineering 7

119902plusmn119897

2(119902plusmn119897

1) as opposed to double integrator of integer order [17

sect35] and Caputo double integrator [5] Nevertheless in someparticular cases the time can be eliminated from expressions(52) For example in case of 1199040

2= 0 one can calculate directly

that these formulas reduce to the following equation

119902plusmn119897

1= (Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

12057211205722

sdot [

[

1199040

1

Γ (1205721)(Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

minus11205722

+Γ (1205722+ 1) 119902

plusmn119897

2

Γ (1205721+ 1205722+ 1)]

]

(53)

Analogously in case of 1205722= 1 it is possible to eliminate

time from (52) and to derive the following equation

119902plusmn119897

1=1

Γ (1205721)(119902plusmn119897

2minus 1199040

2

plusmn119897)

1205721minus1

sdot [

[

1199040

1plusmn119902plusmn119897

2minus 1199040

2

119897

1199040

2

1205721

+ (119902plusmn119897

2minus 1199040

2

119897)

2

1

1205721(1205721+ 1)]

]

(54)

Note that expressions (52) at 1205721= 1205722= 1 reduce

to the quadratic equation for integral vortex boundaries ofdifferential inclusion corresponding to the double integratorof integer order [17 sect35]

In order to calculate the phase trajectories of Riemann-Liouville double integrator in optimal control mode we willsubstitute control (41) into solutions (35) Then the followingmotion laws can be obtained

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)minus212057211205722

212057211205722 minus 1

sdot1198881(119879)

1198791205721+1205722[1199051205721+1205722 minus 2 (119905 minus 119905

1015840)1205721+1205722

Θ(119905 minus 1199051015840)]

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)minus212057211205722

212057211205722 minus 1

1198881(119879)

1198791205721+1205722

sdotΓ (1205721+ 1205722+ 1)

Γ (1205722+ 1)

[1199051205722 minus 2 (119905 minus 119905

1015840)1205722Θ(119905 minus 119905

1015840)]

(55)

where 1199051015840 is the switching point of control (41) and Θ(119905) is theHeavisidersquos function Formulas (55) are obtained for control(41) which is valid in case of 119888

2= 0 Consequently these

expressions allow some simplification

5 Computational Results and Analysis

In this chapter the computational results are representedand analyzed (including comparison with analogous Caputosystems) Calculations and its visualization were performed

0 02 04 06 08120572

100

10minus2

10minus4

u

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

Figure 1 The dependencies for norm of control from 120572 at 1199011015840 = 1(logarithmic scale used for ordinates)

05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 2 The dependencies for norm of control from 120572 at 1199011015840 = 2(logarithmic scale used for ordinates)

in MatLab 79 Mittag-Leffler function values were calculatedusing special procedure [20] Integrals were calculated byGauss-Kronrod method [21]

51 Single Integrator We will consider the task of systemtransfer from any state with 1199040 = 0 in the final state 119902119879 = 0

Figures 1ndash3 show the dependencies of control norm fromdifferentiation index calculated by formula (32) at 119879 = 100and different values 1199040 for 1199011015840 = 1 1199011015840 = 2 and 1199011015840 = 32correspondingly (dash-dot lines) Also the analogous depen-dencies are shown for Caputo single integrator calculated byformulas from [5 8] at different 1199020 for the same values of 1199011015840(solid lines) It is seen that curves for Riemann-Liouville andCaputo single integrator differ from each other qualitativelybut converge to the same point at 120572 = 1 In addition thecontrol norm increases with 1199040 or 1199020 growing for both ofintegrator types

In Figure 4 the dependencies of control norm fromcontrol time 119879 are shown at different values of differentiationindex and 1199040 = 1 (for Riemann-Liouville integrator) and1199020= 1 (for Caputo integrator) The solid lines correspond

to 1199011015840 = 1 the dotted lines correspond to 1199011015840 = 32 and thedash-dotted lines correspond to 1199011015840 = 2 It is clear that all ofthese curves decreasemonotonically which correspond to thecase of Caputo integrator [8] and integrator of integer order[4] And the norm of control increases with growing of 120572 and1199011015840

8 Journal of Control Science and Engineering

04 05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 3 The dependencies for norm of control from 120572 at 1199011015840 = 32(logarithmic scale used for ordinates)

0 20 40 60 80

10minus2

10minus1

10minus3

u

T

120572 = 07120572 = 09

120572 = 05

120572 = 03 120572 = 01

Figure 4 The dependencies for norm of control from 119879 Logarith-mic scale used for ordinates

Consider now the minimal control time 119879lowast dependencyfrom differentiation index 120572 In case of 1199040 = 0 and 119902119879 = 0 (16)subject to formula (32) can be solved explicitly and have onlyone real root

119879lowast= (

10038161003816100381610038161003816119904010038161003816100381610038161003816

119897)

1199011015840

(1199011015840(120572 minus 1) + 1) (56)

In Figure 5 the computational results are shown for theobtained formula used at different 1199011015840 at 119897 = 1 and1199040= 1 (dash-dotted lines) Also the similar dependencies

(calculated by formulas from [8]) are represented for Caputointegrator at 119897 = 1 and 1199020 = 1 (solid lines) As can be seenfrom Figure 5 the dependency at 1199011015840 = 1 is linear for both ofintegrator types In case of 1199011015840 gt 1 the dependency for Caputointegrator becomes nonlinear

52 One-Dimensional System of General Type Consider (asabove) the task of system transfer from any state with 1199040 = 0to the state 119902119879 = 0

Figures 6 and 7 represent the dependencies of controlnorm from differentiation index at 1199011015840 = 1 and 1199011015840 = 2calculated by formulas (24) and (26) correspondingly (dash-doted lines) The analogical dependencies for Caputo one-dimensional system are also shown (solid lines) calculated byformulas from [5 8] The curves were calculated for differentvalues of parameter 119886 and the following values of otherparameters 119879 = 2 119887 = 1 1199040 = 1 (RL-system) and 1199020 = 1

0 02 04 06 080

02

04

06

08

120572

Tlowast

p998400 = 1

p998400 = 32

p998400 = 2

p998400 = 3

Figure 5 The dependencies for minimal control time from 120572

0 02 04 06 080

05

1

120572

a = 1

a = 01

a = 05

u

Figure 6 The dependencies for control norm from 120572 at 1199011015840 = 1 anddifferent values of parameter 119886

06 07 08 090

05

1

120572

a = 1

a = 01a = 05

u

Figure 7 The dependencies for control norm from 120572 at 1199011015840 = 2 anddifferent values of parameter 119886

(C-system) It is seen from Figures 6 and 7 that differencein derivative type has qualitatively effect on behaviour ofdependencies This behaviour changes at 119886 lt 1 and 119886 = 1In case of 1199011015840 = 1 the curves for C-system decrease andcurves for RL-system increase at 119886 lt 1 (analogically to singleintegrator) At 119886 = 1 the curves for both of systems increaseFor1199011015840 = 2 the dependencies increase for both of system types

The dependencies of control norm from control time aresimilar in general for single integrator and are not shownhere

The dependencies of minimal control time from 120572 atdifferent values of parameter 119886 are shown in Figures 8 and 9for 1199011015840 = 1 and 1199011015840 = 2 correspondingly The following valuesof other parameters were chosen 119897 = 2 1199040 = 1 (RL-system)

Journal of Control Science and Engineering 9

0 02 04 06 080

02

04

06

120572

a = 1

a = 05

a = 01

Tlowast

Figure 8The dependencies forminimal control time from120572 at1199011015840 =1 and different values of parameter 119886

06 065 07 075 08 085 09 0950

01

02

03

120572

a = 1

a = 05a = 01

Tlowast

Figure 9The dependencies forminimal control time from120572 at1199011015840 =2 and different values of parameter 119886

and 1199020 = 1 (C-system) Dash-dotted lines correspond to RL-system and solid lines correspond to C-system It is seen thatin case of RL-system the curves have the quasi-linear trendand differ qualitatively from the curves for C-system

53 Double Integrator As in case of one-dimensional sys-tems we will consider the task of system transfer from anystate with nonzero values of phase coordinates into the originat phase plane wewill suppose that 1199040

1= 0 and 119902119879

1= 119902119879

2= 1199040

2=

0The dependencies of control norm from one of differ-

entiation indices for Riemann-Liouville double integrator atseveral fixed values of other indexes are shown in Figures10 and 11 (dash-dotted lines) The curves were calculated byexpression (40) for 1199011015840 = 1 119879 = 100 and 1199040

1= 1 Also the

similar dependencies for Caputo double integrator are shownat these figures calculated for the same values of parameters1199011015840 and 119879 and for 1199020

1= 1 (solid lines) The dependencies for

1199011015840= 2 are similar and not demonstrated here It is seen

from Figure 10 that curves corresponding to different systems(RL- and C-systems) at identical values of 120572

2converge into

the same point at 1205721= 1 In Figure 11 the distance between

different curves for C- and RL-systems decreases when 1205721

increasesThe dependencies of control norm from control time for

double integrator are analogous generally to that in case ofone-dimensional systems and are not demonstrated here

Let us consider the dependency of minimal control timefrom differentiation indices 120572

1and 120572

2 In case of 1199011015840 = 1 this

0 02 04 06 08

100

10minus2

10minus4

u

1205721

1205722 = 09

1205722 = 011205722 = 05

Figure 10 The dependencies for control norm from 1205721at 1199011015840 = 1

and several values of 1205722

0 02 04 06 08

10minus2

10minus4

u

1205722

1205721 = 091205721 = 01 1205721 = 05

Figure 11The dependencies for control norm from 1205722at 1199011015840 = 1 and

several values of 1205721

dependency is given by formula (43) In case of 1199011015840 = 2 withadditional assumption 119902119879

1= 119902119879

2= 1199040

2= 0 the explicit formula

also can be found for such dependency (as solution of (16)subject to (40))

119879lowast= (radic21205721+ 21205722+ 1 (120572

1+ 21205722+ 1) Γ (120572

1+ 1205722)

Γ (1205721+ 1)

sdot

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897)

2(21205722+1)

(57)

In Figures 12ndash15 the dependencies are shown for minimalcontrol time from one of differentiation indices at fixedvalue of other indices The curves are calculated at 119897 = 2for 1199011015840 = 1 (Figures 12 and 13) and 1199011015840 = 2 (Figures 14and 15) Dash-dotted lines correspond to Riemann-Liouvilledouble integrator and are calculated for 1199040

1= 1 Solid lines

correspond toCaputo double integrator and are calculated for1199020

1= 1 As seen from these figures the dependencies for RL-

and C-systems differ qualitatively from each otherLet us now analyze the qualitative dynamics of Riemann-

Liouville double integrator its boundary trajectories andphase trajectories in optimal controlmodeNote that last typeof trajectories in case of Riemann-Liouville double integratorwill not have an obvious initial point at phase plane since theinitial condition (4) defines not the system state but the valueof some integral functional at initial timeThe system state in

10 Journal of Control Science and Engineering

0 02 04 06 08

Tlowast

102

100

10minus2

1205721

1205722 = 01

1205722 = 05 1205722 = 09

Figure 12 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

100

10minus2

1205722

1205721 = 01

1205721 = 05

1205721 = 09

Figure 13 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

105

100

1205721

1205722 = 071205722 = 09

Figure 14 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 2 Logarithmic scale used for

ordinates

this case can be undefined as follows from solutions (35) andexpressions (55)

The boundary trajectories for studied system are repre-sented in Figure 16 for Riemann-Liouville double integrator(dash-dotted lines) and for analogous Caputo system (solidlines) For the last system calculations were performed usingformulas from [5] It is clear that trajectories for both ofsystems are very close to each other In Figure 17 the fragmentof trajectories is shown at enlarged scale

The phase trajectories of Riemann-Liouville double inte-grator in optimal control mode were calculated using expres-sions (55) In Figure 18 the results are shown for different

055 06 065 07 075 08 085 09 0950

1

2

Tlowast

1205722

1205721 = 03 1205721 = 09

1205721 = 07

Figure 15 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 2 Logarithmic scale used for

ordinates

0 200 400

q2

q1

50

0

minus50

minus600 minus400 minus200

Figure 16 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

q2

q1

minus50

minus48

minus46

minus400 minus380 minus360 minus340

Figure 17 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

0 005

0

q1

1205721 = 071205722 = 04

1205721 = 01 1205722 = 021205721 = 01 1205722 = 1

1205721 = 05 1205722 = 1

1205721 = 03 1205722 = 05

q2

minus0005

minus001

minus0015minus005

Figure 18 The phase trajectories in optimal control mode forRiemann-Liouville double integrator

Journal of Control Science and Engineering 11

0

q2

q1

minus1

minus1

minus2

minus2

minus3

minus3

minus4

minus4minus5

minus5minus6

1205721 = 03 1205722 = 05

1205721 = 07 1205722 = 04

1205721 = 05 1205722 = 1

1205721 = 01 1205722 = 02

1205721 = 01 1205722 = 1

Figure 19 The phase trajectories in optimal control mode forCaputo double integrator

0 02 04 06 08 1

0

q1

q2 1205722 = 03 1205722 = 05

1205722 = 08

minus001

minus002

Figure 20 The phase trajectories in optimal control mode forRiemann-Liouville and Caputo double integrator in case of 120572

1= 1

values of differentiation indices 119879 = 100 and 11990401= 1 In

Figure 19 the phase trajectories of Caputo double integrator inoptimal control mode are represented which were calculatedby formulas from [5] at 119879 = 100 and 1199020

1= 1 It is clear

from Figures 18 and 19 that both of systems demonstrate theovercontrolling effect In case of 120572

1= 1 this effect does not

appear and phase trajectories coincide for both of systemsstudied (Figure 20)

6 Summary

In this paper the optimal control problem is investigated forlinear dynamical systems of fractional order described byequations with Riemann-Liouville derivative The problemreduced to themoment problem For the last problem the cor-rectness and solvability conditions derived Some examplesof one- and two-dimensional systems were considered andexplicit solutions of corresponding optimal control problemswere found The properties of these solutions were analyzedand compared with analogous integer-order systems andfractional-order systems described by equationswithCaputoderivative

Some computational results are demonstrated that showthe dependency of control norm from differentiation indicesant control time and the dependency ofminimal control timefrom differentiation indices Qualitative dynamics of doubleintegrator are analyzed and show that boundary trajectoriesof Riemann-Liouville system of this type are very close to

the same trajectories for corresponding Caputo system It isshown also that the overcontrolling effect appears in optimalcontrol mode of system studied

Obtained results can be useful in investigation of optimalcontrol problems for fractional-order systems and in designof particular control systems

Competing Interests

The authors declare that they have no competing interests

References

[1] A G Butkovskii S S Postnov and E A Postnova ldquoFractionalintegro-differential calculus and its control-theoretical applica-tions II Fractional dynamic systems modeling and hardwareimplementationrdquo Automation and Remote Control vol 74 no5 pp 725ndash749 2013

[2] R Kamocki ldquoPontryagin maximum principle for fractionalordinary optimal control problemsrdquo Mathematical Methods inthe Applied Sciences vol 37 no 11 pp 1668ndash1686 2014

[3] R Kamocki and M Majewski ldquoFractional linear control sys-tems with Caputo derivative and their optimizationrdquo OptimalControl Applications and Methods vol 36 no 6 pp 953ndash9672015

[4] A G Butkovskiy Distributed Control Systems American Else-vier New York NY USA 1969

[5] V A Kubyshkin and S S Postnov ldquoOptimal control problemfor a linear stationary fractional order system in the form of aproblem of moments problem setting and a studyrdquoAutomationand Remote Control vol 75 no 5 pp 805ndash817 2014

[6] V A Kubyshkin and S S Postnov ldquoOptimal control prob-lem for linear fractional-order systemsrdquo in Proceedings of theInternational Conference on Fractional Differentiation and ItsApplications (ICFDA rsquo14) 6 pages IEEE Catania Italy June2014 Paper ID 3189701

[7] V A Kubyshkin and S S Postnov ldquoAnalysis of two optimalcontrol problems for a fractional-order pendulum by themethod of momentsrdquo Automation and Remote Control vol 76no 7 pp 1302ndash1314 2015

[8] V A Kubyshkin and S S Postnov ldquoThe optimal controlproblem for linear systems of non-integer order with lumpedand distributed parameters Discontinuityrdquo Nonlinearity andComplexity vol 4 no 4 pp 429ndash443 2015

[9] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 ElsevierAmsterdam The Netherlands 2006

[10] R Almeida and D F M Torres ldquoCalculus of variations withfractional derivatives and fractional integralsrdquo Applied Mathe-matics Letters vol 22 no 12 pp 1816ndash1820 2009

[11] N Heymans and I Podlubny ldquoPhysical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivativesrdquo Rheologica Acta vol 45 no 5pp 765ndash771 2006

[12] V E Tarasov Fractional Dynamics Springer Berlin Germany2010

[13] A N Kolmogorov and S V Fomin Elements of the Theory ofFunctions and Functional Analysis Dover Publications Mine-ola NY USA 1999

[14] N I AkhiezerTheClassical Moment Problem and Some RelatedQuestions in Analysis Hafner New York NY USA 1965

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

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2 Journal of Control Science and Engineering

derivative Another effective and quite universal approachto the search of optimal control is moment method [4]Based on these methods in [5ndash8] the approach was devel-oped to investigation of optimal control problems for lineardynamic systems of fractional order with lumped parametersdescribed by equations with Caputo derivative In [6 8]this approach was generalized on systems with distributedparameters described by diffusion-like equationwithCaputotime-derivative

In this paper the moment method is applied to investi-gation of optimal control problems for dynamical systemsof fractional order with lumped parameters described byequations with Riemann-Liouville derivative The distinctivefeature of problems considered is the posing of nonlocalinitial conditions The optimal control problem for multidi-mensional linear time-invariant system is considered in gen-eral form Possibility of this problem reduction to momentproblem is demonstrated Then correctness and solvabilityare analyzed for the last problem Further one-dimensionalsystem of general view and single and double integrators isinvestigated in detail Explicit solutions of optimal controlproblem are obtained and analyzed including comparisonwith analogous integer-order systems and systems describedby equations with Caputo derivative

2 Problem Statement

Let the system state and control be defined on a half segment(0 119879]119879 gt 0 by vector-functions 119902(119905) = (119902

1(119905) 119902

119873(119905)) and

119906(119905) = (1199061(119905) 119906

119873(119905)) correspondingly

We will consider dynamical systems described by thefollowing equation

0119863120572119894

119905119902119894(119905) = 119886

119894119895119902119895(119905) + 119887

119894119895119906119895(119905) + 119891

119894(119905) (1)

where 119891119894(119905) are perturbations (known) and 119886

119894119895are known

coefficients 119894 119895 = 1119873 The repeated indices suppose sum-mationThe fractional derivative of arbitrary order 120572

119894isin (0 1)

from function 119902119894(119905) is comprehended in our investigation as

left-side Riemann-Liouville derivative [9]

RL0119863120572119894

119905119902119894(119905) =

1

Γ (1 minus 120572119894)

119889

119889119905int

119905

0

119902119894(120591) 119889120591

(119905 minus 120591)120572119894 (2)

Also we will compare obtained results with results describedin [5 6 8] for system (1) with Caputo left-side derivative [9]

119862

0119863120572119894

119905119902119894(119905) =

1

Γ (1 minus 120572119894)int

119905

0

119889119902119894(120591)

119889120591

119889120591

(119905 minus 120591)120572119894 (3)

Definition 1 System (1) with fractional derivative operatorin sense of Riemann-Liouville (Caputo) will be namedas Riemann-Liouville (Caputo) system or RL-system (C-system)

The initial conditions for RL-system are determined innonlocal form [9]

[01198681minus120572119894

119905119902119894(119905)]10038161003816100381610038161003816119905=0+

= 1199040

119894 119894 = 1119873 (4)

where

0119868120572119894

119905119902119894(119905) =

1

Γ (120572119894)int

119905

0

119902119894(120591) 119889120591

(119905 minus 120591)1minus120572119894

(5)

is a left-side Riemann-Liouville integral of order 120572119894[9] The

substitution 119905 = 0+ is comprehended in sense of limitof the expression in square brackets at 119905 rarr 0+ Suchinitial conditions differ from ordinary (local) form used inoptimal control problems Inherently in this case no initialstate is determined (defined by phase coordinates values ininitial time point) but the value of some integral functionalfrom phase coordinates One of the first papers in whichthe optimal control problem with nonlocal initial conditionsis considered was [10] Later such statement was studiedby many authors particularly in [2] Initial conditions oftype (4) can have a quite definite physical interpretation[11 12] For example in some problems of viscoelasticity thestress and strain appear to be coupled by integrodifferentialoperator of fractional orderThen the initial condition of type(4) for one of these functions can be rewritten in local formfor other functions [11]

The final state of system (1) is determined in ordinaryform

119902119894(119879) = 119902

119879

119894 119894 = 1119873 (6)

Let the control belong to the space 119871119901(0 119879] 1 lt 119901 lt infinwith norm [4 13]

119906 (119905) = (int

119879

0

119873

sum

119894=1

1003816100381610038161003816119906119894 (119905)1003816100381610038161003816119901

119889119905)

1119901

(7)

We will consider also the limit case when 119901 rarr infin and thecontrol norm is determined as follows [4 13]

119906 (119905) = vrai max119905isin(0119879]

max119894

1003816100381610038161003816119906119894 (119905)1003816100381610038161003816 (8)

Let us believe that functions 119902119894(119905) 119894 = 1119873 possessing

all properties required for existence of solutions of equationsstudied further on are in particular summable

We will study the two following statements of optimalcontrol problem

Problem 2 (OCP A) Find a control 119906(119905) 119905 isin (0 119879] suchthat system (1) transfers in given final state (6) from the statedetermined by (4) and norm of control will be minimal withthe assigned control time 119879

Problem 3 (OCP B) Find a control 119906(119905) 119905 isin (0 119879] suchthat system (1) transfers in given final state (6) from thestate determined by (4) and control time 119879 will be minimalprovided 119906 le 119897 119897 gt 0 where 119897 is the assigned constant

3 The Moment Problem

31 Preliminaries The classical 119897-problem ofmoments can beformulated in the following way

Journal of Control Science and Engineering 3

Problem 4 Let us have a systemof functions119892119894(119905) isin 119871

1199011015840

(0 119879]119894 = 1119873 1 lt 1199011015840 lt infin Let us also have the assigned numbers119888119894(called moments) 119894 = 1119873 and 119897 gt 0 We should find

function 119906(119905) isin 119871119901(0 119879] 1 lt 119901 lt infin that satisfies thefollowing conditions

int

119879

0

119892119894(120591) 119906 (120591) 119889120591 = 119888

119894(119879) (9)

119906 (119905) le 119897 (10)

1

119901+1

1199011015840= 1 (11)

In order for the moment problem (9) and (10) to besolvable the existence of number 120582

119873gt 0 and numbers

120585lowast

1 120585

lowast

119873that give a solution to the following equivalent

conditional minimum problem is necessary and sufficient[4 14 15]

Problem 5 Find

min1205851 120585119873

(int

119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840

119889119905)

11199011015840

= (int

119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585lowast

119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840

119889119905)

11199011015840

=1

120582119873

(12)

with additional condition119873

sum

119894=1

120585119894119888119894= 1 (13)

In case problem (12) and (13) is solvable the optimalcontrol for OCP A will be given by the following expression[4]

119906 (119905) = 1205821199011015840

119873

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585lowast

119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840minus1

sign[119873

sum

119894=1

120585lowast

119894119892119894(119905)]

119905 isin (0 119879]

(14)

OCP B may be resolved by the following formula

119906 (119905) = 1198971199011015840

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585lowast

119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840minus1

sign[119873

sum

119894=1

120585lowast

119894119892119894(119905)]

119905 isin (0 119879lowast]

(15)

where 119879lowast is the minimal nonnegative real root of equation

120582119873(119879lowast) = 119897 (16)

For solvability of Problem 4 it is necessary and sufficientto satisfy one of two equivalent conditions [4] (1) 120582

119873gt 0

and (2) functions 119892119894(119905) are linearly independent The single

question is the possibility of correct posing of Problem 4which depends on the existence of norm of functions 119892

119894(119905)

in space 1198711199011015840

(0 119879] and the existence of at least one nonzerocomponent in number set 119888

119894

Definition 6 The moment problem (9) and (10) is calledcorrect if the norm of functions 119892

119894(119905) is determined in space

1198711199011015840

(0 119879] 1 lt 1199011015840 lt infin and there exists at least one nonzerocomponent in number set 119888

119894

It is known that optimal control problems in form ofProblems 2 and 3 can be reduced to Problem 4 in case ofinteger-order systems with lumped parameters [4] It wasshown in [5ndash8] that the same is valid for Caputo systemsBelow we will demonstrate that for Riemann-Liouville sys-tems Problems 2 and 3 also can be reduced to Problem 4

32 The Correctness and Solvability of Moment Problemfor Multidimensional Fractional-Order System The generalsolution of (1) in case of RL-system at 120572

1= sdot sdot sdot = 120572

119873= 120572

can be represented by the following expression [9 sect742]

119902 (119905) = 119890A1199051205721199040+ int

119905

0

119890A(119905minus120591)120572

119891 (120591) 119889120591 + int

119905

0

119890A(119905minus120591)120572

119906 (120591) 119889120591 (17)

where 119890A119905120572= 119911120572minus1suminfin

119896=0A119896(119905120572119896Γ((119896 + 1)120572)) is matrix 120572-

exponent [9] A is a matrix of coefficients 119886119894119895 and 1199040 is the

vector of values 1199040119894 defined by initial conditions (4) It is also

known that general solution of (1) in case of Caputo systemcan be written as follows [9 sect742]

119902 (119905) = 119902 (0) + int

119905

0

119890A(119905minus120591)120572

[A119902 (0) + 119891 (120591)] 119889120591

+ int

119905

0

119890A(119905minus120591)120572

119906 (120591) 119889120591

(18)

As in case of integer-order systems solutions (17) and(18) at 119905 = 119879 can be written in the form of expression (9)and consequently can be represented in the form of momentproblem (in case of Caputo system it is demonstrated indetail in [5ndash8]) And in both of these solutions the com-ponents of matrix 120572-exponent act as functions 119892

119894(119905) Other

terms in expressions form the moments Consequently themoment problem for Riemann-Liouville system differs fromthe problem for analogousCaputo systemonly by expressions(and values) for the moments So the theorems concerningcorrectness and solvability of the moment problem provedfor Caputo systems [5ndash8] will be valid also for analogousRiemann-Liouville systems Thus if there exists at least onenonzero moment in the set 119888

119894 119894 = 1119873 we have the following

conditions for multidimensional Riemann-Liouville system

(1) In case of equal differentiation indices 1205721= sdot sdot sdot =

120572119873= 120572 themoment problem of type (9) derived from

(17) at 119905 = 119879will be correct and solvable for all120572whichsatisfy the following condition

120572 gt1199011015840minus 1

1199011015840 (19)

(2) In case of system (1) at 119886119894119895= 120575119894119895+1

119906(119905) = (0 0119906119873(119905)) and119891

119894(119905) = 0 119894 119895 = 1119873 themoment problem

of type (9) derived from (17) at 119905 = 119879 will be correct

4 Journal of Control Science and Engineering

and solvable for every 1205721 120572

119873minus1and 120572

119873which

satisfy the following condition

120572119873gt1199011015840minus 1

1199011015840 (20)

33 The Moment Problem for One-Dimensional System ofFractional-Order Problem Setting and Study In case of119873 =1 the solution of (1) with initial condition (4) can be writtenas follows [9 sect411] (the subscripts are omitted)

119902 (119905) = 1199040119905120572minus1119864120572120572(119886119905120572)

+ int

119905

0

119906 (120591) + 119891 (120591)

(119905 minus 120591)1minus120572119864120572120572[119886 (119905 minus 120591)

120572] 119889120591

(21)

where 119864120572120572(119905) is two-parameter Mittag-Leffler function [9

sect18] By direct calculation one can obtain that at 119905 = 119879expression (21) with regard to (6) may be written over as (9)with the following symbols

119892 (120591) =119864120572120572[119886 (119879 minus 120591)

120572]

(119879 minus 120591)1minus120572

(22)

119888 (119879) = 119902119879minus 1199040119879120572minus1119864120572120572(119886119879120572)

minus int

119879

0

119864120572120572[119886 (119879 minus 120591)

120572]

(119879 minus 120591)1minus120572

119891 (120591) 119889120591

(23)

Expression (22) is identical to analogous expression forone-dimensional Caputo system (formula (17) at 119887 = 1 in[5]) So expressions that result from (12) and (13) will matchwith analogous expressions obtained for the Caputo system[5 6 8] (the difference will appear only after substitution ofexpression (23) for the moment) Consequently if we keepin mind the noted difference we can use the solutions forProblem 5 obtained for Caputo system As shown in [5 6 8]the analytical solution of Problem 4 for one-dimensionalsystem can be obtained for arbitrary 1199011015840 gt 1 (if condition (19)is satisfied)

Let 119891(119905) = 0 Using (13) we can reduce Problem 5 tosimple integral calculation which can be carried out similarlyto case of Caputo system

Consider the case 119906(119905) isin 119871infin(0 119879] Direct calculationby formula (12) with regard to (13) and (22) leads us to thefollowing expression (which matches with analogous resultfor Caputo system [5 6 8])

120582 =|119888 (119879)| 119886

119864120572(119886119879120572) minus 1

(24)

where 119864120572(119886119879120572) = 119864

1205721(119886119879120572) Using (24) we can obtain

from (14) and (15) the solutions of OCP A and OCP Bcorrespondingly

119906 (119905) =|119888 (119879)| 119886

119864120572(119886119879120572) minus 1

119905 isin (0 119879]

119906 (119905) = 119897 sign (119888 (119879lowast)) 119905 isin (0 119879lowast] (25)

where 119879lowast may be received from (16) with (24) It is seen thatcontrols (25) have not one switching point It is similar tothe system of order [120572] + 1 behaviour in accordance withFeldbaumrsquos theorem about 119899 intervals [16]

In case of 119906(119905) isin 119871119901(0 119879] 1 lt 119901 lt infin we can obtain thefollowing from (12) and (13) subject to (22)

120582 =|119888 (119879)|

119865119879

(26)

where

119865119879= [

[

int

119879

0

(119864120572120572(119886 (119879 minus 119905)

120572)

(119879 minus 119905)1minus120572

)

1199011015840

119889119905]

]

11199011015840

(27)

Taking into account (14) one can obtain further for OCP Athe following

119906 (119905) =119888 (119879)

1198651199011015840

119879

(119864120572120572(119886 (119879 minus 119905)

120572)

(119879 minus 119905)1minus120572

)

1199011015840minus1

(28)

For OCP B the solution can be expressed by the followingformula

119906 (119905) = 1198971199011015840

(119864120572120572(119886 (119879lowastminus 119905)120572

)

(119879lowast minus 119905)1minus120572

)

1199011015840minus1

sign (119888 (119879lowast)) (29)

where 119879lowast can be calculated from (16) using (26)Let us now consider a single integrator the special case

of one-dimensional system at 119886 = 0 Instead of (22) and(23) we will have more simple expressions written in termsof elementary functions

119892 (120591) =1

Γ (120572)

1

(119879 minus 120591)1minus120572 (30)

119888 (119879) = 119902119879minus1199040119879120572minus1

Γ (120572) (31)

Taking into account formula (30) one can obtain thefollowing from (12) and (13)

120582 =|119888 (119879)| Γ (120572) (119901

1015840(120572 minus 1) + 1)

11199011015840

119879120572minus1+11199011015840

(32)

From formula (32) using (14) we can derive the followingexpression for control

119906 (119905) =119888 (119879) (119901

1015840(120572 minus 1) + 1) Γ (120572)

1198791199011015840(120572minus1)+1

(119879 minus 119905)(1199011015840minus1)(120572minus1)

119905 isin (0 119879]

(33)

Analogously in case of OCP B we can derive the followingusing (15)

119906 (119905) = 1198971199011015840

(1

Γ (120572)

1

(119879lowast minus 119905)1minus120572)

1199011015840minus1

sign (119888 (119879lowast))

119905 isin (0 119879lowast]

(34)

where 119879lowast can be calculated from (16) using (32)

Journal of Control Science and Engineering 5

Note that expressions (32) (33) and (34) provide anexplicit solution of optimal control problem for Riemann-Liouville single integrator at arbitrary 1199011015840 ge 1

34 The Moment Problem for Double Integrator of FractionalOrder Problem Setting and Study Two-dimensional system(1) at 119891

1(119905) = 119891

2(119905) = 0 119886

11= 11988621= 11988622= 0 119886

12= 1 119906

1(119905) =

0 and 1199062(119905) = 119906(119905) represents itself as a double integrator

Solution (17) subject to initial conditions (4) for this systemcan be written in the following form

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)

+1

Γ (1205721+ 1205722)int

119905

0

119906 (120591) 119889120591

(119905 minus 120591)1minus1205721minus1205722

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)+1

Γ (1205722)int

119905

0

119906 (120591) 119889120591

(119905 minus 120591)1minus1205722

(35)

It is clear that solutions (35) at 119905 = 119879 can be written inform of expression (9) with the following functions 119892

119894(120591) and

moments

1198921(120591) =

1

Γ (1205721+ 1205722)

1

(119879 minus 120591)1minus1205721minus1205722

1198922(120591) =

1

Γ (1205722)

1

(119879 minus 120591)1minus1205722

(36)

1198881(119879) = 119902

119879

1minus1199040

21198791205721+1205722minus1

Γ (1205721+ 1205722)minus1199040

11198791205721minus1

Γ (1205721) (37)

1198882(119879) = 119902

119879

2minus1199040

21198791205722minus1

Γ (1205722) (38)

The functions 11989212(120591) defined by (36) are identical to

the analogous functions obtained in [5] for Caputo doubleintegrator On the contrary the moments defined by (37)-(38) differ in general from the moments obtained in [5]for Caputo double integrator Consequently as for one-dimensional system the form of solution for moment prob-lem will match with that for Caputo double integrator So wecan use the solutions of OCP A and OCP B obtained in [5]taking into account the moments (37)

Firstly consider the case when 119906(119905) isin 119871infin(0 119879] It isshown [5] that in this case theminimization problem (12) and(13) reduces to the following algebraic equation

21205721

(1205721+ 1205722) Γ (1205722+ 1)[Γ (1205721+ 1205722)

Γ (1205722)]

12057221205721

sdot (11988821205852minus 1

1198881

)

minus12057221205721minus1 11988821205852minus 1 minus 120572

21205721

1198881

12058512057221205721

2

=1198791205722

Γ (1205722+ 1)minus1198882

1198881

1198791205721+1205722

Γ (1205721+ 1205722+ 1)

(39)

There is no explicit solution of (39) that can be foundat arbitrary 120572

1and 120572

2 In some particular cases the explicit

solution exists [5] for example in case of zero-secondmoment 119888

2(119879) = 0 Then the solutions of (12) and (13) lead

us to the following expression

1205822=212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

1198881

1198791205721+1205722 (40)

The solution of OCP A will be written as

119906 (119905) = minus212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

1198881

1198791205721+1205722

sdot sign[(119879 minus 119905)1205721 minus 2

minus120572112057221198791205721

1198881

] 119905 isin (0 119879]

(41)

The OCP B explicit solution can be found at additionalassumption 119902119879

1= 0

119906 (119905) = 119897 sign[2minus12057211205722 (119879

lowast)1205721minus (119879lowastminus 119905)1205721

11990201

]

119905 isin (0 119879lowast]

(42)

The minimal control time 119879lowast can be calculated based on(16) and (40)

119879lowast= (212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897Γ (1205721))

1(1+1205722)

(43)

In case of 119906(119905) isin 1198712(0 119879] calculations by (12) and (13) givean expression

1205822=radic1205741

120572111987912057422

11988512 (44)

where 1205741= 1205721+ 21205722minus 1 120574

2= 21205721+ 21205722minus 1 120574

3= 21205722minus 1

and 119885 = |120574112057421198882

1Γ2(1205721+ 1205722) minus 2120574

2120574311988811198882Γ(1205721+ 1205722)Γ(1205722)1198791205721 +

120574112057431198882

2Γ2(1205722)11987921205721 |

According to (14) the OCP A solution will be written as

119906 (119905)

=120574112057431198882Γ (1205722) (1205742(119879 minus 119905)

1205721 minus 12057411198791205721)

120572211198791205741 (119879 minus 119905)

1minus1205722[12057421198881Γ (1205721+ 1205722)

12057431198882Γ (1205722) 1198791205721

sdot1205741(119879 minus 119905)

1205721 minus 12057431198791205721

1205742(119879 minus 119905)

1205721 minus 12057411198791205721minus 1]

(45)

TheOCPB solution can be derived from (15) and (16) andwill be represented by the following expression

6 Journal of Control Science and Engineering

119906 (119905) =119897212057431198882Γ (1205722) (119879lowast)1205721(1205742(119879lowastminus 119905)1205721minus 1205741(119879lowast)1205721)

119885 (119879lowast minus 119905)1minus1205722

[12057421198881Γ (1205721+ 1205722)

12057431198882Γ (1205722) (119879lowast)

1205721

1205741(119879lowastminus 119905)1205721minus 1205743(119879lowast)1205721

1205742(119879lowast minus 119905)

1205721 minus 1205741(119879lowast)1205721minus 1] (46)

where 119879lowast can be found as least positive real root of theequation

11988512=1205721119897

radic120574111987912057422 (47)

Note also that for Caputo double integrator the case when1198882(119879) = 0 corresponds to the case when 1199020

2= 119902119879

2[5] From

formula (38) one can see that the condition 1198882(119879) = 0 is not

so clear

4 Qualitative Analysis of Results

In this chapter the obtained solutions of OCP are analyzedIt will be shown that at integer (equal to 1) values ofdifferentiation indices these solutions reduce to the knownresults for corresponding integer-order systems For doubleintegrator analytical expressions derived for system phasetrajectories in different modes

41 Optimal Control Behaviour at Integer-Order Differentia-tion Indices For one-dimensional system of first order onecan obtain the solution of moment problem and explicitexpressions for optimal controls For example in case ofsingle integrator at 119906(119905) isin 119871infin(0 119879] the OCP A solution isexpressed by the following formula

1199061(119905) =

119902119879minus 1199020

119879 119905 isin (0 119879] (48)

The OCP B solution gives the following [4]

1199061(119905) = 119897 sign (119902119879 minus 1199020) 119905 isin (0 119879lowast]

119879lowast

1=

10038161003816100381610038161003816119902119879minus 119902010038161003816100381610038161003816

119897

(49)

It is easy to see that solution of OCP A defined by (33)(taking into account (31)) at 120572 = 1 is identical to expression(48) The same is true for OCP B solution expressions (34)and (16) subject to (32) at 120572 = 1 give formulas (49)

For double integrator of first order at arbitrary initial andfinal conditions the solutions of OCP A and OCP B lead toquadratic equation for 120585

2[4] Equation (39) reduces to that

equation at 1205721= 1205722= 1 The solution of OCP A for double

integrator of first order at 11990202= 119902119879

2= 0 for 119906(119905) isin 119871infin(0 119879] is

given by the following formula [4]

1199062(119905) =

4 (119902119879

1minus 1199020

1)

1198792sign(1198792 minus 119905

1199021198791minus 11990201

) (50)

By direct calculation one can obtain that formula (41) (ieOCP A solution for Riemann-Liouville double integrator)

subject to (37) at 11990202= 119902119879

2= 0 and 120572

1= 1205722= 1 reduces to

(50)Consider OCP B in case of 119902119879

1= 0 Then the solution of

the problem for double integrator of first order will be givenby the following formulas

119879lowast

2= 2radic

1199020

1

119897

1199062(119905) = 119897 sign( 119905

11990201

minus1

radic11989711990201

) 119905 isin (0 119879lowast

2]

(51)

Expressions (42) and (43) at 1205721= 1205722= 1 transform

into formulas (51) which can be proved by correspondingsubstitution

Thus as for Caputo systems [5 6 8] all the resultsobtained for single and double integrators of fractional orderreduce to corresponding formulas for integer-order systemswhen differentiation indices are equated to 1

42 Investigation of Qualitative Dynamics for Double Integra-tor For two-dimensional systems the analysis of qualitativedynamics is interesting itself We will calculate below theboundary trajectories for double integrator and its trajecto-ries corresponding to optimal control mode Consider 119906(119905) isin119871infin(0 119879]

Definition 7 The boundary trajectories of some system arethe phase trajectories corresponding to boundary values ofcontrol 119906(119905) = plusmn119897

In case of differentiation indices equal to 1 the boundarytrajectories of some system represent the boundaries ofintegral vortex for differential inclusion corresponding to thissystem [17] This manifold bounds the phase space regionwhich contains all admissible trajectories of the systemIn case of fractional-order systems (Caputo and Riemann-Liouville) this property is also valid caused by comparisontheorem [18 19]

Substituting the boundary values of control 119906(119905) =plusmn119897 to solutions (35) one can obtain the following explicitexpressions for boundary trajectories

119902plusmn119897

1(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)plusmn

1198971199051205721+1205722

Γ (1205721+ 1205722+ 1)

119902plusmn119897

2(119905) =

1199040

21199051205722minus1

Γ (1205722)plusmn

1198971199051205722

Γ (1205722+ 1)

(52)

It is seen that expressions (52) at arbitrary1205721and1205722donot

allow eliminating time and obtaining the explicit dependency

Journal of Control Science and Engineering 7

119902plusmn119897

2(119902plusmn119897

1) as opposed to double integrator of integer order [17

sect35] and Caputo double integrator [5] Nevertheless in someparticular cases the time can be eliminated from expressions(52) For example in case of 1199040

2= 0 one can calculate directly

that these formulas reduce to the following equation

119902plusmn119897

1= (Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

12057211205722

sdot [

[

1199040

1

Γ (1205721)(Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

minus11205722

+Γ (1205722+ 1) 119902

plusmn119897

2

Γ (1205721+ 1205722+ 1)]

]

(53)

Analogously in case of 1205722= 1 it is possible to eliminate

time from (52) and to derive the following equation

119902plusmn119897

1=1

Γ (1205721)(119902plusmn119897

2minus 1199040

2

plusmn119897)

1205721minus1

sdot [

[

1199040

1plusmn119902plusmn119897

2minus 1199040

2

119897

1199040

2

1205721

+ (119902plusmn119897

2minus 1199040

2

119897)

2

1

1205721(1205721+ 1)]

]

(54)

Note that expressions (52) at 1205721= 1205722= 1 reduce

to the quadratic equation for integral vortex boundaries ofdifferential inclusion corresponding to the double integratorof integer order [17 sect35]

In order to calculate the phase trajectories of Riemann-Liouville double integrator in optimal control mode we willsubstitute control (41) into solutions (35) Then the followingmotion laws can be obtained

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)minus212057211205722

212057211205722 minus 1

sdot1198881(119879)

1198791205721+1205722[1199051205721+1205722 minus 2 (119905 minus 119905

1015840)1205721+1205722

Θ(119905 minus 1199051015840)]

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)minus212057211205722

212057211205722 minus 1

1198881(119879)

1198791205721+1205722

sdotΓ (1205721+ 1205722+ 1)

Γ (1205722+ 1)

[1199051205722 minus 2 (119905 minus 119905

1015840)1205722Θ(119905 minus 119905

1015840)]

(55)

where 1199051015840 is the switching point of control (41) and Θ(119905) is theHeavisidersquos function Formulas (55) are obtained for control(41) which is valid in case of 119888

2= 0 Consequently these

expressions allow some simplification

5 Computational Results and Analysis

In this chapter the computational results are representedand analyzed (including comparison with analogous Caputosystems) Calculations and its visualization were performed

0 02 04 06 08120572

100

10minus2

10minus4

u

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

Figure 1 The dependencies for norm of control from 120572 at 1199011015840 = 1(logarithmic scale used for ordinates)

05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 2 The dependencies for norm of control from 120572 at 1199011015840 = 2(logarithmic scale used for ordinates)

in MatLab 79 Mittag-Leffler function values were calculatedusing special procedure [20] Integrals were calculated byGauss-Kronrod method [21]

51 Single Integrator We will consider the task of systemtransfer from any state with 1199040 = 0 in the final state 119902119879 = 0

Figures 1ndash3 show the dependencies of control norm fromdifferentiation index calculated by formula (32) at 119879 = 100and different values 1199040 for 1199011015840 = 1 1199011015840 = 2 and 1199011015840 = 32correspondingly (dash-dot lines) Also the analogous depen-dencies are shown for Caputo single integrator calculated byformulas from [5 8] at different 1199020 for the same values of 1199011015840(solid lines) It is seen that curves for Riemann-Liouville andCaputo single integrator differ from each other qualitativelybut converge to the same point at 120572 = 1 In addition thecontrol norm increases with 1199040 or 1199020 growing for both ofintegrator types

In Figure 4 the dependencies of control norm fromcontrol time 119879 are shown at different values of differentiationindex and 1199040 = 1 (for Riemann-Liouville integrator) and1199020= 1 (for Caputo integrator) The solid lines correspond

to 1199011015840 = 1 the dotted lines correspond to 1199011015840 = 32 and thedash-dotted lines correspond to 1199011015840 = 2 It is clear that all ofthese curves decreasemonotonically which correspond to thecase of Caputo integrator [8] and integrator of integer order[4] And the norm of control increases with growing of 120572 and1199011015840

8 Journal of Control Science and Engineering

04 05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 3 The dependencies for norm of control from 120572 at 1199011015840 = 32(logarithmic scale used for ordinates)

0 20 40 60 80

10minus2

10minus1

10minus3

u

T

120572 = 07120572 = 09

120572 = 05

120572 = 03 120572 = 01

Figure 4 The dependencies for norm of control from 119879 Logarith-mic scale used for ordinates

Consider now the minimal control time 119879lowast dependencyfrom differentiation index 120572 In case of 1199040 = 0 and 119902119879 = 0 (16)subject to formula (32) can be solved explicitly and have onlyone real root

119879lowast= (

10038161003816100381610038161003816119904010038161003816100381610038161003816

119897)

1199011015840

(1199011015840(120572 minus 1) + 1) (56)

In Figure 5 the computational results are shown for theobtained formula used at different 1199011015840 at 119897 = 1 and1199040= 1 (dash-dotted lines) Also the similar dependencies

(calculated by formulas from [8]) are represented for Caputointegrator at 119897 = 1 and 1199020 = 1 (solid lines) As can be seenfrom Figure 5 the dependency at 1199011015840 = 1 is linear for both ofintegrator types In case of 1199011015840 gt 1 the dependency for Caputointegrator becomes nonlinear

52 One-Dimensional System of General Type Consider (asabove) the task of system transfer from any state with 1199040 = 0to the state 119902119879 = 0

Figures 6 and 7 represent the dependencies of controlnorm from differentiation index at 1199011015840 = 1 and 1199011015840 = 2calculated by formulas (24) and (26) correspondingly (dash-doted lines) The analogical dependencies for Caputo one-dimensional system are also shown (solid lines) calculated byformulas from [5 8] The curves were calculated for differentvalues of parameter 119886 and the following values of otherparameters 119879 = 2 119887 = 1 1199040 = 1 (RL-system) and 1199020 = 1

0 02 04 06 080

02

04

06

08

120572

Tlowast

p998400 = 1

p998400 = 32

p998400 = 2

p998400 = 3

Figure 5 The dependencies for minimal control time from 120572

0 02 04 06 080

05

1

120572

a = 1

a = 01

a = 05

u

Figure 6 The dependencies for control norm from 120572 at 1199011015840 = 1 anddifferent values of parameter 119886

06 07 08 090

05

1

120572

a = 1

a = 01a = 05

u

Figure 7 The dependencies for control norm from 120572 at 1199011015840 = 2 anddifferent values of parameter 119886

(C-system) It is seen from Figures 6 and 7 that differencein derivative type has qualitatively effect on behaviour ofdependencies This behaviour changes at 119886 lt 1 and 119886 = 1In case of 1199011015840 = 1 the curves for C-system decrease andcurves for RL-system increase at 119886 lt 1 (analogically to singleintegrator) At 119886 = 1 the curves for both of systems increaseFor1199011015840 = 2 the dependencies increase for both of system types

The dependencies of control norm from control time aresimilar in general for single integrator and are not shownhere

The dependencies of minimal control time from 120572 atdifferent values of parameter 119886 are shown in Figures 8 and 9for 1199011015840 = 1 and 1199011015840 = 2 correspondingly The following valuesof other parameters were chosen 119897 = 2 1199040 = 1 (RL-system)

Journal of Control Science and Engineering 9

0 02 04 06 080

02

04

06

120572

a = 1

a = 05

a = 01

Tlowast

Figure 8The dependencies forminimal control time from120572 at1199011015840 =1 and different values of parameter 119886

06 065 07 075 08 085 09 0950

01

02

03

120572

a = 1

a = 05a = 01

Tlowast

Figure 9The dependencies forminimal control time from120572 at1199011015840 =2 and different values of parameter 119886

and 1199020 = 1 (C-system) Dash-dotted lines correspond to RL-system and solid lines correspond to C-system It is seen thatin case of RL-system the curves have the quasi-linear trendand differ qualitatively from the curves for C-system

53 Double Integrator As in case of one-dimensional sys-tems we will consider the task of system transfer from anystate with nonzero values of phase coordinates into the originat phase plane wewill suppose that 1199040

1= 0 and 119902119879

1= 119902119879

2= 1199040

2=

0The dependencies of control norm from one of differ-

entiation indices for Riemann-Liouville double integrator atseveral fixed values of other indexes are shown in Figures10 and 11 (dash-dotted lines) The curves were calculated byexpression (40) for 1199011015840 = 1 119879 = 100 and 1199040

1= 1 Also the

similar dependencies for Caputo double integrator are shownat these figures calculated for the same values of parameters1199011015840 and 119879 and for 1199020

1= 1 (solid lines) The dependencies for

1199011015840= 2 are similar and not demonstrated here It is seen

from Figure 10 that curves corresponding to different systems(RL- and C-systems) at identical values of 120572

2converge into

the same point at 1205721= 1 In Figure 11 the distance between

different curves for C- and RL-systems decreases when 1205721

increasesThe dependencies of control norm from control time for

double integrator are analogous generally to that in case ofone-dimensional systems and are not demonstrated here

Let us consider the dependency of minimal control timefrom differentiation indices 120572

1and 120572

2 In case of 1199011015840 = 1 this

0 02 04 06 08

100

10minus2

10minus4

u

1205721

1205722 = 09

1205722 = 011205722 = 05

Figure 10 The dependencies for control norm from 1205721at 1199011015840 = 1

and several values of 1205722

0 02 04 06 08

10minus2

10minus4

u

1205722

1205721 = 091205721 = 01 1205721 = 05

Figure 11The dependencies for control norm from 1205722at 1199011015840 = 1 and

several values of 1205721

dependency is given by formula (43) In case of 1199011015840 = 2 withadditional assumption 119902119879

1= 119902119879

2= 1199040

2= 0 the explicit formula

also can be found for such dependency (as solution of (16)subject to (40))

119879lowast= (radic21205721+ 21205722+ 1 (120572

1+ 21205722+ 1) Γ (120572

1+ 1205722)

Γ (1205721+ 1)

sdot

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897)

2(21205722+1)

(57)

In Figures 12ndash15 the dependencies are shown for minimalcontrol time from one of differentiation indices at fixedvalue of other indices The curves are calculated at 119897 = 2for 1199011015840 = 1 (Figures 12 and 13) and 1199011015840 = 2 (Figures 14and 15) Dash-dotted lines correspond to Riemann-Liouvilledouble integrator and are calculated for 1199040

1= 1 Solid lines

correspond toCaputo double integrator and are calculated for1199020

1= 1 As seen from these figures the dependencies for RL-

and C-systems differ qualitatively from each otherLet us now analyze the qualitative dynamics of Riemann-

Liouville double integrator its boundary trajectories andphase trajectories in optimal controlmodeNote that last typeof trajectories in case of Riemann-Liouville double integratorwill not have an obvious initial point at phase plane since theinitial condition (4) defines not the system state but the valueof some integral functional at initial timeThe system state in

10 Journal of Control Science and Engineering

0 02 04 06 08

Tlowast

102

100

10minus2

1205721

1205722 = 01

1205722 = 05 1205722 = 09

Figure 12 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

100

10minus2

1205722

1205721 = 01

1205721 = 05

1205721 = 09

Figure 13 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

105

100

1205721

1205722 = 071205722 = 09

Figure 14 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 2 Logarithmic scale used for

ordinates

this case can be undefined as follows from solutions (35) andexpressions (55)

The boundary trajectories for studied system are repre-sented in Figure 16 for Riemann-Liouville double integrator(dash-dotted lines) and for analogous Caputo system (solidlines) For the last system calculations were performed usingformulas from [5] It is clear that trajectories for both ofsystems are very close to each other In Figure 17 the fragmentof trajectories is shown at enlarged scale

The phase trajectories of Riemann-Liouville double inte-grator in optimal control mode were calculated using expres-sions (55) In Figure 18 the results are shown for different

055 06 065 07 075 08 085 09 0950

1

2

Tlowast

1205722

1205721 = 03 1205721 = 09

1205721 = 07

Figure 15 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 2 Logarithmic scale used for

ordinates

0 200 400

q2

q1

50

0

minus50

minus600 minus400 minus200

Figure 16 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

q2

q1

minus50

minus48

minus46

minus400 minus380 minus360 minus340

Figure 17 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

0 005

0

q1

1205721 = 071205722 = 04

1205721 = 01 1205722 = 021205721 = 01 1205722 = 1

1205721 = 05 1205722 = 1

1205721 = 03 1205722 = 05

q2

minus0005

minus001

minus0015minus005

Figure 18 The phase trajectories in optimal control mode forRiemann-Liouville double integrator

Journal of Control Science and Engineering 11

0

q2

q1

minus1

minus1

minus2

minus2

minus3

minus3

minus4

minus4minus5

minus5minus6

1205721 = 03 1205722 = 05

1205721 = 07 1205722 = 04

1205721 = 05 1205722 = 1

1205721 = 01 1205722 = 02

1205721 = 01 1205722 = 1

Figure 19 The phase trajectories in optimal control mode forCaputo double integrator

0 02 04 06 08 1

0

q1

q2 1205722 = 03 1205722 = 05

1205722 = 08

minus001

minus002

Figure 20 The phase trajectories in optimal control mode forRiemann-Liouville and Caputo double integrator in case of 120572

1= 1

values of differentiation indices 119879 = 100 and 11990401= 1 In

Figure 19 the phase trajectories of Caputo double integrator inoptimal control mode are represented which were calculatedby formulas from [5] at 119879 = 100 and 1199020

1= 1 It is clear

from Figures 18 and 19 that both of systems demonstrate theovercontrolling effect In case of 120572

1= 1 this effect does not

appear and phase trajectories coincide for both of systemsstudied (Figure 20)

6 Summary

In this paper the optimal control problem is investigated forlinear dynamical systems of fractional order described byequations with Riemann-Liouville derivative The problemreduced to themoment problem For the last problem the cor-rectness and solvability conditions derived Some examplesof one- and two-dimensional systems were considered andexplicit solutions of corresponding optimal control problemswere found The properties of these solutions were analyzedand compared with analogous integer-order systems andfractional-order systems described by equationswithCaputoderivative

Some computational results are demonstrated that showthe dependency of control norm from differentiation indicesant control time and the dependency ofminimal control timefrom differentiation indices Qualitative dynamics of doubleintegrator are analyzed and show that boundary trajectoriesof Riemann-Liouville system of this type are very close to

the same trajectories for corresponding Caputo system It isshown also that the overcontrolling effect appears in optimalcontrol mode of system studied

Obtained results can be useful in investigation of optimalcontrol problems for fractional-order systems and in designof particular control systems

Competing Interests

The authors declare that they have no competing interests

References

[1] A G Butkovskii S S Postnov and E A Postnova ldquoFractionalintegro-differential calculus and its control-theoretical applica-tions II Fractional dynamic systems modeling and hardwareimplementationrdquo Automation and Remote Control vol 74 no5 pp 725ndash749 2013

[2] R Kamocki ldquoPontryagin maximum principle for fractionalordinary optimal control problemsrdquo Mathematical Methods inthe Applied Sciences vol 37 no 11 pp 1668ndash1686 2014

[3] R Kamocki and M Majewski ldquoFractional linear control sys-tems with Caputo derivative and their optimizationrdquo OptimalControl Applications and Methods vol 36 no 6 pp 953ndash9672015

[4] A G Butkovskiy Distributed Control Systems American Else-vier New York NY USA 1969

[5] V A Kubyshkin and S S Postnov ldquoOptimal control problemfor a linear stationary fractional order system in the form of aproblem of moments problem setting and a studyrdquoAutomationand Remote Control vol 75 no 5 pp 805ndash817 2014

[6] V A Kubyshkin and S S Postnov ldquoOptimal control prob-lem for linear fractional-order systemsrdquo in Proceedings of theInternational Conference on Fractional Differentiation and ItsApplications (ICFDA rsquo14) 6 pages IEEE Catania Italy June2014 Paper ID 3189701

[7] V A Kubyshkin and S S Postnov ldquoAnalysis of two optimalcontrol problems for a fractional-order pendulum by themethod of momentsrdquo Automation and Remote Control vol 76no 7 pp 1302ndash1314 2015

[8] V A Kubyshkin and S S Postnov ldquoThe optimal controlproblem for linear systems of non-integer order with lumpedand distributed parameters Discontinuityrdquo Nonlinearity andComplexity vol 4 no 4 pp 429ndash443 2015

[9] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 ElsevierAmsterdam The Netherlands 2006

[10] R Almeida and D F M Torres ldquoCalculus of variations withfractional derivatives and fractional integralsrdquo Applied Mathe-matics Letters vol 22 no 12 pp 1816ndash1820 2009

[11] N Heymans and I Podlubny ldquoPhysical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivativesrdquo Rheologica Acta vol 45 no 5pp 765ndash771 2006

[12] V E Tarasov Fractional Dynamics Springer Berlin Germany2010

[13] A N Kolmogorov and S V Fomin Elements of the Theory ofFunctions and Functional Analysis Dover Publications Mine-ola NY USA 1999

[14] N I AkhiezerTheClassical Moment Problem and Some RelatedQuestions in Analysis Hafner New York NY USA 1965

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

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International Journal of

Journal of Control Science and Engineering 3

Problem 4 Let us have a systemof functions119892119894(119905) isin 119871

1199011015840

(0 119879]119894 = 1119873 1 lt 1199011015840 lt infin Let us also have the assigned numbers119888119894(called moments) 119894 = 1119873 and 119897 gt 0 We should find

function 119906(119905) isin 119871119901(0 119879] 1 lt 119901 lt infin that satisfies thefollowing conditions

int

119879

0

119892119894(120591) 119906 (120591) 119889120591 = 119888

119894(119879) (9)

119906 (119905) le 119897 (10)

1

119901+1

1199011015840= 1 (11)

In order for the moment problem (9) and (10) to besolvable the existence of number 120582

119873gt 0 and numbers

120585lowast

1 120585

lowast

119873that give a solution to the following equivalent

conditional minimum problem is necessary and sufficient[4 14 15]

Problem 5 Find

min1205851 120585119873

(int

119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840

119889119905)

11199011015840

= (int

119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585lowast

119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840

119889119905)

11199011015840

=1

120582119873

(12)

with additional condition119873

sum

119894=1

120585119894119888119894= 1 (13)

In case problem (12) and (13) is solvable the optimalcontrol for OCP A will be given by the following expression[4]

119906 (119905) = 1205821199011015840

119873

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585lowast

119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840minus1

sign[119873

sum

119894=1

120585lowast

119894119892119894(119905)]

119905 isin (0 119879]

(14)

OCP B may be resolved by the following formula

119906 (119905) = 1198971199011015840

1003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119894=1

120585lowast

119894119892119894(119905)

1003816100381610038161003816100381610038161003816100381610038161003816

1199011015840minus1

sign[119873

sum

119894=1

120585lowast

119894119892119894(119905)]

119905 isin (0 119879lowast]

(15)

where 119879lowast is the minimal nonnegative real root of equation

120582119873(119879lowast) = 119897 (16)

For solvability of Problem 4 it is necessary and sufficientto satisfy one of two equivalent conditions [4] (1) 120582

119873gt 0

and (2) functions 119892119894(119905) are linearly independent The single

question is the possibility of correct posing of Problem 4which depends on the existence of norm of functions 119892

119894(119905)

in space 1198711199011015840

(0 119879] and the existence of at least one nonzerocomponent in number set 119888

119894

Definition 6 The moment problem (9) and (10) is calledcorrect if the norm of functions 119892

119894(119905) is determined in space

1198711199011015840

(0 119879] 1 lt 1199011015840 lt infin and there exists at least one nonzerocomponent in number set 119888

119894

It is known that optimal control problems in form ofProblems 2 and 3 can be reduced to Problem 4 in case ofinteger-order systems with lumped parameters [4] It wasshown in [5ndash8] that the same is valid for Caputo systemsBelow we will demonstrate that for Riemann-Liouville sys-tems Problems 2 and 3 also can be reduced to Problem 4

32 The Correctness and Solvability of Moment Problemfor Multidimensional Fractional-Order System The generalsolution of (1) in case of RL-system at 120572

1= sdot sdot sdot = 120572

119873= 120572

can be represented by the following expression [9 sect742]

119902 (119905) = 119890A1199051205721199040+ int

119905

0

119890A(119905minus120591)120572

119891 (120591) 119889120591 + int

119905

0

119890A(119905minus120591)120572

119906 (120591) 119889120591 (17)

where 119890A119905120572= 119911120572minus1suminfin

119896=0A119896(119905120572119896Γ((119896 + 1)120572)) is matrix 120572-

exponent [9] A is a matrix of coefficients 119886119894119895 and 1199040 is the

vector of values 1199040119894 defined by initial conditions (4) It is also

known that general solution of (1) in case of Caputo systemcan be written as follows [9 sect742]

119902 (119905) = 119902 (0) + int

119905

0

119890A(119905minus120591)120572

[A119902 (0) + 119891 (120591)] 119889120591

+ int

119905

0

119890A(119905minus120591)120572

119906 (120591) 119889120591

(18)

As in case of integer-order systems solutions (17) and(18) at 119905 = 119879 can be written in the form of expression (9)and consequently can be represented in the form of momentproblem (in case of Caputo system it is demonstrated indetail in [5ndash8]) And in both of these solutions the com-ponents of matrix 120572-exponent act as functions 119892

119894(119905) Other

terms in expressions form the moments Consequently themoment problem for Riemann-Liouville system differs fromthe problem for analogousCaputo systemonly by expressions(and values) for the moments So the theorems concerningcorrectness and solvability of the moment problem provedfor Caputo systems [5ndash8] will be valid also for analogousRiemann-Liouville systems Thus if there exists at least onenonzero moment in the set 119888

119894 119894 = 1119873 we have the following

conditions for multidimensional Riemann-Liouville system

(1) In case of equal differentiation indices 1205721= sdot sdot sdot =

120572119873= 120572 themoment problem of type (9) derived from

(17) at 119905 = 119879will be correct and solvable for all120572whichsatisfy the following condition

120572 gt1199011015840minus 1

1199011015840 (19)

(2) In case of system (1) at 119886119894119895= 120575119894119895+1

119906(119905) = (0 0119906119873(119905)) and119891

119894(119905) = 0 119894 119895 = 1119873 themoment problem

of type (9) derived from (17) at 119905 = 119879 will be correct

4 Journal of Control Science and Engineering

and solvable for every 1205721 120572

119873minus1and 120572

119873which

satisfy the following condition

120572119873gt1199011015840minus 1

1199011015840 (20)

33 The Moment Problem for One-Dimensional System ofFractional-Order Problem Setting and Study In case of119873 =1 the solution of (1) with initial condition (4) can be writtenas follows [9 sect411] (the subscripts are omitted)

119902 (119905) = 1199040119905120572minus1119864120572120572(119886119905120572)

+ int

119905

0

119906 (120591) + 119891 (120591)

(119905 minus 120591)1minus120572119864120572120572[119886 (119905 minus 120591)

120572] 119889120591

(21)

where 119864120572120572(119905) is two-parameter Mittag-Leffler function [9

sect18] By direct calculation one can obtain that at 119905 = 119879expression (21) with regard to (6) may be written over as (9)with the following symbols

119892 (120591) =119864120572120572[119886 (119879 minus 120591)

120572]

(119879 minus 120591)1minus120572

(22)

119888 (119879) = 119902119879minus 1199040119879120572minus1119864120572120572(119886119879120572)

minus int

119879

0

119864120572120572[119886 (119879 minus 120591)

120572]

(119879 minus 120591)1minus120572

119891 (120591) 119889120591

(23)

Expression (22) is identical to analogous expression forone-dimensional Caputo system (formula (17) at 119887 = 1 in[5]) So expressions that result from (12) and (13) will matchwith analogous expressions obtained for the Caputo system[5 6 8] (the difference will appear only after substitution ofexpression (23) for the moment) Consequently if we keepin mind the noted difference we can use the solutions forProblem 5 obtained for Caputo system As shown in [5 6 8]the analytical solution of Problem 4 for one-dimensionalsystem can be obtained for arbitrary 1199011015840 gt 1 (if condition (19)is satisfied)

Let 119891(119905) = 0 Using (13) we can reduce Problem 5 tosimple integral calculation which can be carried out similarlyto case of Caputo system

Consider the case 119906(119905) isin 119871infin(0 119879] Direct calculationby formula (12) with regard to (13) and (22) leads us to thefollowing expression (which matches with analogous resultfor Caputo system [5 6 8])

120582 =|119888 (119879)| 119886

119864120572(119886119879120572) minus 1

(24)

where 119864120572(119886119879120572) = 119864

1205721(119886119879120572) Using (24) we can obtain

from (14) and (15) the solutions of OCP A and OCP Bcorrespondingly

119906 (119905) =|119888 (119879)| 119886

119864120572(119886119879120572) minus 1

119905 isin (0 119879]

119906 (119905) = 119897 sign (119888 (119879lowast)) 119905 isin (0 119879lowast] (25)

where 119879lowast may be received from (16) with (24) It is seen thatcontrols (25) have not one switching point It is similar tothe system of order [120572] + 1 behaviour in accordance withFeldbaumrsquos theorem about 119899 intervals [16]

In case of 119906(119905) isin 119871119901(0 119879] 1 lt 119901 lt infin we can obtain thefollowing from (12) and (13) subject to (22)

120582 =|119888 (119879)|

119865119879

(26)

where

119865119879= [

[

int

119879

0

(119864120572120572(119886 (119879 minus 119905)

120572)

(119879 minus 119905)1minus120572

)

1199011015840

119889119905]

]

11199011015840

(27)

Taking into account (14) one can obtain further for OCP Athe following

119906 (119905) =119888 (119879)

1198651199011015840

119879

(119864120572120572(119886 (119879 minus 119905)

120572)

(119879 minus 119905)1minus120572

)

1199011015840minus1

(28)

For OCP B the solution can be expressed by the followingformula

119906 (119905) = 1198971199011015840

(119864120572120572(119886 (119879lowastminus 119905)120572

)

(119879lowast minus 119905)1minus120572

)

1199011015840minus1

sign (119888 (119879lowast)) (29)

where 119879lowast can be calculated from (16) using (26)Let us now consider a single integrator the special case

of one-dimensional system at 119886 = 0 Instead of (22) and(23) we will have more simple expressions written in termsof elementary functions

119892 (120591) =1

Γ (120572)

1

(119879 minus 120591)1minus120572 (30)

119888 (119879) = 119902119879minus1199040119879120572minus1

Γ (120572) (31)

Taking into account formula (30) one can obtain thefollowing from (12) and (13)

120582 =|119888 (119879)| Γ (120572) (119901

1015840(120572 minus 1) + 1)

11199011015840

119879120572minus1+11199011015840

(32)

From formula (32) using (14) we can derive the followingexpression for control

119906 (119905) =119888 (119879) (119901

1015840(120572 minus 1) + 1) Γ (120572)

1198791199011015840(120572minus1)+1

(119879 minus 119905)(1199011015840minus1)(120572minus1)

119905 isin (0 119879]

(33)

Analogously in case of OCP B we can derive the followingusing (15)

119906 (119905) = 1198971199011015840

(1

Γ (120572)

1

(119879lowast minus 119905)1minus120572)

1199011015840minus1

sign (119888 (119879lowast))

119905 isin (0 119879lowast]

(34)

where 119879lowast can be calculated from (16) using (32)

Journal of Control Science and Engineering 5

Note that expressions (32) (33) and (34) provide anexplicit solution of optimal control problem for Riemann-Liouville single integrator at arbitrary 1199011015840 ge 1

34 The Moment Problem for Double Integrator of FractionalOrder Problem Setting and Study Two-dimensional system(1) at 119891

1(119905) = 119891

2(119905) = 0 119886

11= 11988621= 11988622= 0 119886

12= 1 119906

1(119905) =

0 and 1199062(119905) = 119906(119905) represents itself as a double integrator

Solution (17) subject to initial conditions (4) for this systemcan be written in the following form

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)

+1

Γ (1205721+ 1205722)int

119905

0

119906 (120591) 119889120591

(119905 minus 120591)1minus1205721minus1205722

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)+1

Γ (1205722)int

119905

0

119906 (120591) 119889120591

(119905 minus 120591)1minus1205722

(35)

It is clear that solutions (35) at 119905 = 119879 can be written inform of expression (9) with the following functions 119892

119894(120591) and

moments

1198921(120591) =

1

Γ (1205721+ 1205722)

1

(119879 minus 120591)1minus1205721minus1205722

1198922(120591) =

1

Γ (1205722)

1

(119879 minus 120591)1minus1205722

(36)

1198881(119879) = 119902

119879

1minus1199040

21198791205721+1205722minus1

Γ (1205721+ 1205722)minus1199040

11198791205721minus1

Γ (1205721) (37)

1198882(119879) = 119902

119879

2minus1199040

21198791205722minus1

Γ (1205722) (38)

The functions 11989212(120591) defined by (36) are identical to

the analogous functions obtained in [5] for Caputo doubleintegrator On the contrary the moments defined by (37)-(38) differ in general from the moments obtained in [5]for Caputo double integrator Consequently as for one-dimensional system the form of solution for moment prob-lem will match with that for Caputo double integrator So wecan use the solutions of OCP A and OCP B obtained in [5]taking into account the moments (37)

Firstly consider the case when 119906(119905) isin 119871infin(0 119879] It isshown [5] that in this case theminimization problem (12) and(13) reduces to the following algebraic equation

21205721

(1205721+ 1205722) Γ (1205722+ 1)[Γ (1205721+ 1205722)

Γ (1205722)]

12057221205721

sdot (11988821205852minus 1

1198881

)

minus12057221205721minus1 11988821205852minus 1 minus 120572

21205721

1198881

12058512057221205721

2

=1198791205722

Γ (1205722+ 1)minus1198882

1198881

1198791205721+1205722

Γ (1205721+ 1205722+ 1)

(39)

There is no explicit solution of (39) that can be foundat arbitrary 120572

1and 120572

2 In some particular cases the explicit

solution exists [5] for example in case of zero-secondmoment 119888

2(119879) = 0 Then the solutions of (12) and (13) lead

us to the following expression

1205822=212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

1198881

1198791205721+1205722 (40)

The solution of OCP A will be written as

119906 (119905) = minus212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

1198881

1198791205721+1205722

sdot sign[(119879 minus 119905)1205721 minus 2

minus120572112057221198791205721

1198881

] 119905 isin (0 119879]

(41)

The OCP B explicit solution can be found at additionalassumption 119902119879

1= 0

119906 (119905) = 119897 sign[2minus12057211205722 (119879

lowast)1205721minus (119879lowastminus 119905)1205721

11990201

]

119905 isin (0 119879lowast]

(42)

The minimal control time 119879lowast can be calculated based on(16) and (40)

119879lowast= (212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897Γ (1205721))

1(1+1205722)

(43)

In case of 119906(119905) isin 1198712(0 119879] calculations by (12) and (13) givean expression

1205822=radic1205741

120572111987912057422

11988512 (44)

where 1205741= 1205721+ 21205722minus 1 120574

2= 21205721+ 21205722minus 1 120574

3= 21205722minus 1

and 119885 = |120574112057421198882

1Γ2(1205721+ 1205722) minus 2120574

2120574311988811198882Γ(1205721+ 1205722)Γ(1205722)1198791205721 +

120574112057431198882

2Γ2(1205722)11987921205721 |

According to (14) the OCP A solution will be written as

119906 (119905)

=120574112057431198882Γ (1205722) (1205742(119879 minus 119905)

1205721 minus 12057411198791205721)

120572211198791205741 (119879 minus 119905)

1minus1205722[12057421198881Γ (1205721+ 1205722)

12057431198882Γ (1205722) 1198791205721

sdot1205741(119879 minus 119905)

1205721 minus 12057431198791205721

1205742(119879 minus 119905)

1205721 minus 12057411198791205721minus 1]

(45)

TheOCPB solution can be derived from (15) and (16) andwill be represented by the following expression

6 Journal of Control Science and Engineering

119906 (119905) =119897212057431198882Γ (1205722) (119879lowast)1205721(1205742(119879lowastminus 119905)1205721minus 1205741(119879lowast)1205721)

119885 (119879lowast minus 119905)1minus1205722

[12057421198881Γ (1205721+ 1205722)

12057431198882Γ (1205722) (119879lowast)

1205721

1205741(119879lowastminus 119905)1205721minus 1205743(119879lowast)1205721

1205742(119879lowast minus 119905)

1205721 minus 1205741(119879lowast)1205721minus 1] (46)

where 119879lowast can be found as least positive real root of theequation

11988512=1205721119897

radic120574111987912057422 (47)

Note also that for Caputo double integrator the case when1198882(119879) = 0 corresponds to the case when 1199020

2= 119902119879

2[5] From

formula (38) one can see that the condition 1198882(119879) = 0 is not

so clear

4 Qualitative Analysis of Results

In this chapter the obtained solutions of OCP are analyzedIt will be shown that at integer (equal to 1) values ofdifferentiation indices these solutions reduce to the knownresults for corresponding integer-order systems For doubleintegrator analytical expressions derived for system phasetrajectories in different modes

41 Optimal Control Behaviour at Integer-Order Differentia-tion Indices For one-dimensional system of first order onecan obtain the solution of moment problem and explicitexpressions for optimal controls For example in case ofsingle integrator at 119906(119905) isin 119871infin(0 119879] the OCP A solution isexpressed by the following formula

1199061(119905) =

119902119879minus 1199020

119879 119905 isin (0 119879] (48)

The OCP B solution gives the following [4]

1199061(119905) = 119897 sign (119902119879 minus 1199020) 119905 isin (0 119879lowast]

119879lowast

1=

10038161003816100381610038161003816119902119879minus 119902010038161003816100381610038161003816

119897

(49)

It is easy to see that solution of OCP A defined by (33)(taking into account (31)) at 120572 = 1 is identical to expression(48) The same is true for OCP B solution expressions (34)and (16) subject to (32) at 120572 = 1 give formulas (49)

For double integrator of first order at arbitrary initial andfinal conditions the solutions of OCP A and OCP B lead toquadratic equation for 120585

2[4] Equation (39) reduces to that

equation at 1205721= 1205722= 1 The solution of OCP A for double

integrator of first order at 11990202= 119902119879

2= 0 for 119906(119905) isin 119871infin(0 119879] is

given by the following formula [4]

1199062(119905) =

4 (119902119879

1minus 1199020

1)

1198792sign(1198792 minus 119905

1199021198791minus 11990201

) (50)

By direct calculation one can obtain that formula (41) (ieOCP A solution for Riemann-Liouville double integrator)

subject to (37) at 11990202= 119902119879

2= 0 and 120572

1= 1205722= 1 reduces to

(50)Consider OCP B in case of 119902119879

1= 0 Then the solution of

the problem for double integrator of first order will be givenby the following formulas

119879lowast

2= 2radic

1199020

1

119897

1199062(119905) = 119897 sign( 119905

11990201

minus1

radic11989711990201

) 119905 isin (0 119879lowast

2]

(51)

Expressions (42) and (43) at 1205721= 1205722= 1 transform

into formulas (51) which can be proved by correspondingsubstitution

Thus as for Caputo systems [5 6 8] all the resultsobtained for single and double integrators of fractional orderreduce to corresponding formulas for integer-order systemswhen differentiation indices are equated to 1

42 Investigation of Qualitative Dynamics for Double Integra-tor For two-dimensional systems the analysis of qualitativedynamics is interesting itself We will calculate below theboundary trajectories for double integrator and its trajecto-ries corresponding to optimal control mode Consider 119906(119905) isin119871infin(0 119879]

Definition 7 The boundary trajectories of some system arethe phase trajectories corresponding to boundary values ofcontrol 119906(119905) = plusmn119897

In case of differentiation indices equal to 1 the boundarytrajectories of some system represent the boundaries ofintegral vortex for differential inclusion corresponding to thissystem [17] This manifold bounds the phase space regionwhich contains all admissible trajectories of the systemIn case of fractional-order systems (Caputo and Riemann-Liouville) this property is also valid caused by comparisontheorem [18 19]

Substituting the boundary values of control 119906(119905) =plusmn119897 to solutions (35) one can obtain the following explicitexpressions for boundary trajectories

119902plusmn119897

1(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)plusmn

1198971199051205721+1205722

Γ (1205721+ 1205722+ 1)

119902plusmn119897

2(119905) =

1199040

21199051205722minus1

Γ (1205722)plusmn

1198971199051205722

Γ (1205722+ 1)

(52)

It is seen that expressions (52) at arbitrary1205721and1205722donot

allow eliminating time and obtaining the explicit dependency

Journal of Control Science and Engineering 7

119902plusmn119897

2(119902plusmn119897

1) as opposed to double integrator of integer order [17

sect35] and Caputo double integrator [5] Nevertheless in someparticular cases the time can be eliminated from expressions(52) For example in case of 1199040

2= 0 one can calculate directly

that these formulas reduce to the following equation

119902plusmn119897

1= (Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

12057211205722

sdot [

[

1199040

1

Γ (1205721)(Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

minus11205722

+Γ (1205722+ 1) 119902

plusmn119897

2

Γ (1205721+ 1205722+ 1)]

]

(53)

Analogously in case of 1205722= 1 it is possible to eliminate

time from (52) and to derive the following equation

119902plusmn119897

1=1

Γ (1205721)(119902plusmn119897

2minus 1199040

2

plusmn119897)

1205721minus1

sdot [

[

1199040

1plusmn119902plusmn119897

2minus 1199040

2

119897

1199040

2

1205721

+ (119902plusmn119897

2minus 1199040

2

119897)

2

1

1205721(1205721+ 1)]

]

(54)

Note that expressions (52) at 1205721= 1205722= 1 reduce

to the quadratic equation for integral vortex boundaries ofdifferential inclusion corresponding to the double integratorof integer order [17 sect35]

In order to calculate the phase trajectories of Riemann-Liouville double integrator in optimal control mode we willsubstitute control (41) into solutions (35) Then the followingmotion laws can be obtained

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)minus212057211205722

212057211205722 minus 1

sdot1198881(119879)

1198791205721+1205722[1199051205721+1205722 minus 2 (119905 minus 119905

1015840)1205721+1205722

Θ(119905 minus 1199051015840)]

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)minus212057211205722

212057211205722 minus 1

1198881(119879)

1198791205721+1205722

sdotΓ (1205721+ 1205722+ 1)

Γ (1205722+ 1)

[1199051205722 minus 2 (119905 minus 119905

1015840)1205722Θ(119905 minus 119905

1015840)]

(55)

where 1199051015840 is the switching point of control (41) and Θ(119905) is theHeavisidersquos function Formulas (55) are obtained for control(41) which is valid in case of 119888

2= 0 Consequently these

expressions allow some simplification

5 Computational Results and Analysis

In this chapter the computational results are representedand analyzed (including comparison with analogous Caputosystems) Calculations and its visualization were performed

0 02 04 06 08120572

100

10minus2

10minus4

u

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

Figure 1 The dependencies for norm of control from 120572 at 1199011015840 = 1(logarithmic scale used for ordinates)

05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 2 The dependencies for norm of control from 120572 at 1199011015840 = 2(logarithmic scale used for ordinates)

in MatLab 79 Mittag-Leffler function values were calculatedusing special procedure [20] Integrals were calculated byGauss-Kronrod method [21]

51 Single Integrator We will consider the task of systemtransfer from any state with 1199040 = 0 in the final state 119902119879 = 0

Figures 1ndash3 show the dependencies of control norm fromdifferentiation index calculated by formula (32) at 119879 = 100and different values 1199040 for 1199011015840 = 1 1199011015840 = 2 and 1199011015840 = 32correspondingly (dash-dot lines) Also the analogous depen-dencies are shown for Caputo single integrator calculated byformulas from [5 8] at different 1199020 for the same values of 1199011015840(solid lines) It is seen that curves for Riemann-Liouville andCaputo single integrator differ from each other qualitativelybut converge to the same point at 120572 = 1 In addition thecontrol norm increases with 1199040 or 1199020 growing for both ofintegrator types

In Figure 4 the dependencies of control norm fromcontrol time 119879 are shown at different values of differentiationindex and 1199040 = 1 (for Riemann-Liouville integrator) and1199020= 1 (for Caputo integrator) The solid lines correspond

to 1199011015840 = 1 the dotted lines correspond to 1199011015840 = 32 and thedash-dotted lines correspond to 1199011015840 = 2 It is clear that all ofthese curves decreasemonotonically which correspond to thecase of Caputo integrator [8] and integrator of integer order[4] And the norm of control increases with growing of 120572 and1199011015840

8 Journal of Control Science and Engineering

04 05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 3 The dependencies for norm of control from 120572 at 1199011015840 = 32(logarithmic scale used for ordinates)

0 20 40 60 80

10minus2

10minus1

10minus3

u

T

120572 = 07120572 = 09

120572 = 05

120572 = 03 120572 = 01

Figure 4 The dependencies for norm of control from 119879 Logarith-mic scale used for ordinates

Consider now the minimal control time 119879lowast dependencyfrom differentiation index 120572 In case of 1199040 = 0 and 119902119879 = 0 (16)subject to formula (32) can be solved explicitly and have onlyone real root

119879lowast= (

10038161003816100381610038161003816119904010038161003816100381610038161003816

119897)

1199011015840

(1199011015840(120572 minus 1) + 1) (56)

In Figure 5 the computational results are shown for theobtained formula used at different 1199011015840 at 119897 = 1 and1199040= 1 (dash-dotted lines) Also the similar dependencies

(calculated by formulas from [8]) are represented for Caputointegrator at 119897 = 1 and 1199020 = 1 (solid lines) As can be seenfrom Figure 5 the dependency at 1199011015840 = 1 is linear for both ofintegrator types In case of 1199011015840 gt 1 the dependency for Caputointegrator becomes nonlinear

52 One-Dimensional System of General Type Consider (asabove) the task of system transfer from any state with 1199040 = 0to the state 119902119879 = 0

Figures 6 and 7 represent the dependencies of controlnorm from differentiation index at 1199011015840 = 1 and 1199011015840 = 2calculated by formulas (24) and (26) correspondingly (dash-doted lines) The analogical dependencies for Caputo one-dimensional system are also shown (solid lines) calculated byformulas from [5 8] The curves were calculated for differentvalues of parameter 119886 and the following values of otherparameters 119879 = 2 119887 = 1 1199040 = 1 (RL-system) and 1199020 = 1

0 02 04 06 080

02

04

06

08

120572

Tlowast

p998400 = 1

p998400 = 32

p998400 = 2

p998400 = 3

Figure 5 The dependencies for minimal control time from 120572

0 02 04 06 080

05

1

120572

a = 1

a = 01

a = 05

u

Figure 6 The dependencies for control norm from 120572 at 1199011015840 = 1 anddifferent values of parameter 119886

06 07 08 090

05

1

120572

a = 1

a = 01a = 05

u

Figure 7 The dependencies for control norm from 120572 at 1199011015840 = 2 anddifferent values of parameter 119886

(C-system) It is seen from Figures 6 and 7 that differencein derivative type has qualitatively effect on behaviour ofdependencies This behaviour changes at 119886 lt 1 and 119886 = 1In case of 1199011015840 = 1 the curves for C-system decrease andcurves for RL-system increase at 119886 lt 1 (analogically to singleintegrator) At 119886 = 1 the curves for both of systems increaseFor1199011015840 = 2 the dependencies increase for both of system types

The dependencies of control norm from control time aresimilar in general for single integrator and are not shownhere

The dependencies of minimal control time from 120572 atdifferent values of parameter 119886 are shown in Figures 8 and 9for 1199011015840 = 1 and 1199011015840 = 2 correspondingly The following valuesof other parameters were chosen 119897 = 2 1199040 = 1 (RL-system)

Journal of Control Science and Engineering 9

0 02 04 06 080

02

04

06

120572

a = 1

a = 05

a = 01

Tlowast

Figure 8The dependencies forminimal control time from120572 at1199011015840 =1 and different values of parameter 119886

06 065 07 075 08 085 09 0950

01

02

03

120572

a = 1

a = 05a = 01

Tlowast

Figure 9The dependencies forminimal control time from120572 at1199011015840 =2 and different values of parameter 119886

and 1199020 = 1 (C-system) Dash-dotted lines correspond to RL-system and solid lines correspond to C-system It is seen thatin case of RL-system the curves have the quasi-linear trendand differ qualitatively from the curves for C-system

53 Double Integrator As in case of one-dimensional sys-tems we will consider the task of system transfer from anystate with nonzero values of phase coordinates into the originat phase plane wewill suppose that 1199040

1= 0 and 119902119879

1= 119902119879

2= 1199040

2=

0The dependencies of control norm from one of differ-

entiation indices for Riemann-Liouville double integrator atseveral fixed values of other indexes are shown in Figures10 and 11 (dash-dotted lines) The curves were calculated byexpression (40) for 1199011015840 = 1 119879 = 100 and 1199040

1= 1 Also the

similar dependencies for Caputo double integrator are shownat these figures calculated for the same values of parameters1199011015840 and 119879 and for 1199020

1= 1 (solid lines) The dependencies for

1199011015840= 2 are similar and not demonstrated here It is seen

from Figure 10 that curves corresponding to different systems(RL- and C-systems) at identical values of 120572

2converge into

the same point at 1205721= 1 In Figure 11 the distance between

different curves for C- and RL-systems decreases when 1205721

increasesThe dependencies of control norm from control time for

double integrator are analogous generally to that in case ofone-dimensional systems and are not demonstrated here

Let us consider the dependency of minimal control timefrom differentiation indices 120572

1and 120572

2 In case of 1199011015840 = 1 this

0 02 04 06 08

100

10minus2

10minus4

u

1205721

1205722 = 09

1205722 = 011205722 = 05

Figure 10 The dependencies for control norm from 1205721at 1199011015840 = 1

and several values of 1205722

0 02 04 06 08

10minus2

10minus4

u

1205722

1205721 = 091205721 = 01 1205721 = 05

Figure 11The dependencies for control norm from 1205722at 1199011015840 = 1 and

several values of 1205721

dependency is given by formula (43) In case of 1199011015840 = 2 withadditional assumption 119902119879

1= 119902119879

2= 1199040

2= 0 the explicit formula

also can be found for such dependency (as solution of (16)subject to (40))

119879lowast= (radic21205721+ 21205722+ 1 (120572

1+ 21205722+ 1) Γ (120572

1+ 1205722)

Γ (1205721+ 1)

sdot

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897)

2(21205722+1)

(57)

In Figures 12ndash15 the dependencies are shown for minimalcontrol time from one of differentiation indices at fixedvalue of other indices The curves are calculated at 119897 = 2for 1199011015840 = 1 (Figures 12 and 13) and 1199011015840 = 2 (Figures 14and 15) Dash-dotted lines correspond to Riemann-Liouvilledouble integrator and are calculated for 1199040

1= 1 Solid lines

correspond toCaputo double integrator and are calculated for1199020

1= 1 As seen from these figures the dependencies for RL-

and C-systems differ qualitatively from each otherLet us now analyze the qualitative dynamics of Riemann-

Liouville double integrator its boundary trajectories andphase trajectories in optimal controlmodeNote that last typeof trajectories in case of Riemann-Liouville double integratorwill not have an obvious initial point at phase plane since theinitial condition (4) defines not the system state but the valueof some integral functional at initial timeThe system state in

10 Journal of Control Science and Engineering

0 02 04 06 08

Tlowast

102

100

10minus2

1205721

1205722 = 01

1205722 = 05 1205722 = 09

Figure 12 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

100

10minus2

1205722

1205721 = 01

1205721 = 05

1205721 = 09

Figure 13 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

105

100

1205721

1205722 = 071205722 = 09

Figure 14 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 2 Logarithmic scale used for

ordinates

this case can be undefined as follows from solutions (35) andexpressions (55)

The boundary trajectories for studied system are repre-sented in Figure 16 for Riemann-Liouville double integrator(dash-dotted lines) and for analogous Caputo system (solidlines) For the last system calculations were performed usingformulas from [5] It is clear that trajectories for both ofsystems are very close to each other In Figure 17 the fragmentof trajectories is shown at enlarged scale

The phase trajectories of Riemann-Liouville double inte-grator in optimal control mode were calculated using expres-sions (55) In Figure 18 the results are shown for different

055 06 065 07 075 08 085 09 0950

1

2

Tlowast

1205722

1205721 = 03 1205721 = 09

1205721 = 07

Figure 15 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 2 Logarithmic scale used for

ordinates

0 200 400

q2

q1

50

0

minus50

minus600 minus400 minus200

Figure 16 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

q2

q1

minus50

minus48

minus46

minus400 minus380 minus360 minus340

Figure 17 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

0 005

0

q1

1205721 = 071205722 = 04

1205721 = 01 1205722 = 021205721 = 01 1205722 = 1

1205721 = 05 1205722 = 1

1205721 = 03 1205722 = 05

q2

minus0005

minus001

minus0015minus005

Figure 18 The phase trajectories in optimal control mode forRiemann-Liouville double integrator

Journal of Control Science and Engineering 11

0

q2

q1

minus1

minus1

minus2

minus2

minus3

minus3

minus4

minus4minus5

minus5minus6

1205721 = 03 1205722 = 05

1205721 = 07 1205722 = 04

1205721 = 05 1205722 = 1

1205721 = 01 1205722 = 02

1205721 = 01 1205722 = 1

Figure 19 The phase trajectories in optimal control mode forCaputo double integrator

0 02 04 06 08 1

0

q1

q2 1205722 = 03 1205722 = 05

1205722 = 08

minus001

minus002

Figure 20 The phase trajectories in optimal control mode forRiemann-Liouville and Caputo double integrator in case of 120572

1= 1

values of differentiation indices 119879 = 100 and 11990401= 1 In

Figure 19 the phase trajectories of Caputo double integrator inoptimal control mode are represented which were calculatedby formulas from [5] at 119879 = 100 and 1199020

1= 1 It is clear

from Figures 18 and 19 that both of systems demonstrate theovercontrolling effect In case of 120572

1= 1 this effect does not

appear and phase trajectories coincide for both of systemsstudied (Figure 20)

6 Summary

In this paper the optimal control problem is investigated forlinear dynamical systems of fractional order described byequations with Riemann-Liouville derivative The problemreduced to themoment problem For the last problem the cor-rectness and solvability conditions derived Some examplesof one- and two-dimensional systems were considered andexplicit solutions of corresponding optimal control problemswere found The properties of these solutions were analyzedand compared with analogous integer-order systems andfractional-order systems described by equationswithCaputoderivative

Some computational results are demonstrated that showthe dependency of control norm from differentiation indicesant control time and the dependency ofminimal control timefrom differentiation indices Qualitative dynamics of doubleintegrator are analyzed and show that boundary trajectoriesof Riemann-Liouville system of this type are very close to

the same trajectories for corresponding Caputo system It isshown also that the overcontrolling effect appears in optimalcontrol mode of system studied

Obtained results can be useful in investigation of optimalcontrol problems for fractional-order systems and in designof particular control systems

Competing Interests

The authors declare that they have no competing interests

References

[1] A G Butkovskii S S Postnov and E A Postnova ldquoFractionalintegro-differential calculus and its control-theoretical applica-tions II Fractional dynamic systems modeling and hardwareimplementationrdquo Automation and Remote Control vol 74 no5 pp 725ndash749 2013

[2] R Kamocki ldquoPontryagin maximum principle for fractionalordinary optimal control problemsrdquo Mathematical Methods inthe Applied Sciences vol 37 no 11 pp 1668ndash1686 2014

[3] R Kamocki and M Majewski ldquoFractional linear control sys-tems with Caputo derivative and their optimizationrdquo OptimalControl Applications and Methods vol 36 no 6 pp 953ndash9672015

[4] A G Butkovskiy Distributed Control Systems American Else-vier New York NY USA 1969

[5] V A Kubyshkin and S S Postnov ldquoOptimal control problemfor a linear stationary fractional order system in the form of aproblem of moments problem setting and a studyrdquoAutomationand Remote Control vol 75 no 5 pp 805ndash817 2014

[6] V A Kubyshkin and S S Postnov ldquoOptimal control prob-lem for linear fractional-order systemsrdquo in Proceedings of theInternational Conference on Fractional Differentiation and ItsApplications (ICFDA rsquo14) 6 pages IEEE Catania Italy June2014 Paper ID 3189701

[7] V A Kubyshkin and S S Postnov ldquoAnalysis of two optimalcontrol problems for a fractional-order pendulum by themethod of momentsrdquo Automation and Remote Control vol 76no 7 pp 1302ndash1314 2015

[8] V A Kubyshkin and S S Postnov ldquoThe optimal controlproblem for linear systems of non-integer order with lumpedand distributed parameters Discontinuityrdquo Nonlinearity andComplexity vol 4 no 4 pp 429ndash443 2015

[9] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 ElsevierAmsterdam The Netherlands 2006

[10] R Almeida and D F M Torres ldquoCalculus of variations withfractional derivatives and fractional integralsrdquo Applied Mathe-matics Letters vol 22 no 12 pp 1816ndash1820 2009

[11] N Heymans and I Podlubny ldquoPhysical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivativesrdquo Rheologica Acta vol 45 no 5pp 765ndash771 2006

[12] V E Tarasov Fractional Dynamics Springer Berlin Germany2010

[13] A N Kolmogorov and S V Fomin Elements of the Theory ofFunctions and Functional Analysis Dover Publications Mine-ola NY USA 1999

[14] N I AkhiezerTheClassical Moment Problem and Some RelatedQuestions in Analysis Hafner New York NY USA 1965

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

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International Journal of

4 Journal of Control Science and Engineering

and solvable for every 1205721 120572

119873minus1and 120572

119873which

satisfy the following condition

120572119873gt1199011015840minus 1

1199011015840 (20)

33 The Moment Problem for One-Dimensional System ofFractional-Order Problem Setting and Study In case of119873 =1 the solution of (1) with initial condition (4) can be writtenas follows [9 sect411] (the subscripts are omitted)

119902 (119905) = 1199040119905120572minus1119864120572120572(119886119905120572)

+ int

119905

0

119906 (120591) + 119891 (120591)

(119905 minus 120591)1minus120572119864120572120572[119886 (119905 minus 120591)

120572] 119889120591

(21)

where 119864120572120572(119905) is two-parameter Mittag-Leffler function [9

sect18] By direct calculation one can obtain that at 119905 = 119879expression (21) with regard to (6) may be written over as (9)with the following symbols

119892 (120591) =119864120572120572[119886 (119879 minus 120591)

120572]

(119879 minus 120591)1minus120572

(22)

119888 (119879) = 119902119879minus 1199040119879120572minus1119864120572120572(119886119879120572)

minus int

119879

0

119864120572120572[119886 (119879 minus 120591)

120572]

(119879 minus 120591)1minus120572

119891 (120591) 119889120591

(23)

Expression (22) is identical to analogous expression forone-dimensional Caputo system (formula (17) at 119887 = 1 in[5]) So expressions that result from (12) and (13) will matchwith analogous expressions obtained for the Caputo system[5 6 8] (the difference will appear only after substitution ofexpression (23) for the moment) Consequently if we keepin mind the noted difference we can use the solutions forProblem 5 obtained for Caputo system As shown in [5 6 8]the analytical solution of Problem 4 for one-dimensionalsystem can be obtained for arbitrary 1199011015840 gt 1 (if condition (19)is satisfied)

Let 119891(119905) = 0 Using (13) we can reduce Problem 5 tosimple integral calculation which can be carried out similarlyto case of Caputo system

Consider the case 119906(119905) isin 119871infin(0 119879] Direct calculationby formula (12) with regard to (13) and (22) leads us to thefollowing expression (which matches with analogous resultfor Caputo system [5 6 8])

120582 =|119888 (119879)| 119886

119864120572(119886119879120572) minus 1

(24)

where 119864120572(119886119879120572) = 119864

1205721(119886119879120572) Using (24) we can obtain

from (14) and (15) the solutions of OCP A and OCP Bcorrespondingly

119906 (119905) =|119888 (119879)| 119886

119864120572(119886119879120572) minus 1

119905 isin (0 119879]

119906 (119905) = 119897 sign (119888 (119879lowast)) 119905 isin (0 119879lowast] (25)

where 119879lowast may be received from (16) with (24) It is seen thatcontrols (25) have not one switching point It is similar tothe system of order [120572] + 1 behaviour in accordance withFeldbaumrsquos theorem about 119899 intervals [16]

In case of 119906(119905) isin 119871119901(0 119879] 1 lt 119901 lt infin we can obtain thefollowing from (12) and (13) subject to (22)

120582 =|119888 (119879)|

119865119879

(26)

where

119865119879= [

[

int

119879

0

(119864120572120572(119886 (119879 minus 119905)

120572)

(119879 minus 119905)1minus120572

)

1199011015840

119889119905]

]

11199011015840

(27)

Taking into account (14) one can obtain further for OCP Athe following

119906 (119905) =119888 (119879)

1198651199011015840

119879

(119864120572120572(119886 (119879 minus 119905)

120572)

(119879 minus 119905)1minus120572

)

1199011015840minus1

(28)

For OCP B the solution can be expressed by the followingformula

119906 (119905) = 1198971199011015840

(119864120572120572(119886 (119879lowastminus 119905)120572

)

(119879lowast minus 119905)1minus120572

)

1199011015840minus1

sign (119888 (119879lowast)) (29)

where 119879lowast can be calculated from (16) using (26)Let us now consider a single integrator the special case

of one-dimensional system at 119886 = 0 Instead of (22) and(23) we will have more simple expressions written in termsof elementary functions

119892 (120591) =1

Γ (120572)

1

(119879 minus 120591)1minus120572 (30)

119888 (119879) = 119902119879minus1199040119879120572minus1

Γ (120572) (31)

Taking into account formula (30) one can obtain thefollowing from (12) and (13)

120582 =|119888 (119879)| Γ (120572) (119901

1015840(120572 minus 1) + 1)

11199011015840

119879120572minus1+11199011015840

(32)

From formula (32) using (14) we can derive the followingexpression for control

119906 (119905) =119888 (119879) (119901

1015840(120572 minus 1) + 1) Γ (120572)

1198791199011015840(120572minus1)+1

(119879 minus 119905)(1199011015840minus1)(120572minus1)

119905 isin (0 119879]

(33)

Analogously in case of OCP B we can derive the followingusing (15)

119906 (119905) = 1198971199011015840

(1

Γ (120572)

1

(119879lowast minus 119905)1minus120572)

1199011015840minus1

sign (119888 (119879lowast))

119905 isin (0 119879lowast]

(34)

where 119879lowast can be calculated from (16) using (32)

Journal of Control Science and Engineering 5

Note that expressions (32) (33) and (34) provide anexplicit solution of optimal control problem for Riemann-Liouville single integrator at arbitrary 1199011015840 ge 1

34 The Moment Problem for Double Integrator of FractionalOrder Problem Setting and Study Two-dimensional system(1) at 119891

1(119905) = 119891

2(119905) = 0 119886

11= 11988621= 11988622= 0 119886

12= 1 119906

1(119905) =

0 and 1199062(119905) = 119906(119905) represents itself as a double integrator

Solution (17) subject to initial conditions (4) for this systemcan be written in the following form

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)

+1

Γ (1205721+ 1205722)int

119905

0

119906 (120591) 119889120591

(119905 minus 120591)1minus1205721minus1205722

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)+1

Γ (1205722)int

119905

0

119906 (120591) 119889120591

(119905 minus 120591)1minus1205722

(35)

It is clear that solutions (35) at 119905 = 119879 can be written inform of expression (9) with the following functions 119892

119894(120591) and

moments

1198921(120591) =

1

Γ (1205721+ 1205722)

1

(119879 minus 120591)1minus1205721minus1205722

1198922(120591) =

1

Γ (1205722)

1

(119879 minus 120591)1minus1205722

(36)

1198881(119879) = 119902

119879

1minus1199040

21198791205721+1205722minus1

Γ (1205721+ 1205722)minus1199040

11198791205721minus1

Γ (1205721) (37)

1198882(119879) = 119902

119879

2minus1199040

21198791205722minus1

Γ (1205722) (38)

The functions 11989212(120591) defined by (36) are identical to

the analogous functions obtained in [5] for Caputo doubleintegrator On the contrary the moments defined by (37)-(38) differ in general from the moments obtained in [5]for Caputo double integrator Consequently as for one-dimensional system the form of solution for moment prob-lem will match with that for Caputo double integrator So wecan use the solutions of OCP A and OCP B obtained in [5]taking into account the moments (37)

Firstly consider the case when 119906(119905) isin 119871infin(0 119879] It isshown [5] that in this case theminimization problem (12) and(13) reduces to the following algebraic equation

21205721

(1205721+ 1205722) Γ (1205722+ 1)[Γ (1205721+ 1205722)

Γ (1205722)]

12057221205721

sdot (11988821205852minus 1

1198881

)

minus12057221205721minus1 11988821205852minus 1 minus 120572

21205721

1198881

12058512057221205721

2

=1198791205722

Γ (1205722+ 1)minus1198882

1198881

1198791205721+1205722

Γ (1205721+ 1205722+ 1)

(39)

There is no explicit solution of (39) that can be foundat arbitrary 120572

1and 120572

2 In some particular cases the explicit

solution exists [5] for example in case of zero-secondmoment 119888

2(119879) = 0 Then the solutions of (12) and (13) lead

us to the following expression

1205822=212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

1198881

1198791205721+1205722 (40)

The solution of OCP A will be written as

119906 (119905) = minus212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

1198881

1198791205721+1205722

sdot sign[(119879 minus 119905)1205721 minus 2

minus120572112057221198791205721

1198881

] 119905 isin (0 119879]

(41)

The OCP B explicit solution can be found at additionalassumption 119902119879

1= 0

119906 (119905) = 119897 sign[2minus12057211205722 (119879

lowast)1205721minus (119879lowastminus 119905)1205721

11990201

]

119905 isin (0 119879lowast]

(42)

The minimal control time 119879lowast can be calculated based on(16) and (40)

119879lowast= (212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897Γ (1205721))

1(1+1205722)

(43)

In case of 119906(119905) isin 1198712(0 119879] calculations by (12) and (13) givean expression

1205822=radic1205741

120572111987912057422

11988512 (44)

where 1205741= 1205721+ 21205722minus 1 120574

2= 21205721+ 21205722minus 1 120574

3= 21205722minus 1

and 119885 = |120574112057421198882

1Γ2(1205721+ 1205722) minus 2120574

2120574311988811198882Γ(1205721+ 1205722)Γ(1205722)1198791205721 +

120574112057431198882

2Γ2(1205722)11987921205721 |

According to (14) the OCP A solution will be written as

119906 (119905)

=120574112057431198882Γ (1205722) (1205742(119879 minus 119905)

1205721 minus 12057411198791205721)

120572211198791205741 (119879 minus 119905)

1minus1205722[12057421198881Γ (1205721+ 1205722)

12057431198882Γ (1205722) 1198791205721

sdot1205741(119879 minus 119905)

1205721 minus 12057431198791205721

1205742(119879 minus 119905)

1205721 minus 12057411198791205721minus 1]

(45)

TheOCPB solution can be derived from (15) and (16) andwill be represented by the following expression

6 Journal of Control Science and Engineering

119906 (119905) =119897212057431198882Γ (1205722) (119879lowast)1205721(1205742(119879lowastminus 119905)1205721minus 1205741(119879lowast)1205721)

119885 (119879lowast minus 119905)1minus1205722

[12057421198881Γ (1205721+ 1205722)

12057431198882Γ (1205722) (119879lowast)

1205721

1205741(119879lowastminus 119905)1205721minus 1205743(119879lowast)1205721

1205742(119879lowast minus 119905)

1205721 minus 1205741(119879lowast)1205721minus 1] (46)

where 119879lowast can be found as least positive real root of theequation

11988512=1205721119897

radic120574111987912057422 (47)

Note also that for Caputo double integrator the case when1198882(119879) = 0 corresponds to the case when 1199020

2= 119902119879

2[5] From

formula (38) one can see that the condition 1198882(119879) = 0 is not

so clear

4 Qualitative Analysis of Results

In this chapter the obtained solutions of OCP are analyzedIt will be shown that at integer (equal to 1) values ofdifferentiation indices these solutions reduce to the knownresults for corresponding integer-order systems For doubleintegrator analytical expressions derived for system phasetrajectories in different modes

41 Optimal Control Behaviour at Integer-Order Differentia-tion Indices For one-dimensional system of first order onecan obtain the solution of moment problem and explicitexpressions for optimal controls For example in case ofsingle integrator at 119906(119905) isin 119871infin(0 119879] the OCP A solution isexpressed by the following formula

1199061(119905) =

119902119879minus 1199020

119879 119905 isin (0 119879] (48)

The OCP B solution gives the following [4]

1199061(119905) = 119897 sign (119902119879 minus 1199020) 119905 isin (0 119879lowast]

119879lowast

1=

10038161003816100381610038161003816119902119879minus 119902010038161003816100381610038161003816

119897

(49)

It is easy to see that solution of OCP A defined by (33)(taking into account (31)) at 120572 = 1 is identical to expression(48) The same is true for OCP B solution expressions (34)and (16) subject to (32) at 120572 = 1 give formulas (49)

For double integrator of first order at arbitrary initial andfinal conditions the solutions of OCP A and OCP B lead toquadratic equation for 120585

2[4] Equation (39) reduces to that

equation at 1205721= 1205722= 1 The solution of OCP A for double

integrator of first order at 11990202= 119902119879

2= 0 for 119906(119905) isin 119871infin(0 119879] is

given by the following formula [4]

1199062(119905) =

4 (119902119879

1minus 1199020

1)

1198792sign(1198792 minus 119905

1199021198791minus 11990201

) (50)

By direct calculation one can obtain that formula (41) (ieOCP A solution for Riemann-Liouville double integrator)

subject to (37) at 11990202= 119902119879

2= 0 and 120572

1= 1205722= 1 reduces to

(50)Consider OCP B in case of 119902119879

1= 0 Then the solution of

the problem for double integrator of first order will be givenby the following formulas

119879lowast

2= 2radic

1199020

1

119897

1199062(119905) = 119897 sign( 119905

11990201

minus1

radic11989711990201

) 119905 isin (0 119879lowast

2]

(51)

Expressions (42) and (43) at 1205721= 1205722= 1 transform

into formulas (51) which can be proved by correspondingsubstitution

Thus as for Caputo systems [5 6 8] all the resultsobtained for single and double integrators of fractional orderreduce to corresponding formulas for integer-order systemswhen differentiation indices are equated to 1

42 Investigation of Qualitative Dynamics for Double Integra-tor For two-dimensional systems the analysis of qualitativedynamics is interesting itself We will calculate below theboundary trajectories for double integrator and its trajecto-ries corresponding to optimal control mode Consider 119906(119905) isin119871infin(0 119879]

Definition 7 The boundary trajectories of some system arethe phase trajectories corresponding to boundary values ofcontrol 119906(119905) = plusmn119897

In case of differentiation indices equal to 1 the boundarytrajectories of some system represent the boundaries ofintegral vortex for differential inclusion corresponding to thissystem [17] This manifold bounds the phase space regionwhich contains all admissible trajectories of the systemIn case of fractional-order systems (Caputo and Riemann-Liouville) this property is also valid caused by comparisontheorem [18 19]

Substituting the boundary values of control 119906(119905) =plusmn119897 to solutions (35) one can obtain the following explicitexpressions for boundary trajectories

119902plusmn119897

1(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)plusmn

1198971199051205721+1205722

Γ (1205721+ 1205722+ 1)

119902plusmn119897

2(119905) =

1199040

21199051205722minus1

Γ (1205722)plusmn

1198971199051205722

Γ (1205722+ 1)

(52)

It is seen that expressions (52) at arbitrary1205721and1205722donot

allow eliminating time and obtaining the explicit dependency

Journal of Control Science and Engineering 7

119902plusmn119897

2(119902plusmn119897

1) as opposed to double integrator of integer order [17

sect35] and Caputo double integrator [5] Nevertheless in someparticular cases the time can be eliminated from expressions(52) For example in case of 1199040

2= 0 one can calculate directly

that these formulas reduce to the following equation

119902plusmn119897

1= (Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

12057211205722

sdot [

[

1199040

1

Γ (1205721)(Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

minus11205722

+Γ (1205722+ 1) 119902

plusmn119897

2

Γ (1205721+ 1205722+ 1)]

]

(53)

Analogously in case of 1205722= 1 it is possible to eliminate

time from (52) and to derive the following equation

119902plusmn119897

1=1

Γ (1205721)(119902plusmn119897

2minus 1199040

2

plusmn119897)

1205721minus1

sdot [

[

1199040

1plusmn119902plusmn119897

2minus 1199040

2

119897

1199040

2

1205721

+ (119902plusmn119897

2minus 1199040

2

119897)

2

1

1205721(1205721+ 1)]

]

(54)

Note that expressions (52) at 1205721= 1205722= 1 reduce

to the quadratic equation for integral vortex boundaries ofdifferential inclusion corresponding to the double integratorof integer order [17 sect35]

In order to calculate the phase trajectories of Riemann-Liouville double integrator in optimal control mode we willsubstitute control (41) into solutions (35) Then the followingmotion laws can be obtained

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)minus212057211205722

212057211205722 minus 1

sdot1198881(119879)

1198791205721+1205722[1199051205721+1205722 minus 2 (119905 minus 119905

1015840)1205721+1205722

Θ(119905 minus 1199051015840)]

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)minus212057211205722

212057211205722 minus 1

1198881(119879)

1198791205721+1205722

sdotΓ (1205721+ 1205722+ 1)

Γ (1205722+ 1)

[1199051205722 minus 2 (119905 minus 119905

1015840)1205722Θ(119905 minus 119905

1015840)]

(55)

where 1199051015840 is the switching point of control (41) and Θ(119905) is theHeavisidersquos function Formulas (55) are obtained for control(41) which is valid in case of 119888

2= 0 Consequently these

expressions allow some simplification

5 Computational Results and Analysis

In this chapter the computational results are representedand analyzed (including comparison with analogous Caputosystems) Calculations and its visualization were performed

0 02 04 06 08120572

100

10minus2

10minus4

u

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

Figure 1 The dependencies for norm of control from 120572 at 1199011015840 = 1(logarithmic scale used for ordinates)

05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 2 The dependencies for norm of control from 120572 at 1199011015840 = 2(logarithmic scale used for ordinates)

in MatLab 79 Mittag-Leffler function values were calculatedusing special procedure [20] Integrals were calculated byGauss-Kronrod method [21]

51 Single Integrator We will consider the task of systemtransfer from any state with 1199040 = 0 in the final state 119902119879 = 0

Figures 1ndash3 show the dependencies of control norm fromdifferentiation index calculated by formula (32) at 119879 = 100and different values 1199040 for 1199011015840 = 1 1199011015840 = 2 and 1199011015840 = 32correspondingly (dash-dot lines) Also the analogous depen-dencies are shown for Caputo single integrator calculated byformulas from [5 8] at different 1199020 for the same values of 1199011015840(solid lines) It is seen that curves for Riemann-Liouville andCaputo single integrator differ from each other qualitativelybut converge to the same point at 120572 = 1 In addition thecontrol norm increases with 1199040 or 1199020 growing for both ofintegrator types

In Figure 4 the dependencies of control norm fromcontrol time 119879 are shown at different values of differentiationindex and 1199040 = 1 (for Riemann-Liouville integrator) and1199020= 1 (for Caputo integrator) The solid lines correspond

to 1199011015840 = 1 the dotted lines correspond to 1199011015840 = 32 and thedash-dotted lines correspond to 1199011015840 = 2 It is clear that all ofthese curves decreasemonotonically which correspond to thecase of Caputo integrator [8] and integrator of integer order[4] And the norm of control increases with growing of 120572 and1199011015840

8 Journal of Control Science and Engineering

04 05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 3 The dependencies for norm of control from 120572 at 1199011015840 = 32(logarithmic scale used for ordinates)

0 20 40 60 80

10minus2

10minus1

10minus3

u

T

120572 = 07120572 = 09

120572 = 05

120572 = 03 120572 = 01

Figure 4 The dependencies for norm of control from 119879 Logarith-mic scale used for ordinates

Consider now the minimal control time 119879lowast dependencyfrom differentiation index 120572 In case of 1199040 = 0 and 119902119879 = 0 (16)subject to formula (32) can be solved explicitly and have onlyone real root

119879lowast= (

10038161003816100381610038161003816119904010038161003816100381610038161003816

119897)

1199011015840

(1199011015840(120572 minus 1) + 1) (56)

In Figure 5 the computational results are shown for theobtained formula used at different 1199011015840 at 119897 = 1 and1199040= 1 (dash-dotted lines) Also the similar dependencies

(calculated by formulas from [8]) are represented for Caputointegrator at 119897 = 1 and 1199020 = 1 (solid lines) As can be seenfrom Figure 5 the dependency at 1199011015840 = 1 is linear for both ofintegrator types In case of 1199011015840 gt 1 the dependency for Caputointegrator becomes nonlinear

52 One-Dimensional System of General Type Consider (asabove) the task of system transfer from any state with 1199040 = 0to the state 119902119879 = 0

Figures 6 and 7 represent the dependencies of controlnorm from differentiation index at 1199011015840 = 1 and 1199011015840 = 2calculated by formulas (24) and (26) correspondingly (dash-doted lines) The analogical dependencies for Caputo one-dimensional system are also shown (solid lines) calculated byformulas from [5 8] The curves were calculated for differentvalues of parameter 119886 and the following values of otherparameters 119879 = 2 119887 = 1 1199040 = 1 (RL-system) and 1199020 = 1

0 02 04 06 080

02

04

06

08

120572

Tlowast

p998400 = 1

p998400 = 32

p998400 = 2

p998400 = 3

Figure 5 The dependencies for minimal control time from 120572

0 02 04 06 080

05

1

120572

a = 1

a = 01

a = 05

u

Figure 6 The dependencies for control norm from 120572 at 1199011015840 = 1 anddifferent values of parameter 119886

06 07 08 090

05

1

120572

a = 1

a = 01a = 05

u

Figure 7 The dependencies for control norm from 120572 at 1199011015840 = 2 anddifferent values of parameter 119886

(C-system) It is seen from Figures 6 and 7 that differencein derivative type has qualitatively effect on behaviour ofdependencies This behaviour changes at 119886 lt 1 and 119886 = 1In case of 1199011015840 = 1 the curves for C-system decrease andcurves for RL-system increase at 119886 lt 1 (analogically to singleintegrator) At 119886 = 1 the curves for both of systems increaseFor1199011015840 = 2 the dependencies increase for both of system types

The dependencies of control norm from control time aresimilar in general for single integrator and are not shownhere

The dependencies of minimal control time from 120572 atdifferent values of parameter 119886 are shown in Figures 8 and 9for 1199011015840 = 1 and 1199011015840 = 2 correspondingly The following valuesof other parameters were chosen 119897 = 2 1199040 = 1 (RL-system)

Journal of Control Science and Engineering 9

0 02 04 06 080

02

04

06

120572

a = 1

a = 05

a = 01

Tlowast

Figure 8The dependencies forminimal control time from120572 at1199011015840 =1 and different values of parameter 119886

06 065 07 075 08 085 09 0950

01

02

03

120572

a = 1

a = 05a = 01

Tlowast

Figure 9The dependencies forminimal control time from120572 at1199011015840 =2 and different values of parameter 119886

and 1199020 = 1 (C-system) Dash-dotted lines correspond to RL-system and solid lines correspond to C-system It is seen thatin case of RL-system the curves have the quasi-linear trendand differ qualitatively from the curves for C-system

53 Double Integrator As in case of one-dimensional sys-tems we will consider the task of system transfer from anystate with nonzero values of phase coordinates into the originat phase plane wewill suppose that 1199040

1= 0 and 119902119879

1= 119902119879

2= 1199040

2=

0The dependencies of control norm from one of differ-

entiation indices for Riemann-Liouville double integrator atseveral fixed values of other indexes are shown in Figures10 and 11 (dash-dotted lines) The curves were calculated byexpression (40) for 1199011015840 = 1 119879 = 100 and 1199040

1= 1 Also the

similar dependencies for Caputo double integrator are shownat these figures calculated for the same values of parameters1199011015840 and 119879 and for 1199020

1= 1 (solid lines) The dependencies for

1199011015840= 2 are similar and not demonstrated here It is seen

from Figure 10 that curves corresponding to different systems(RL- and C-systems) at identical values of 120572

2converge into

the same point at 1205721= 1 In Figure 11 the distance between

different curves for C- and RL-systems decreases when 1205721

increasesThe dependencies of control norm from control time for

double integrator are analogous generally to that in case ofone-dimensional systems and are not demonstrated here

Let us consider the dependency of minimal control timefrom differentiation indices 120572

1and 120572

2 In case of 1199011015840 = 1 this

0 02 04 06 08

100

10minus2

10minus4

u

1205721

1205722 = 09

1205722 = 011205722 = 05

Figure 10 The dependencies for control norm from 1205721at 1199011015840 = 1

and several values of 1205722

0 02 04 06 08

10minus2

10minus4

u

1205722

1205721 = 091205721 = 01 1205721 = 05

Figure 11The dependencies for control norm from 1205722at 1199011015840 = 1 and

several values of 1205721

dependency is given by formula (43) In case of 1199011015840 = 2 withadditional assumption 119902119879

1= 119902119879

2= 1199040

2= 0 the explicit formula

also can be found for such dependency (as solution of (16)subject to (40))

119879lowast= (radic21205721+ 21205722+ 1 (120572

1+ 21205722+ 1) Γ (120572

1+ 1205722)

Γ (1205721+ 1)

sdot

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897)

2(21205722+1)

(57)

In Figures 12ndash15 the dependencies are shown for minimalcontrol time from one of differentiation indices at fixedvalue of other indices The curves are calculated at 119897 = 2for 1199011015840 = 1 (Figures 12 and 13) and 1199011015840 = 2 (Figures 14and 15) Dash-dotted lines correspond to Riemann-Liouvilledouble integrator and are calculated for 1199040

1= 1 Solid lines

correspond toCaputo double integrator and are calculated for1199020

1= 1 As seen from these figures the dependencies for RL-

and C-systems differ qualitatively from each otherLet us now analyze the qualitative dynamics of Riemann-

Liouville double integrator its boundary trajectories andphase trajectories in optimal controlmodeNote that last typeof trajectories in case of Riemann-Liouville double integratorwill not have an obvious initial point at phase plane since theinitial condition (4) defines not the system state but the valueof some integral functional at initial timeThe system state in

10 Journal of Control Science and Engineering

0 02 04 06 08

Tlowast

102

100

10minus2

1205721

1205722 = 01

1205722 = 05 1205722 = 09

Figure 12 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

100

10minus2

1205722

1205721 = 01

1205721 = 05

1205721 = 09

Figure 13 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

105

100

1205721

1205722 = 071205722 = 09

Figure 14 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 2 Logarithmic scale used for

ordinates

this case can be undefined as follows from solutions (35) andexpressions (55)

The boundary trajectories for studied system are repre-sented in Figure 16 for Riemann-Liouville double integrator(dash-dotted lines) and for analogous Caputo system (solidlines) For the last system calculations were performed usingformulas from [5] It is clear that trajectories for both ofsystems are very close to each other In Figure 17 the fragmentof trajectories is shown at enlarged scale

The phase trajectories of Riemann-Liouville double inte-grator in optimal control mode were calculated using expres-sions (55) In Figure 18 the results are shown for different

055 06 065 07 075 08 085 09 0950

1

2

Tlowast

1205722

1205721 = 03 1205721 = 09

1205721 = 07

Figure 15 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 2 Logarithmic scale used for

ordinates

0 200 400

q2

q1

50

0

minus50

minus600 minus400 minus200

Figure 16 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

q2

q1

minus50

minus48

minus46

minus400 minus380 minus360 minus340

Figure 17 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

0 005

0

q1

1205721 = 071205722 = 04

1205721 = 01 1205722 = 021205721 = 01 1205722 = 1

1205721 = 05 1205722 = 1

1205721 = 03 1205722 = 05

q2

minus0005

minus001

minus0015minus005

Figure 18 The phase trajectories in optimal control mode forRiemann-Liouville double integrator

Journal of Control Science and Engineering 11

0

q2

q1

minus1

minus1

minus2

minus2

minus3

minus3

minus4

minus4minus5

minus5minus6

1205721 = 03 1205722 = 05

1205721 = 07 1205722 = 04

1205721 = 05 1205722 = 1

1205721 = 01 1205722 = 02

1205721 = 01 1205722 = 1

Figure 19 The phase trajectories in optimal control mode forCaputo double integrator

0 02 04 06 08 1

0

q1

q2 1205722 = 03 1205722 = 05

1205722 = 08

minus001

minus002

Figure 20 The phase trajectories in optimal control mode forRiemann-Liouville and Caputo double integrator in case of 120572

1= 1

values of differentiation indices 119879 = 100 and 11990401= 1 In

Figure 19 the phase trajectories of Caputo double integrator inoptimal control mode are represented which were calculatedby formulas from [5] at 119879 = 100 and 1199020

1= 1 It is clear

from Figures 18 and 19 that both of systems demonstrate theovercontrolling effect In case of 120572

1= 1 this effect does not

appear and phase trajectories coincide for both of systemsstudied (Figure 20)

6 Summary

In this paper the optimal control problem is investigated forlinear dynamical systems of fractional order described byequations with Riemann-Liouville derivative The problemreduced to themoment problem For the last problem the cor-rectness and solvability conditions derived Some examplesof one- and two-dimensional systems were considered andexplicit solutions of corresponding optimal control problemswere found The properties of these solutions were analyzedand compared with analogous integer-order systems andfractional-order systems described by equationswithCaputoderivative

Some computational results are demonstrated that showthe dependency of control norm from differentiation indicesant control time and the dependency ofminimal control timefrom differentiation indices Qualitative dynamics of doubleintegrator are analyzed and show that boundary trajectoriesof Riemann-Liouville system of this type are very close to

the same trajectories for corresponding Caputo system It isshown also that the overcontrolling effect appears in optimalcontrol mode of system studied

Obtained results can be useful in investigation of optimalcontrol problems for fractional-order systems and in designof particular control systems

Competing Interests

The authors declare that they have no competing interests

References

[1] A G Butkovskii S S Postnov and E A Postnova ldquoFractionalintegro-differential calculus and its control-theoretical applica-tions II Fractional dynamic systems modeling and hardwareimplementationrdquo Automation and Remote Control vol 74 no5 pp 725ndash749 2013

[2] R Kamocki ldquoPontryagin maximum principle for fractionalordinary optimal control problemsrdquo Mathematical Methods inthe Applied Sciences vol 37 no 11 pp 1668ndash1686 2014

[3] R Kamocki and M Majewski ldquoFractional linear control sys-tems with Caputo derivative and their optimizationrdquo OptimalControl Applications and Methods vol 36 no 6 pp 953ndash9672015

[4] A G Butkovskiy Distributed Control Systems American Else-vier New York NY USA 1969

[5] V A Kubyshkin and S S Postnov ldquoOptimal control problemfor a linear stationary fractional order system in the form of aproblem of moments problem setting and a studyrdquoAutomationand Remote Control vol 75 no 5 pp 805ndash817 2014

[6] V A Kubyshkin and S S Postnov ldquoOptimal control prob-lem for linear fractional-order systemsrdquo in Proceedings of theInternational Conference on Fractional Differentiation and ItsApplications (ICFDA rsquo14) 6 pages IEEE Catania Italy June2014 Paper ID 3189701

[7] V A Kubyshkin and S S Postnov ldquoAnalysis of two optimalcontrol problems for a fractional-order pendulum by themethod of momentsrdquo Automation and Remote Control vol 76no 7 pp 1302ndash1314 2015

[8] V A Kubyshkin and S S Postnov ldquoThe optimal controlproblem for linear systems of non-integer order with lumpedand distributed parameters Discontinuityrdquo Nonlinearity andComplexity vol 4 no 4 pp 429ndash443 2015

[9] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 ElsevierAmsterdam The Netherlands 2006

[10] R Almeida and D F M Torres ldquoCalculus of variations withfractional derivatives and fractional integralsrdquo Applied Mathe-matics Letters vol 22 no 12 pp 1816ndash1820 2009

[11] N Heymans and I Podlubny ldquoPhysical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivativesrdquo Rheologica Acta vol 45 no 5pp 765ndash771 2006

[12] V E Tarasov Fractional Dynamics Springer Berlin Germany2010

[13] A N Kolmogorov and S V Fomin Elements of the Theory ofFunctions and Functional Analysis Dover Publications Mine-ola NY USA 1999

[14] N I AkhiezerTheClassical Moment Problem and Some RelatedQuestions in Analysis Hafner New York NY USA 1965

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

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Journal of Control Science and Engineering 5

Note that expressions (32) (33) and (34) provide anexplicit solution of optimal control problem for Riemann-Liouville single integrator at arbitrary 1199011015840 ge 1

34 The Moment Problem for Double Integrator of FractionalOrder Problem Setting and Study Two-dimensional system(1) at 119891

1(119905) = 119891

2(119905) = 0 119886

11= 11988621= 11988622= 0 119886

12= 1 119906

1(119905) =

0 and 1199062(119905) = 119906(119905) represents itself as a double integrator

Solution (17) subject to initial conditions (4) for this systemcan be written in the following form

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)

+1

Γ (1205721+ 1205722)int

119905

0

119906 (120591) 119889120591

(119905 minus 120591)1minus1205721minus1205722

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)+1

Γ (1205722)int

119905

0

119906 (120591) 119889120591

(119905 minus 120591)1minus1205722

(35)

It is clear that solutions (35) at 119905 = 119879 can be written inform of expression (9) with the following functions 119892

119894(120591) and

moments

1198921(120591) =

1

Γ (1205721+ 1205722)

1

(119879 minus 120591)1minus1205721minus1205722

1198922(120591) =

1

Γ (1205722)

1

(119879 minus 120591)1minus1205722

(36)

1198881(119879) = 119902

119879

1minus1199040

21198791205721+1205722minus1

Γ (1205721+ 1205722)minus1199040

11198791205721minus1

Γ (1205721) (37)

1198882(119879) = 119902

119879

2minus1199040

21198791205722minus1

Γ (1205722) (38)

The functions 11989212(120591) defined by (36) are identical to

the analogous functions obtained in [5] for Caputo doubleintegrator On the contrary the moments defined by (37)-(38) differ in general from the moments obtained in [5]for Caputo double integrator Consequently as for one-dimensional system the form of solution for moment prob-lem will match with that for Caputo double integrator So wecan use the solutions of OCP A and OCP B obtained in [5]taking into account the moments (37)

Firstly consider the case when 119906(119905) isin 119871infin(0 119879] It isshown [5] that in this case theminimization problem (12) and(13) reduces to the following algebraic equation

21205721

(1205721+ 1205722) Γ (1205722+ 1)[Γ (1205721+ 1205722)

Γ (1205722)]

12057221205721

sdot (11988821205852minus 1

1198881

)

minus12057221205721minus1 11988821205852minus 1 minus 120572

21205721

1198881

12058512057221205721

2

=1198791205722

Γ (1205722+ 1)minus1198882

1198881

1198791205721+1205722

Γ (1205721+ 1205722+ 1)

(39)

There is no explicit solution of (39) that can be foundat arbitrary 120572

1and 120572

2 In some particular cases the explicit

solution exists [5] for example in case of zero-secondmoment 119888

2(119879) = 0 Then the solutions of (12) and (13) lead

us to the following expression

1205822=212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

1198881

1198791205721+1205722 (40)

The solution of OCP A will be written as

119906 (119905) = minus212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

1198881

1198791205721+1205722

sdot sign[(119879 minus 119905)1205721 minus 2

minus120572112057221198791205721

1198881

] 119905 isin (0 119879]

(41)

The OCP B explicit solution can be found at additionalassumption 119902119879

1= 0

119906 (119905) = 119897 sign[2minus12057211205722 (119879

lowast)1205721minus (119879lowastminus 119905)1205721

11990201

]

119905 isin (0 119879lowast]

(42)

The minimal control time 119879lowast can be calculated based on(16) and (40)

119879lowast= (212057211205722Γ (120572

1+ 1205722+ 1)

212057211205722 minus 1

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897Γ (1205721))

1(1+1205722)

(43)

In case of 119906(119905) isin 1198712(0 119879] calculations by (12) and (13) givean expression

1205822=radic1205741

120572111987912057422

11988512 (44)

where 1205741= 1205721+ 21205722minus 1 120574

2= 21205721+ 21205722minus 1 120574

3= 21205722minus 1

and 119885 = |120574112057421198882

1Γ2(1205721+ 1205722) minus 2120574

2120574311988811198882Γ(1205721+ 1205722)Γ(1205722)1198791205721 +

120574112057431198882

2Γ2(1205722)11987921205721 |

According to (14) the OCP A solution will be written as

119906 (119905)

=120574112057431198882Γ (1205722) (1205742(119879 minus 119905)

1205721 minus 12057411198791205721)

120572211198791205741 (119879 minus 119905)

1minus1205722[12057421198881Γ (1205721+ 1205722)

12057431198882Γ (1205722) 1198791205721

sdot1205741(119879 minus 119905)

1205721 minus 12057431198791205721

1205742(119879 minus 119905)

1205721 minus 12057411198791205721minus 1]

(45)

TheOCPB solution can be derived from (15) and (16) andwill be represented by the following expression

6 Journal of Control Science and Engineering

119906 (119905) =119897212057431198882Γ (1205722) (119879lowast)1205721(1205742(119879lowastminus 119905)1205721minus 1205741(119879lowast)1205721)

119885 (119879lowast minus 119905)1minus1205722

[12057421198881Γ (1205721+ 1205722)

12057431198882Γ (1205722) (119879lowast)

1205721

1205741(119879lowastminus 119905)1205721minus 1205743(119879lowast)1205721

1205742(119879lowast minus 119905)

1205721 minus 1205741(119879lowast)1205721minus 1] (46)

where 119879lowast can be found as least positive real root of theequation

11988512=1205721119897

radic120574111987912057422 (47)

Note also that for Caputo double integrator the case when1198882(119879) = 0 corresponds to the case when 1199020

2= 119902119879

2[5] From

formula (38) one can see that the condition 1198882(119879) = 0 is not

so clear

4 Qualitative Analysis of Results

In this chapter the obtained solutions of OCP are analyzedIt will be shown that at integer (equal to 1) values ofdifferentiation indices these solutions reduce to the knownresults for corresponding integer-order systems For doubleintegrator analytical expressions derived for system phasetrajectories in different modes

41 Optimal Control Behaviour at Integer-Order Differentia-tion Indices For one-dimensional system of first order onecan obtain the solution of moment problem and explicitexpressions for optimal controls For example in case ofsingle integrator at 119906(119905) isin 119871infin(0 119879] the OCP A solution isexpressed by the following formula

1199061(119905) =

119902119879minus 1199020

119879 119905 isin (0 119879] (48)

The OCP B solution gives the following [4]

1199061(119905) = 119897 sign (119902119879 minus 1199020) 119905 isin (0 119879lowast]

119879lowast

1=

10038161003816100381610038161003816119902119879minus 119902010038161003816100381610038161003816

119897

(49)

It is easy to see that solution of OCP A defined by (33)(taking into account (31)) at 120572 = 1 is identical to expression(48) The same is true for OCP B solution expressions (34)and (16) subject to (32) at 120572 = 1 give formulas (49)

For double integrator of first order at arbitrary initial andfinal conditions the solutions of OCP A and OCP B lead toquadratic equation for 120585

2[4] Equation (39) reduces to that

equation at 1205721= 1205722= 1 The solution of OCP A for double

integrator of first order at 11990202= 119902119879

2= 0 for 119906(119905) isin 119871infin(0 119879] is

given by the following formula [4]

1199062(119905) =

4 (119902119879

1minus 1199020

1)

1198792sign(1198792 minus 119905

1199021198791minus 11990201

) (50)

By direct calculation one can obtain that formula (41) (ieOCP A solution for Riemann-Liouville double integrator)

subject to (37) at 11990202= 119902119879

2= 0 and 120572

1= 1205722= 1 reduces to

(50)Consider OCP B in case of 119902119879

1= 0 Then the solution of

the problem for double integrator of first order will be givenby the following formulas

119879lowast

2= 2radic

1199020

1

119897

1199062(119905) = 119897 sign( 119905

11990201

minus1

radic11989711990201

) 119905 isin (0 119879lowast

2]

(51)

Expressions (42) and (43) at 1205721= 1205722= 1 transform

into formulas (51) which can be proved by correspondingsubstitution

Thus as for Caputo systems [5 6 8] all the resultsobtained for single and double integrators of fractional orderreduce to corresponding formulas for integer-order systemswhen differentiation indices are equated to 1

42 Investigation of Qualitative Dynamics for Double Integra-tor For two-dimensional systems the analysis of qualitativedynamics is interesting itself We will calculate below theboundary trajectories for double integrator and its trajecto-ries corresponding to optimal control mode Consider 119906(119905) isin119871infin(0 119879]

Definition 7 The boundary trajectories of some system arethe phase trajectories corresponding to boundary values ofcontrol 119906(119905) = plusmn119897

In case of differentiation indices equal to 1 the boundarytrajectories of some system represent the boundaries ofintegral vortex for differential inclusion corresponding to thissystem [17] This manifold bounds the phase space regionwhich contains all admissible trajectories of the systemIn case of fractional-order systems (Caputo and Riemann-Liouville) this property is also valid caused by comparisontheorem [18 19]

Substituting the boundary values of control 119906(119905) =plusmn119897 to solutions (35) one can obtain the following explicitexpressions for boundary trajectories

119902plusmn119897

1(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)plusmn

1198971199051205721+1205722

Γ (1205721+ 1205722+ 1)

119902plusmn119897

2(119905) =

1199040

21199051205722minus1

Γ (1205722)plusmn

1198971199051205722

Γ (1205722+ 1)

(52)

It is seen that expressions (52) at arbitrary1205721and1205722donot

allow eliminating time and obtaining the explicit dependency

Journal of Control Science and Engineering 7

119902plusmn119897

2(119902plusmn119897

1) as opposed to double integrator of integer order [17

sect35] and Caputo double integrator [5] Nevertheless in someparticular cases the time can be eliminated from expressions(52) For example in case of 1199040

2= 0 one can calculate directly

that these formulas reduce to the following equation

119902plusmn119897

1= (Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

12057211205722

sdot [

[

1199040

1

Γ (1205721)(Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

minus11205722

+Γ (1205722+ 1) 119902

plusmn119897

2

Γ (1205721+ 1205722+ 1)]

]

(53)

Analogously in case of 1205722= 1 it is possible to eliminate

time from (52) and to derive the following equation

119902plusmn119897

1=1

Γ (1205721)(119902plusmn119897

2minus 1199040

2

plusmn119897)

1205721minus1

sdot [

[

1199040

1plusmn119902plusmn119897

2minus 1199040

2

119897

1199040

2

1205721

+ (119902plusmn119897

2minus 1199040

2

119897)

2

1

1205721(1205721+ 1)]

]

(54)

Note that expressions (52) at 1205721= 1205722= 1 reduce

to the quadratic equation for integral vortex boundaries ofdifferential inclusion corresponding to the double integratorof integer order [17 sect35]

In order to calculate the phase trajectories of Riemann-Liouville double integrator in optimal control mode we willsubstitute control (41) into solutions (35) Then the followingmotion laws can be obtained

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)minus212057211205722

212057211205722 minus 1

sdot1198881(119879)

1198791205721+1205722[1199051205721+1205722 minus 2 (119905 minus 119905

1015840)1205721+1205722

Θ(119905 minus 1199051015840)]

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)minus212057211205722

212057211205722 minus 1

1198881(119879)

1198791205721+1205722

sdotΓ (1205721+ 1205722+ 1)

Γ (1205722+ 1)

[1199051205722 minus 2 (119905 minus 119905

1015840)1205722Θ(119905 minus 119905

1015840)]

(55)

where 1199051015840 is the switching point of control (41) and Θ(119905) is theHeavisidersquos function Formulas (55) are obtained for control(41) which is valid in case of 119888

2= 0 Consequently these

expressions allow some simplification

5 Computational Results and Analysis

In this chapter the computational results are representedand analyzed (including comparison with analogous Caputosystems) Calculations and its visualization were performed

0 02 04 06 08120572

100

10minus2

10minus4

u

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

Figure 1 The dependencies for norm of control from 120572 at 1199011015840 = 1(logarithmic scale used for ordinates)

05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 2 The dependencies for norm of control from 120572 at 1199011015840 = 2(logarithmic scale used for ordinates)

in MatLab 79 Mittag-Leffler function values were calculatedusing special procedure [20] Integrals were calculated byGauss-Kronrod method [21]

51 Single Integrator We will consider the task of systemtransfer from any state with 1199040 = 0 in the final state 119902119879 = 0

Figures 1ndash3 show the dependencies of control norm fromdifferentiation index calculated by formula (32) at 119879 = 100and different values 1199040 for 1199011015840 = 1 1199011015840 = 2 and 1199011015840 = 32correspondingly (dash-dot lines) Also the analogous depen-dencies are shown for Caputo single integrator calculated byformulas from [5 8] at different 1199020 for the same values of 1199011015840(solid lines) It is seen that curves for Riemann-Liouville andCaputo single integrator differ from each other qualitativelybut converge to the same point at 120572 = 1 In addition thecontrol norm increases with 1199040 or 1199020 growing for both ofintegrator types

In Figure 4 the dependencies of control norm fromcontrol time 119879 are shown at different values of differentiationindex and 1199040 = 1 (for Riemann-Liouville integrator) and1199020= 1 (for Caputo integrator) The solid lines correspond

to 1199011015840 = 1 the dotted lines correspond to 1199011015840 = 32 and thedash-dotted lines correspond to 1199011015840 = 2 It is clear that all ofthese curves decreasemonotonically which correspond to thecase of Caputo integrator [8] and integrator of integer order[4] And the norm of control increases with growing of 120572 and1199011015840

8 Journal of Control Science and Engineering

04 05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 3 The dependencies for norm of control from 120572 at 1199011015840 = 32(logarithmic scale used for ordinates)

0 20 40 60 80

10minus2

10minus1

10minus3

u

T

120572 = 07120572 = 09

120572 = 05

120572 = 03 120572 = 01

Figure 4 The dependencies for norm of control from 119879 Logarith-mic scale used for ordinates

Consider now the minimal control time 119879lowast dependencyfrom differentiation index 120572 In case of 1199040 = 0 and 119902119879 = 0 (16)subject to formula (32) can be solved explicitly and have onlyone real root

119879lowast= (

10038161003816100381610038161003816119904010038161003816100381610038161003816

119897)

1199011015840

(1199011015840(120572 minus 1) + 1) (56)

In Figure 5 the computational results are shown for theobtained formula used at different 1199011015840 at 119897 = 1 and1199040= 1 (dash-dotted lines) Also the similar dependencies

(calculated by formulas from [8]) are represented for Caputointegrator at 119897 = 1 and 1199020 = 1 (solid lines) As can be seenfrom Figure 5 the dependency at 1199011015840 = 1 is linear for both ofintegrator types In case of 1199011015840 gt 1 the dependency for Caputointegrator becomes nonlinear

52 One-Dimensional System of General Type Consider (asabove) the task of system transfer from any state with 1199040 = 0to the state 119902119879 = 0

Figures 6 and 7 represent the dependencies of controlnorm from differentiation index at 1199011015840 = 1 and 1199011015840 = 2calculated by formulas (24) and (26) correspondingly (dash-doted lines) The analogical dependencies for Caputo one-dimensional system are also shown (solid lines) calculated byformulas from [5 8] The curves were calculated for differentvalues of parameter 119886 and the following values of otherparameters 119879 = 2 119887 = 1 1199040 = 1 (RL-system) and 1199020 = 1

0 02 04 06 080

02

04

06

08

120572

Tlowast

p998400 = 1

p998400 = 32

p998400 = 2

p998400 = 3

Figure 5 The dependencies for minimal control time from 120572

0 02 04 06 080

05

1

120572

a = 1

a = 01

a = 05

u

Figure 6 The dependencies for control norm from 120572 at 1199011015840 = 1 anddifferent values of parameter 119886

06 07 08 090

05

1

120572

a = 1

a = 01a = 05

u

Figure 7 The dependencies for control norm from 120572 at 1199011015840 = 2 anddifferent values of parameter 119886

(C-system) It is seen from Figures 6 and 7 that differencein derivative type has qualitatively effect on behaviour ofdependencies This behaviour changes at 119886 lt 1 and 119886 = 1In case of 1199011015840 = 1 the curves for C-system decrease andcurves for RL-system increase at 119886 lt 1 (analogically to singleintegrator) At 119886 = 1 the curves for both of systems increaseFor1199011015840 = 2 the dependencies increase for both of system types

The dependencies of control norm from control time aresimilar in general for single integrator and are not shownhere

The dependencies of minimal control time from 120572 atdifferent values of parameter 119886 are shown in Figures 8 and 9for 1199011015840 = 1 and 1199011015840 = 2 correspondingly The following valuesof other parameters were chosen 119897 = 2 1199040 = 1 (RL-system)

Journal of Control Science and Engineering 9

0 02 04 06 080

02

04

06

120572

a = 1

a = 05

a = 01

Tlowast

Figure 8The dependencies forminimal control time from120572 at1199011015840 =1 and different values of parameter 119886

06 065 07 075 08 085 09 0950

01

02

03

120572

a = 1

a = 05a = 01

Tlowast

Figure 9The dependencies forminimal control time from120572 at1199011015840 =2 and different values of parameter 119886

and 1199020 = 1 (C-system) Dash-dotted lines correspond to RL-system and solid lines correspond to C-system It is seen thatin case of RL-system the curves have the quasi-linear trendand differ qualitatively from the curves for C-system

53 Double Integrator As in case of one-dimensional sys-tems we will consider the task of system transfer from anystate with nonzero values of phase coordinates into the originat phase plane wewill suppose that 1199040

1= 0 and 119902119879

1= 119902119879

2= 1199040

2=

0The dependencies of control norm from one of differ-

entiation indices for Riemann-Liouville double integrator atseveral fixed values of other indexes are shown in Figures10 and 11 (dash-dotted lines) The curves were calculated byexpression (40) for 1199011015840 = 1 119879 = 100 and 1199040

1= 1 Also the

similar dependencies for Caputo double integrator are shownat these figures calculated for the same values of parameters1199011015840 and 119879 and for 1199020

1= 1 (solid lines) The dependencies for

1199011015840= 2 are similar and not demonstrated here It is seen

from Figure 10 that curves corresponding to different systems(RL- and C-systems) at identical values of 120572

2converge into

the same point at 1205721= 1 In Figure 11 the distance between

different curves for C- and RL-systems decreases when 1205721

increasesThe dependencies of control norm from control time for

double integrator are analogous generally to that in case ofone-dimensional systems and are not demonstrated here

Let us consider the dependency of minimal control timefrom differentiation indices 120572

1and 120572

2 In case of 1199011015840 = 1 this

0 02 04 06 08

100

10minus2

10minus4

u

1205721

1205722 = 09

1205722 = 011205722 = 05

Figure 10 The dependencies for control norm from 1205721at 1199011015840 = 1

and several values of 1205722

0 02 04 06 08

10minus2

10minus4

u

1205722

1205721 = 091205721 = 01 1205721 = 05

Figure 11The dependencies for control norm from 1205722at 1199011015840 = 1 and

several values of 1205721

dependency is given by formula (43) In case of 1199011015840 = 2 withadditional assumption 119902119879

1= 119902119879

2= 1199040

2= 0 the explicit formula

also can be found for such dependency (as solution of (16)subject to (40))

119879lowast= (radic21205721+ 21205722+ 1 (120572

1+ 21205722+ 1) Γ (120572

1+ 1205722)

Γ (1205721+ 1)

sdot

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897)

2(21205722+1)

(57)

In Figures 12ndash15 the dependencies are shown for minimalcontrol time from one of differentiation indices at fixedvalue of other indices The curves are calculated at 119897 = 2for 1199011015840 = 1 (Figures 12 and 13) and 1199011015840 = 2 (Figures 14and 15) Dash-dotted lines correspond to Riemann-Liouvilledouble integrator and are calculated for 1199040

1= 1 Solid lines

correspond toCaputo double integrator and are calculated for1199020

1= 1 As seen from these figures the dependencies for RL-

and C-systems differ qualitatively from each otherLet us now analyze the qualitative dynamics of Riemann-

Liouville double integrator its boundary trajectories andphase trajectories in optimal controlmodeNote that last typeof trajectories in case of Riemann-Liouville double integratorwill not have an obvious initial point at phase plane since theinitial condition (4) defines not the system state but the valueof some integral functional at initial timeThe system state in

10 Journal of Control Science and Engineering

0 02 04 06 08

Tlowast

102

100

10minus2

1205721

1205722 = 01

1205722 = 05 1205722 = 09

Figure 12 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

100

10minus2

1205722

1205721 = 01

1205721 = 05

1205721 = 09

Figure 13 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

105

100

1205721

1205722 = 071205722 = 09

Figure 14 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 2 Logarithmic scale used for

ordinates

this case can be undefined as follows from solutions (35) andexpressions (55)

The boundary trajectories for studied system are repre-sented in Figure 16 for Riemann-Liouville double integrator(dash-dotted lines) and for analogous Caputo system (solidlines) For the last system calculations were performed usingformulas from [5] It is clear that trajectories for both ofsystems are very close to each other In Figure 17 the fragmentof trajectories is shown at enlarged scale

The phase trajectories of Riemann-Liouville double inte-grator in optimal control mode were calculated using expres-sions (55) In Figure 18 the results are shown for different

055 06 065 07 075 08 085 09 0950

1

2

Tlowast

1205722

1205721 = 03 1205721 = 09

1205721 = 07

Figure 15 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 2 Logarithmic scale used for

ordinates

0 200 400

q2

q1

50

0

minus50

minus600 minus400 minus200

Figure 16 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

q2

q1

minus50

minus48

minus46

minus400 minus380 minus360 minus340

Figure 17 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

0 005

0

q1

1205721 = 071205722 = 04

1205721 = 01 1205722 = 021205721 = 01 1205722 = 1

1205721 = 05 1205722 = 1

1205721 = 03 1205722 = 05

q2

minus0005

minus001

minus0015minus005

Figure 18 The phase trajectories in optimal control mode forRiemann-Liouville double integrator

Journal of Control Science and Engineering 11

0

q2

q1

minus1

minus1

minus2

minus2

minus3

minus3

minus4

minus4minus5

minus5minus6

1205721 = 03 1205722 = 05

1205721 = 07 1205722 = 04

1205721 = 05 1205722 = 1

1205721 = 01 1205722 = 02

1205721 = 01 1205722 = 1

Figure 19 The phase trajectories in optimal control mode forCaputo double integrator

0 02 04 06 08 1

0

q1

q2 1205722 = 03 1205722 = 05

1205722 = 08

minus001

minus002

Figure 20 The phase trajectories in optimal control mode forRiemann-Liouville and Caputo double integrator in case of 120572

1= 1

values of differentiation indices 119879 = 100 and 11990401= 1 In

Figure 19 the phase trajectories of Caputo double integrator inoptimal control mode are represented which were calculatedby formulas from [5] at 119879 = 100 and 1199020

1= 1 It is clear

from Figures 18 and 19 that both of systems demonstrate theovercontrolling effect In case of 120572

1= 1 this effect does not

appear and phase trajectories coincide for both of systemsstudied (Figure 20)

6 Summary

In this paper the optimal control problem is investigated forlinear dynamical systems of fractional order described byequations with Riemann-Liouville derivative The problemreduced to themoment problem For the last problem the cor-rectness and solvability conditions derived Some examplesof one- and two-dimensional systems were considered andexplicit solutions of corresponding optimal control problemswere found The properties of these solutions were analyzedand compared with analogous integer-order systems andfractional-order systems described by equationswithCaputoderivative

Some computational results are demonstrated that showthe dependency of control norm from differentiation indicesant control time and the dependency ofminimal control timefrom differentiation indices Qualitative dynamics of doubleintegrator are analyzed and show that boundary trajectoriesof Riemann-Liouville system of this type are very close to

the same trajectories for corresponding Caputo system It isshown also that the overcontrolling effect appears in optimalcontrol mode of system studied

Obtained results can be useful in investigation of optimalcontrol problems for fractional-order systems and in designof particular control systems

Competing Interests

The authors declare that they have no competing interests

References

[1] A G Butkovskii S S Postnov and E A Postnova ldquoFractionalintegro-differential calculus and its control-theoretical applica-tions II Fractional dynamic systems modeling and hardwareimplementationrdquo Automation and Remote Control vol 74 no5 pp 725ndash749 2013

[2] R Kamocki ldquoPontryagin maximum principle for fractionalordinary optimal control problemsrdquo Mathematical Methods inthe Applied Sciences vol 37 no 11 pp 1668ndash1686 2014

[3] R Kamocki and M Majewski ldquoFractional linear control sys-tems with Caputo derivative and their optimizationrdquo OptimalControl Applications and Methods vol 36 no 6 pp 953ndash9672015

[4] A G Butkovskiy Distributed Control Systems American Else-vier New York NY USA 1969

[5] V A Kubyshkin and S S Postnov ldquoOptimal control problemfor a linear stationary fractional order system in the form of aproblem of moments problem setting and a studyrdquoAutomationand Remote Control vol 75 no 5 pp 805ndash817 2014

[6] V A Kubyshkin and S S Postnov ldquoOptimal control prob-lem for linear fractional-order systemsrdquo in Proceedings of theInternational Conference on Fractional Differentiation and ItsApplications (ICFDA rsquo14) 6 pages IEEE Catania Italy June2014 Paper ID 3189701

[7] V A Kubyshkin and S S Postnov ldquoAnalysis of two optimalcontrol problems for a fractional-order pendulum by themethod of momentsrdquo Automation and Remote Control vol 76no 7 pp 1302ndash1314 2015

[8] V A Kubyshkin and S S Postnov ldquoThe optimal controlproblem for linear systems of non-integer order with lumpedand distributed parameters Discontinuityrdquo Nonlinearity andComplexity vol 4 no 4 pp 429ndash443 2015

[9] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 ElsevierAmsterdam The Netherlands 2006

[10] R Almeida and D F M Torres ldquoCalculus of variations withfractional derivatives and fractional integralsrdquo Applied Mathe-matics Letters vol 22 no 12 pp 1816ndash1820 2009

[11] N Heymans and I Podlubny ldquoPhysical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivativesrdquo Rheologica Acta vol 45 no 5pp 765ndash771 2006

[12] V E Tarasov Fractional Dynamics Springer Berlin Germany2010

[13] A N Kolmogorov and S V Fomin Elements of the Theory ofFunctions and Functional Analysis Dover Publications Mine-ola NY USA 1999

[14] N I AkhiezerTheClassical Moment Problem and Some RelatedQuestions in Analysis Hafner New York NY USA 1965

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

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DistributedSensor Networks

International Journal of

6 Journal of Control Science and Engineering

119906 (119905) =119897212057431198882Γ (1205722) (119879lowast)1205721(1205742(119879lowastminus 119905)1205721minus 1205741(119879lowast)1205721)

119885 (119879lowast minus 119905)1minus1205722

[12057421198881Γ (1205721+ 1205722)

12057431198882Γ (1205722) (119879lowast)

1205721

1205741(119879lowastminus 119905)1205721minus 1205743(119879lowast)1205721

1205742(119879lowast minus 119905)

1205721 minus 1205741(119879lowast)1205721minus 1] (46)

where 119879lowast can be found as least positive real root of theequation

11988512=1205721119897

radic120574111987912057422 (47)

Note also that for Caputo double integrator the case when1198882(119879) = 0 corresponds to the case when 1199020

2= 119902119879

2[5] From

formula (38) one can see that the condition 1198882(119879) = 0 is not

so clear

4 Qualitative Analysis of Results

In this chapter the obtained solutions of OCP are analyzedIt will be shown that at integer (equal to 1) values ofdifferentiation indices these solutions reduce to the knownresults for corresponding integer-order systems For doubleintegrator analytical expressions derived for system phasetrajectories in different modes

41 Optimal Control Behaviour at Integer-Order Differentia-tion Indices For one-dimensional system of first order onecan obtain the solution of moment problem and explicitexpressions for optimal controls For example in case ofsingle integrator at 119906(119905) isin 119871infin(0 119879] the OCP A solution isexpressed by the following formula

1199061(119905) =

119902119879minus 1199020

119879 119905 isin (0 119879] (48)

The OCP B solution gives the following [4]

1199061(119905) = 119897 sign (119902119879 minus 1199020) 119905 isin (0 119879lowast]

119879lowast

1=

10038161003816100381610038161003816119902119879minus 119902010038161003816100381610038161003816

119897

(49)

It is easy to see that solution of OCP A defined by (33)(taking into account (31)) at 120572 = 1 is identical to expression(48) The same is true for OCP B solution expressions (34)and (16) subject to (32) at 120572 = 1 give formulas (49)

For double integrator of first order at arbitrary initial andfinal conditions the solutions of OCP A and OCP B lead toquadratic equation for 120585

2[4] Equation (39) reduces to that

equation at 1205721= 1205722= 1 The solution of OCP A for double

integrator of first order at 11990202= 119902119879

2= 0 for 119906(119905) isin 119871infin(0 119879] is

given by the following formula [4]

1199062(119905) =

4 (119902119879

1minus 1199020

1)

1198792sign(1198792 minus 119905

1199021198791minus 11990201

) (50)

By direct calculation one can obtain that formula (41) (ieOCP A solution for Riemann-Liouville double integrator)

subject to (37) at 11990202= 119902119879

2= 0 and 120572

1= 1205722= 1 reduces to

(50)Consider OCP B in case of 119902119879

1= 0 Then the solution of

the problem for double integrator of first order will be givenby the following formulas

119879lowast

2= 2radic

1199020

1

119897

1199062(119905) = 119897 sign( 119905

11990201

minus1

radic11989711990201

) 119905 isin (0 119879lowast

2]

(51)

Expressions (42) and (43) at 1205721= 1205722= 1 transform

into formulas (51) which can be proved by correspondingsubstitution

Thus as for Caputo systems [5 6 8] all the resultsobtained for single and double integrators of fractional orderreduce to corresponding formulas for integer-order systemswhen differentiation indices are equated to 1

42 Investigation of Qualitative Dynamics for Double Integra-tor For two-dimensional systems the analysis of qualitativedynamics is interesting itself We will calculate below theboundary trajectories for double integrator and its trajecto-ries corresponding to optimal control mode Consider 119906(119905) isin119871infin(0 119879]

Definition 7 The boundary trajectories of some system arethe phase trajectories corresponding to boundary values ofcontrol 119906(119905) = plusmn119897

In case of differentiation indices equal to 1 the boundarytrajectories of some system represent the boundaries ofintegral vortex for differential inclusion corresponding to thissystem [17] This manifold bounds the phase space regionwhich contains all admissible trajectories of the systemIn case of fractional-order systems (Caputo and Riemann-Liouville) this property is also valid caused by comparisontheorem [18 19]

Substituting the boundary values of control 119906(119905) =plusmn119897 to solutions (35) one can obtain the following explicitexpressions for boundary trajectories

119902plusmn119897

1(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)plusmn

1198971199051205721+1205722

Γ (1205721+ 1205722+ 1)

119902plusmn119897

2(119905) =

1199040

21199051205722minus1

Γ (1205722)plusmn

1198971199051205722

Γ (1205722+ 1)

(52)

It is seen that expressions (52) at arbitrary1205721and1205722donot

allow eliminating time and obtaining the explicit dependency

Journal of Control Science and Engineering 7

119902plusmn119897

2(119902plusmn119897

1) as opposed to double integrator of integer order [17

sect35] and Caputo double integrator [5] Nevertheless in someparticular cases the time can be eliminated from expressions(52) For example in case of 1199040

2= 0 one can calculate directly

that these formulas reduce to the following equation

119902plusmn119897

1= (Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

12057211205722

sdot [

[

1199040

1

Γ (1205721)(Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

minus11205722

+Γ (1205722+ 1) 119902

plusmn119897

2

Γ (1205721+ 1205722+ 1)]

]

(53)

Analogously in case of 1205722= 1 it is possible to eliminate

time from (52) and to derive the following equation

119902plusmn119897

1=1

Γ (1205721)(119902plusmn119897

2minus 1199040

2

plusmn119897)

1205721minus1

sdot [

[

1199040

1plusmn119902plusmn119897

2minus 1199040

2

119897

1199040

2

1205721

+ (119902plusmn119897

2minus 1199040

2

119897)

2

1

1205721(1205721+ 1)]

]

(54)

Note that expressions (52) at 1205721= 1205722= 1 reduce

to the quadratic equation for integral vortex boundaries ofdifferential inclusion corresponding to the double integratorof integer order [17 sect35]

In order to calculate the phase trajectories of Riemann-Liouville double integrator in optimal control mode we willsubstitute control (41) into solutions (35) Then the followingmotion laws can be obtained

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)minus212057211205722

212057211205722 minus 1

sdot1198881(119879)

1198791205721+1205722[1199051205721+1205722 minus 2 (119905 minus 119905

1015840)1205721+1205722

Θ(119905 minus 1199051015840)]

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)minus212057211205722

212057211205722 minus 1

1198881(119879)

1198791205721+1205722

sdotΓ (1205721+ 1205722+ 1)

Γ (1205722+ 1)

[1199051205722 minus 2 (119905 minus 119905

1015840)1205722Θ(119905 minus 119905

1015840)]

(55)

where 1199051015840 is the switching point of control (41) and Θ(119905) is theHeavisidersquos function Formulas (55) are obtained for control(41) which is valid in case of 119888

2= 0 Consequently these

expressions allow some simplification

5 Computational Results and Analysis

In this chapter the computational results are representedand analyzed (including comparison with analogous Caputosystems) Calculations and its visualization were performed

0 02 04 06 08120572

100

10minus2

10minus4

u

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

Figure 1 The dependencies for norm of control from 120572 at 1199011015840 = 1(logarithmic scale used for ordinates)

05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 2 The dependencies for norm of control from 120572 at 1199011015840 = 2(logarithmic scale used for ordinates)

in MatLab 79 Mittag-Leffler function values were calculatedusing special procedure [20] Integrals were calculated byGauss-Kronrod method [21]

51 Single Integrator We will consider the task of systemtransfer from any state with 1199040 = 0 in the final state 119902119879 = 0

Figures 1ndash3 show the dependencies of control norm fromdifferentiation index calculated by formula (32) at 119879 = 100and different values 1199040 for 1199011015840 = 1 1199011015840 = 2 and 1199011015840 = 32correspondingly (dash-dot lines) Also the analogous depen-dencies are shown for Caputo single integrator calculated byformulas from [5 8] at different 1199020 for the same values of 1199011015840(solid lines) It is seen that curves for Riemann-Liouville andCaputo single integrator differ from each other qualitativelybut converge to the same point at 120572 = 1 In addition thecontrol norm increases with 1199040 or 1199020 growing for both ofintegrator types

In Figure 4 the dependencies of control norm fromcontrol time 119879 are shown at different values of differentiationindex and 1199040 = 1 (for Riemann-Liouville integrator) and1199020= 1 (for Caputo integrator) The solid lines correspond

to 1199011015840 = 1 the dotted lines correspond to 1199011015840 = 32 and thedash-dotted lines correspond to 1199011015840 = 2 It is clear that all ofthese curves decreasemonotonically which correspond to thecase of Caputo integrator [8] and integrator of integer order[4] And the norm of control increases with growing of 120572 and1199011015840

8 Journal of Control Science and Engineering

04 05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 3 The dependencies for norm of control from 120572 at 1199011015840 = 32(logarithmic scale used for ordinates)

0 20 40 60 80

10minus2

10minus1

10minus3

u

T

120572 = 07120572 = 09

120572 = 05

120572 = 03 120572 = 01

Figure 4 The dependencies for norm of control from 119879 Logarith-mic scale used for ordinates

Consider now the minimal control time 119879lowast dependencyfrom differentiation index 120572 In case of 1199040 = 0 and 119902119879 = 0 (16)subject to formula (32) can be solved explicitly and have onlyone real root

119879lowast= (

10038161003816100381610038161003816119904010038161003816100381610038161003816

119897)

1199011015840

(1199011015840(120572 minus 1) + 1) (56)

In Figure 5 the computational results are shown for theobtained formula used at different 1199011015840 at 119897 = 1 and1199040= 1 (dash-dotted lines) Also the similar dependencies

(calculated by formulas from [8]) are represented for Caputointegrator at 119897 = 1 and 1199020 = 1 (solid lines) As can be seenfrom Figure 5 the dependency at 1199011015840 = 1 is linear for both ofintegrator types In case of 1199011015840 gt 1 the dependency for Caputointegrator becomes nonlinear

52 One-Dimensional System of General Type Consider (asabove) the task of system transfer from any state with 1199040 = 0to the state 119902119879 = 0

Figures 6 and 7 represent the dependencies of controlnorm from differentiation index at 1199011015840 = 1 and 1199011015840 = 2calculated by formulas (24) and (26) correspondingly (dash-doted lines) The analogical dependencies for Caputo one-dimensional system are also shown (solid lines) calculated byformulas from [5 8] The curves were calculated for differentvalues of parameter 119886 and the following values of otherparameters 119879 = 2 119887 = 1 1199040 = 1 (RL-system) and 1199020 = 1

0 02 04 06 080

02

04

06

08

120572

Tlowast

p998400 = 1

p998400 = 32

p998400 = 2

p998400 = 3

Figure 5 The dependencies for minimal control time from 120572

0 02 04 06 080

05

1

120572

a = 1

a = 01

a = 05

u

Figure 6 The dependencies for control norm from 120572 at 1199011015840 = 1 anddifferent values of parameter 119886

06 07 08 090

05

1

120572

a = 1

a = 01a = 05

u

Figure 7 The dependencies for control norm from 120572 at 1199011015840 = 2 anddifferent values of parameter 119886

(C-system) It is seen from Figures 6 and 7 that differencein derivative type has qualitatively effect on behaviour ofdependencies This behaviour changes at 119886 lt 1 and 119886 = 1In case of 1199011015840 = 1 the curves for C-system decrease andcurves for RL-system increase at 119886 lt 1 (analogically to singleintegrator) At 119886 = 1 the curves for both of systems increaseFor1199011015840 = 2 the dependencies increase for both of system types

The dependencies of control norm from control time aresimilar in general for single integrator and are not shownhere

The dependencies of minimal control time from 120572 atdifferent values of parameter 119886 are shown in Figures 8 and 9for 1199011015840 = 1 and 1199011015840 = 2 correspondingly The following valuesof other parameters were chosen 119897 = 2 1199040 = 1 (RL-system)

Journal of Control Science and Engineering 9

0 02 04 06 080

02

04

06

120572

a = 1

a = 05

a = 01

Tlowast

Figure 8The dependencies forminimal control time from120572 at1199011015840 =1 and different values of parameter 119886

06 065 07 075 08 085 09 0950

01

02

03

120572

a = 1

a = 05a = 01

Tlowast

Figure 9The dependencies forminimal control time from120572 at1199011015840 =2 and different values of parameter 119886

and 1199020 = 1 (C-system) Dash-dotted lines correspond to RL-system and solid lines correspond to C-system It is seen thatin case of RL-system the curves have the quasi-linear trendand differ qualitatively from the curves for C-system

53 Double Integrator As in case of one-dimensional sys-tems we will consider the task of system transfer from anystate with nonzero values of phase coordinates into the originat phase plane wewill suppose that 1199040

1= 0 and 119902119879

1= 119902119879

2= 1199040

2=

0The dependencies of control norm from one of differ-

entiation indices for Riemann-Liouville double integrator atseveral fixed values of other indexes are shown in Figures10 and 11 (dash-dotted lines) The curves were calculated byexpression (40) for 1199011015840 = 1 119879 = 100 and 1199040

1= 1 Also the

similar dependencies for Caputo double integrator are shownat these figures calculated for the same values of parameters1199011015840 and 119879 and for 1199020

1= 1 (solid lines) The dependencies for

1199011015840= 2 are similar and not demonstrated here It is seen

from Figure 10 that curves corresponding to different systems(RL- and C-systems) at identical values of 120572

2converge into

the same point at 1205721= 1 In Figure 11 the distance between

different curves for C- and RL-systems decreases when 1205721

increasesThe dependencies of control norm from control time for

double integrator are analogous generally to that in case ofone-dimensional systems and are not demonstrated here

Let us consider the dependency of minimal control timefrom differentiation indices 120572

1and 120572

2 In case of 1199011015840 = 1 this

0 02 04 06 08

100

10minus2

10minus4

u

1205721

1205722 = 09

1205722 = 011205722 = 05

Figure 10 The dependencies for control norm from 1205721at 1199011015840 = 1

and several values of 1205722

0 02 04 06 08

10minus2

10minus4

u

1205722

1205721 = 091205721 = 01 1205721 = 05

Figure 11The dependencies for control norm from 1205722at 1199011015840 = 1 and

several values of 1205721

dependency is given by formula (43) In case of 1199011015840 = 2 withadditional assumption 119902119879

1= 119902119879

2= 1199040

2= 0 the explicit formula

also can be found for such dependency (as solution of (16)subject to (40))

119879lowast= (radic21205721+ 21205722+ 1 (120572

1+ 21205722+ 1) Γ (120572

1+ 1205722)

Γ (1205721+ 1)

sdot

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897)

2(21205722+1)

(57)

In Figures 12ndash15 the dependencies are shown for minimalcontrol time from one of differentiation indices at fixedvalue of other indices The curves are calculated at 119897 = 2for 1199011015840 = 1 (Figures 12 and 13) and 1199011015840 = 2 (Figures 14and 15) Dash-dotted lines correspond to Riemann-Liouvilledouble integrator and are calculated for 1199040

1= 1 Solid lines

correspond toCaputo double integrator and are calculated for1199020

1= 1 As seen from these figures the dependencies for RL-

and C-systems differ qualitatively from each otherLet us now analyze the qualitative dynamics of Riemann-

Liouville double integrator its boundary trajectories andphase trajectories in optimal controlmodeNote that last typeof trajectories in case of Riemann-Liouville double integratorwill not have an obvious initial point at phase plane since theinitial condition (4) defines not the system state but the valueof some integral functional at initial timeThe system state in

10 Journal of Control Science and Engineering

0 02 04 06 08

Tlowast

102

100

10minus2

1205721

1205722 = 01

1205722 = 05 1205722 = 09

Figure 12 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

100

10minus2

1205722

1205721 = 01

1205721 = 05

1205721 = 09

Figure 13 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

105

100

1205721

1205722 = 071205722 = 09

Figure 14 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 2 Logarithmic scale used for

ordinates

this case can be undefined as follows from solutions (35) andexpressions (55)

The boundary trajectories for studied system are repre-sented in Figure 16 for Riemann-Liouville double integrator(dash-dotted lines) and for analogous Caputo system (solidlines) For the last system calculations were performed usingformulas from [5] It is clear that trajectories for both ofsystems are very close to each other In Figure 17 the fragmentof trajectories is shown at enlarged scale

The phase trajectories of Riemann-Liouville double inte-grator in optimal control mode were calculated using expres-sions (55) In Figure 18 the results are shown for different

055 06 065 07 075 08 085 09 0950

1

2

Tlowast

1205722

1205721 = 03 1205721 = 09

1205721 = 07

Figure 15 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 2 Logarithmic scale used for

ordinates

0 200 400

q2

q1

50

0

minus50

minus600 minus400 minus200

Figure 16 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

q2

q1

minus50

minus48

minus46

minus400 minus380 minus360 minus340

Figure 17 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

0 005

0

q1

1205721 = 071205722 = 04

1205721 = 01 1205722 = 021205721 = 01 1205722 = 1

1205721 = 05 1205722 = 1

1205721 = 03 1205722 = 05

q2

minus0005

minus001

minus0015minus005

Figure 18 The phase trajectories in optimal control mode forRiemann-Liouville double integrator

Journal of Control Science and Engineering 11

0

q2

q1

minus1

minus1

minus2

minus2

minus3

minus3

minus4

minus4minus5

minus5minus6

1205721 = 03 1205722 = 05

1205721 = 07 1205722 = 04

1205721 = 05 1205722 = 1

1205721 = 01 1205722 = 02

1205721 = 01 1205722 = 1

Figure 19 The phase trajectories in optimal control mode forCaputo double integrator

0 02 04 06 08 1

0

q1

q2 1205722 = 03 1205722 = 05

1205722 = 08

minus001

minus002

Figure 20 The phase trajectories in optimal control mode forRiemann-Liouville and Caputo double integrator in case of 120572

1= 1

values of differentiation indices 119879 = 100 and 11990401= 1 In

Figure 19 the phase trajectories of Caputo double integrator inoptimal control mode are represented which were calculatedby formulas from [5] at 119879 = 100 and 1199020

1= 1 It is clear

from Figures 18 and 19 that both of systems demonstrate theovercontrolling effect In case of 120572

1= 1 this effect does not

appear and phase trajectories coincide for both of systemsstudied (Figure 20)

6 Summary

In this paper the optimal control problem is investigated forlinear dynamical systems of fractional order described byequations with Riemann-Liouville derivative The problemreduced to themoment problem For the last problem the cor-rectness and solvability conditions derived Some examplesof one- and two-dimensional systems were considered andexplicit solutions of corresponding optimal control problemswere found The properties of these solutions were analyzedand compared with analogous integer-order systems andfractional-order systems described by equationswithCaputoderivative

Some computational results are demonstrated that showthe dependency of control norm from differentiation indicesant control time and the dependency ofminimal control timefrom differentiation indices Qualitative dynamics of doubleintegrator are analyzed and show that boundary trajectoriesof Riemann-Liouville system of this type are very close to

the same trajectories for corresponding Caputo system It isshown also that the overcontrolling effect appears in optimalcontrol mode of system studied

Obtained results can be useful in investigation of optimalcontrol problems for fractional-order systems and in designof particular control systems

Competing Interests

The authors declare that they have no competing interests

References

[1] A G Butkovskii S S Postnov and E A Postnova ldquoFractionalintegro-differential calculus and its control-theoretical applica-tions II Fractional dynamic systems modeling and hardwareimplementationrdquo Automation and Remote Control vol 74 no5 pp 725ndash749 2013

[2] R Kamocki ldquoPontryagin maximum principle for fractionalordinary optimal control problemsrdquo Mathematical Methods inthe Applied Sciences vol 37 no 11 pp 1668ndash1686 2014

[3] R Kamocki and M Majewski ldquoFractional linear control sys-tems with Caputo derivative and their optimizationrdquo OptimalControl Applications and Methods vol 36 no 6 pp 953ndash9672015

[4] A G Butkovskiy Distributed Control Systems American Else-vier New York NY USA 1969

[5] V A Kubyshkin and S S Postnov ldquoOptimal control problemfor a linear stationary fractional order system in the form of aproblem of moments problem setting and a studyrdquoAutomationand Remote Control vol 75 no 5 pp 805ndash817 2014

[6] V A Kubyshkin and S S Postnov ldquoOptimal control prob-lem for linear fractional-order systemsrdquo in Proceedings of theInternational Conference on Fractional Differentiation and ItsApplications (ICFDA rsquo14) 6 pages IEEE Catania Italy June2014 Paper ID 3189701

[7] V A Kubyshkin and S S Postnov ldquoAnalysis of two optimalcontrol problems for a fractional-order pendulum by themethod of momentsrdquo Automation and Remote Control vol 76no 7 pp 1302ndash1314 2015

[8] V A Kubyshkin and S S Postnov ldquoThe optimal controlproblem for linear systems of non-integer order with lumpedand distributed parameters Discontinuityrdquo Nonlinearity andComplexity vol 4 no 4 pp 429ndash443 2015

[9] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 ElsevierAmsterdam The Netherlands 2006

[10] R Almeida and D F M Torres ldquoCalculus of variations withfractional derivatives and fractional integralsrdquo Applied Mathe-matics Letters vol 22 no 12 pp 1816ndash1820 2009

[11] N Heymans and I Podlubny ldquoPhysical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivativesrdquo Rheologica Acta vol 45 no 5pp 765ndash771 2006

[12] V E Tarasov Fractional Dynamics Springer Berlin Germany2010

[13] A N Kolmogorov and S V Fomin Elements of the Theory ofFunctions and Functional Analysis Dover Publications Mine-ola NY USA 1999

[14] N I AkhiezerTheClassical Moment Problem and Some RelatedQuestions in Analysis Hafner New York NY USA 1965

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 7

119902plusmn119897

2(119902plusmn119897

1) as opposed to double integrator of integer order [17

sect35] and Caputo double integrator [5] Nevertheless in someparticular cases the time can be eliminated from expressions(52) For example in case of 1199040

2= 0 one can calculate directly

that these formulas reduce to the following equation

119902plusmn119897

1= (Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

12057211205722

sdot [

[

1199040

1

Γ (1205721)(Γ (1205722+ 1) 119902

plusmn119897

2

plusmn119897)

minus11205722

+Γ (1205722+ 1) 119902

plusmn119897

2

Γ (1205721+ 1205722+ 1)]

]

(53)

Analogously in case of 1205722= 1 it is possible to eliminate

time from (52) and to derive the following equation

119902plusmn119897

1=1

Γ (1205721)(119902plusmn119897

2minus 1199040

2

plusmn119897)

1205721minus1

sdot [

[

1199040

1plusmn119902plusmn119897

2minus 1199040

2

119897

1199040

2

1205721

+ (119902plusmn119897

2minus 1199040

2

119897)

2

1

1205721(1205721+ 1)]

]

(54)

Note that expressions (52) at 1205721= 1205722= 1 reduce

to the quadratic equation for integral vortex boundaries ofdifferential inclusion corresponding to the double integratorof integer order [17 sect35]

In order to calculate the phase trajectories of Riemann-Liouville double integrator in optimal control mode we willsubstitute control (41) into solutions (35) Then the followingmotion laws can be obtained

1199021(119905) =

1199040

11199051205721minus1

Γ (1205721)+1199040

21199051205721+1205722minus1

Γ (1205721+ 1205722)minus212057211205722

212057211205722 minus 1

sdot1198881(119879)

1198791205721+1205722[1199051205721+1205722 minus 2 (119905 minus 119905

1015840)1205721+1205722

Θ(119905 minus 1199051015840)]

1199022(119905) =

1199040

21199051205722minus1

Γ (1205722)minus212057211205722

212057211205722 minus 1

1198881(119879)

1198791205721+1205722

sdotΓ (1205721+ 1205722+ 1)

Γ (1205722+ 1)

[1199051205722 minus 2 (119905 minus 119905

1015840)1205722Θ(119905 minus 119905

1015840)]

(55)

where 1199051015840 is the switching point of control (41) and Θ(119905) is theHeavisidersquos function Formulas (55) are obtained for control(41) which is valid in case of 119888

2= 0 Consequently these

expressions allow some simplification

5 Computational Results and Analysis

In this chapter the computational results are representedand analyzed (including comparison with analogous Caputosystems) Calculations and its visualization were performed

0 02 04 06 08120572

100

10minus2

10minus4

u

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

Figure 1 The dependencies for norm of control from 120572 at 1199011015840 = 1(logarithmic scale used for ordinates)

05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 2 The dependencies for norm of control from 120572 at 1199011015840 = 2(logarithmic scale used for ordinates)

in MatLab 79 Mittag-Leffler function values were calculatedusing special procedure [20] Integrals were calculated byGauss-Kronrod method [21]

51 Single Integrator We will consider the task of systemtransfer from any state with 1199040 = 0 in the final state 119902119879 = 0

Figures 1ndash3 show the dependencies of control norm fromdifferentiation index calculated by formula (32) at 119879 = 100and different values 1199040 for 1199011015840 = 1 1199011015840 = 2 and 1199011015840 = 32correspondingly (dash-dot lines) Also the analogous depen-dencies are shown for Caputo single integrator calculated byformulas from [5 8] at different 1199020 for the same values of 1199011015840(solid lines) It is seen that curves for Riemann-Liouville andCaputo single integrator differ from each other qualitativelybut converge to the same point at 120572 = 1 In addition thecontrol norm increases with 1199040 or 1199020 growing for both ofintegrator types

In Figure 4 the dependencies of control norm fromcontrol time 119879 are shown at different values of differentiationindex and 1199040 = 1 (for Riemann-Liouville integrator) and1199020= 1 (for Caputo integrator) The solid lines correspond

to 1199011015840 = 1 the dotted lines correspond to 1199011015840 = 32 and thedash-dotted lines correspond to 1199011015840 = 2 It is clear that all ofthese curves decreasemonotonically which correspond to thecase of Caputo integrator [8] and integrator of integer order[4] And the norm of control increases with growing of 120572 and1199011015840

8 Journal of Control Science and Engineering

04 05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 3 The dependencies for norm of control from 120572 at 1199011015840 = 32(logarithmic scale used for ordinates)

0 20 40 60 80

10minus2

10minus1

10minus3

u

T

120572 = 07120572 = 09

120572 = 05

120572 = 03 120572 = 01

Figure 4 The dependencies for norm of control from 119879 Logarith-mic scale used for ordinates

Consider now the minimal control time 119879lowast dependencyfrom differentiation index 120572 In case of 1199040 = 0 and 119902119879 = 0 (16)subject to formula (32) can be solved explicitly and have onlyone real root

119879lowast= (

10038161003816100381610038161003816119904010038161003816100381610038161003816

119897)

1199011015840

(1199011015840(120572 minus 1) + 1) (56)

In Figure 5 the computational results are shown for theobtained formula used at different 1199011015840 at 119897 = 1 and1199040= 1 (dash-dotted lines) Also the similar dependencies

(calculated by formulas from [8]) are represented for Caputointegrator at 119897 = 1 and 1199020 = 1 (solid lines) As can be seenfrom Figure 5 the dependency at 1199011015840 = 1 is linear for both ofintegrator types In case of 1199011015840 gt 1 the dependency for Caputointegrator becomes nonlinear

52 One-Dimensional System of General Type Consider (asabove) the task of system transfer from any state with 1199040 = 0to the state 119902119879 = 0

Figures 6 and 7 represent the dependencies of controlnorm from differentiation index at 1199011015840 = 1 and 1199011015840 = 2calculated by formulas (24) and (26) correspondingly (dash-doted lines) The analogical dependencies for Caputo one-dimensional system are also shown (solid lines) calculated byformulas from [5 8] The curves were calculated for differentvalues of parameter 119886 and the following values of otherparameters 119879 = 2 119887 = 1 1199040 = 1 (RL-system) and 1199020 = 1

0 02 04 06 080

02

04

06

08

120572

Tlowast

p998400 = 1

p998400 = 32

p998400 = 2

p998400 = 3

Figure 5 The dependencies for minimal control time from 120572

0 02 04 06 080

05

1

120572

a = 1

a = 01

a = 05

u

Figure 6 The dependencies for control norm from 120572 at 1199011015840 = 1 anddifferent values of parameter 119886

06 07 08 090

05

1

120572

a = 1

a = 01a = 05

u

Figure 7 The dependencies for control norm from 120572 at 1199011015840 = 2 anddifferent values of parameter 119886

(C-system) It is seen from Figures 6 and 7 that differencein derivative type has qualitatively effect on behaviour ofdependencies This behaviour changes at 119886 lt 1 and 119886 = 1In case of 1199011015840 = 1 the curves for C-system decrease andcurves for RL-system increase at 119886 lt 1 (analogically to singleintegrator) At 119886 = 1 the curves for both of systems increaseFor1199011015840 = 2 the dependencies increase for both of system types

The dependencies of control norm from control time aresimilar in general for single integrator and are not shownhere

The dependencies of minimal control time from 120572 atdifferent values of parameter 119886 are shown in Figures 8 and 9for 1199011015840 = 1 and 1199011015840 = 2 correspondingly The following valuesof other parameters were chosen 119897 = 2 1199040 = 1 (RL-system)

Journal of Control Science and Engineering 9

0 02 04 06 080

02

04

06

120572

a = 1

a = 05

a = 01

Tlowast

Figure 8The dependencies forminimal control time from120572 at1199011015840 =1 and different values of parameter 119886

06 065 07 075 08 085 09 0950

01

02

03

120572

a = 1

a = 05a = 01

Tlowast

Figure 9The dependencies forminimal control time from120572 at1199011015840 =2 and different values of parameter 119886

and 1199020 = 1 (C-system) Dash-dotted lines correspond to RL-system and solid lines correspond to C-system It is seen thatin case of RL-system the curves have the quasi-linear trendand differ qualitatively from the curves for C-system

53 Double Integrator As in case of one-dimensional sys-tems we will consider the task of system transfer from anystate with nonzero values of phase coordinates into the originat phase plane wewill suppose that 1199040

1= 0 and 119902119879

1= 119902119879

2= 1199040

2=

0The dependencies of control norm from one of differ-

entiation indices for Riemann-Liouville double integrator atseveral fixed values of other indexes are shown in Figures10 and 11 (dash-dotted lines) The curves were calculated byexpression (40) for 1199011015840 = 1 119879 = 100 and 1199040

1= 1 Also the

similar dependencies for Caputo double integrator are shownat these figures calculated for the same values of parameters1199011015840 and 119879 and for 1199020

1= 1 (solid lines) The dependencies for

1199011015840= 2 are similar and not demonstrated here It is seen

from Figure 10 that curves corresponding to different systems(RL- and C-systems) at identical values of 120572

2converge into

the same point at 1205721= 1 In Figure 11 the distance between

different curves for C- and RL-systems decreases when 1205721

increasesThe dependencies of control norm from control time for

double integrator are analogous generally to that in case ofone-dimensional systems and are not demonstrated here

Let us consider the dependency of minimal control timefrom differentiation indices 120572

1and 120572

2 In case of 1199011015840 = 1 this

0 02 04 06 08

100

10minus2

10minus4

u

1205721

1205722 = 09

1205722 = 011205722 = 05

Figure 10 The dependencies for control norm from 1205721at 1199011015840 = 1

and several values of 1205722

0 02 04 06 08

10minus2

10minus4

u

1205722

1205721 = 091205721 = 01 1205721 = 05

Figure 11The dependencies for control norm from 1205722at 1199011015840 = 1 and

several values of 1205721

dependency is given by formula (43) In case of 1199011015840 = 2 withadditional assumption 119902119879

1= 119902119879

2= 1199040

2= 0 the explicit formula

also can be found for such dependency (as solution of (16)subject to (40))

119879lowast= (radic21205721+ 21205722+ 1 (120572

1+ 21205722+ 1) Γ (120572

1+ 1205722)

Γ (1205721+ 1)

sdot

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897)

2(21205722+1)

(57)

In Figures 12ndash15 the dependencies are shown for minimalcontrol time from one of differentiation indices at fixedvalue of other indices The curves are calculated at 119897 = 2for 1199011015840 = 1 (Figures 12 and 13) and 1199011015840 = 2 (Figures 14and 15) Dash-dotted lines correspond to Riemann-Liouvilledouble integrator and are calculated for 1199040

1= 1 Solid lines

correspond toCaputo double integrator and are calculated for1199020

1= 1 As seen from these figures the dependencies for RL-

and C-systems differ qualitatively from each otherLet us now analyze the qualitative dynamics of Riemann-

Liouville double integrator its boundary trajectories andphase trajectories in optimal controlmodeNote that last typeof trajectories in case of Riemann-Liouville double integratorwill not have an obvious initial point at phase plane since theinitial condition (4) defines not the system state but the valueof some integral functional at initial timeThe system state in

10 Journal of Control Science and Engineering

0 02 04 06 08

Tlowast

102

100

10minus2

1205721

1205722 = 01

1205722 = 05 1205722 = 09

Figure 12 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

100

10minus2

1205722

1205721 = 01

1205721 = 05

1205721 = 09

Figure 13 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

105

100

1205721

1205722 = 071205722 = 09

Figure 14 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 2 Logarithmic scale used for

ordinates

this case can be undefined as follows from solutions (35) andexpressions (55)

The boundary trajectories for studied system are repre-sented in Figure 16 for Riemann-Liouville double integrator(dash-dotted lines) and for analogous Caputo system (solidlines) For the last system calculations were performed usingformulas from [5] It is clear that trajectories for both ofsystems are very close to each other In Figure 17 the fragmentof trajectories is shown at enlarged scale

The phase trajectories of Riemann-Liouville double inte-grator in optimal control mode were calculated using expres-sions (55) In Figure 18 the results are shown for different

055 06 065 07 075 08 085 09 0950

1

2

Tlowast

1205722

1205721 = 03 1205721 = 09

1205721 = 07

Figure 15 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 2 Logarithmic scale used for

ordinates

0 200 400

q2

q1

50

0

minus50

minus600 minus400 minus200

Figure 16 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

q2

q1

minus50

minus48

minus46

minus400 minus380 minus360 minus340

Figure 17 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

0 005

0

q1

1205721 = 071205722 = 04

1205721 = 01 1205722 = 021205721 = 01 1205722 = 1

1205721 = 05 1205722 = 1

1205721 = 03 1205722 = 05

q2

minus0005

minus001

minus0015minus005

Figure 18 The phase trajectories in optimal control mode forRiemann-Liouville double integrator

Journal of Control Science and Engineering 11

0

q2

q1

minus1

minus1

minus2

minus2

minus3

minus3

minus4

minus4minus5

minus5minus6

1205721 = 03 1205722 = 05

1205721 = 07 1205722 = 04

1205721 = 05 1205722 = 1

1205721 = 01 1205722 = 02

1205721 = 01 1205722 = 1

Figure 19 The phase trajectories in optimal control mode forCaputo double integrator

0 02 04 06 08 1

0

q1

q2 1205722 = 03 1205722 = 05

1205722 = 08

minus001

minus002

Figure 20 The phase trajectories in optimal control mode forRiemann-Liouville and Caputo double integrator in case of 120572

1= 1

values of differentiation indices 119879 = 100 and 11990401= 1 In

Figure 19 the phase trajectories of Caputo double integrator inoptimal control mode are represented which were calculatedby formulas from [5] at 119879 = 100 and 1199020

1= 1 It is clear

from Figures 18 and 19 that both of systems demonstrate theovercontrolling effect In case of 120572

1= 1 this effect does not

appear and phase trajectories coincide for both of systemsstudied (Figure 20)

6 Summary

In this paper the optimal control problem is investigated forlinear dynamical systems of fractional order described byequations with Riemann-Liouville derivative The problemreduced to themoment problem For the last problem the cor-rectness and solvability conditions derived Some examplesof one- and two-dimensional systems were considered andexplicit solutions of corresponding optimal control problemswere found The properties of these solutions were analyzedand compared with analogous integer-order systems andfractional-order systems described by equationswithCaputoderivative

Some computational results are demonstrated that showthe dependency of control norm from differentiation indicesant control time and the dependency ofminimal control timefrom differentiation indices Qualitative dynamics of doubleintegrator are analyzed and show that boundary trajectoriesof Riemann-Liouville system of this type are very close to

the same trajectories for corresponding Caputo system It isshown also that the overcontrolling effect appears in optimalcontrol mode of system studied

Obtained results can be useful in investigation of optimalcontrol problems for fractional-order systems and in designof particular control systems

Competing Interests

The authors declare that they have no competing interests

References

[1] A G Butkovskii S S Postnov and E A Postnova ldquoFractionalintegro-differential calculus and its control-theoretical applica-tions II Fractional dynamic systems modeling and hardwareimplementationrdquo Automation and Remote Control vol 74 no5 pp 725ndash749 2013

[2] R Kamocki ldquoPontryagin maximum principle for fractionalordinary optimal control problemsrdquo Mathematical Methods inthe Applied Sciences vol 37 no 11 pp 1668ndash1686 2014

[3] R Kamocki and M Majewski ldquoFractional linear control sys-tems with Caputo derivative and their optimizationrdquo OptimalControl Applications and Methods vol 36 no 6 pp 953ndash9672015

[4] A G Butkovskiy Distributed Control Systems American Else-vier New York NY USA 1969

[5] V A Kubyshkin and S S Postnov ldquoOptimal control problemfor a linear stationary fractional order system in the form of aproblem of moments problem setting and a studyrdquoAutomationand Remote Control vol 75 no 5 pp 805ndash817 2014

[6] V A Kubyshkin and S S Postnov ldquoOptimal control prob-lem for linear fractional-order systemsrdquo in Proceedings of theInternational Conference on Fractional Differentiation and ItsApplications (ICFDA rsquo14) 6 pages IEEE Catania Italy June2014 Paper ID 3189701

[7] V A Kubyshkin and S S Postnov ldquoAnalysis of two optimalcontrol problems for a fractional-order pendulum by themethod of momentsrdquo Automation and Remote Control vol 76no 7 pp 1302ndash1314 2015

[8] V A Kubyshkin and S S Postnov ldquoThe optimal controlproblem for linear systems of non-integer order with lumpedand distributed parameters Discontinuityrdquo Nonlinearity andComplexity vol 4 no 4 pp 429ndash443 2015

[9] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 ElsevierAmsterdam The Netherlands 2006

[10] R Almeida and D F M Torres ldquoCalculus of variations withfractional derivatives and fractional integralsrdquo Applied Mathe-matics Letters vol 22 no 12 pp 1816ndash1820 2009

[11] N Heymans and I Podlubny ldquoPhysical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivativesrdquo Rheologica Acta vol 45 no 5pp 765ndash771 2006

[12] V E Tarasov Fractional Dynamics Springer Berlin Germany2010

[13] A N Kolmogorov and S V Fomin Elements of the Theory ofFunctions and Functional Analysis Dover Publications Mine-ola NY USA 1999

[14] N I AkhiezerTheClassical Moment Problem and Some RelatedQuestions in Analysis Hafner New York NY USA 1965

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Control Science and Engineering

04 05 06 07 08 09

s0 = 100

s0 = 10

s0 = 1

q0 = 100

q0 = 10

q0 = 1

100

10minus2

u

120572

Figure 3 The dependencies for norm of control from 120572 at 1199011015840 = 32(logarithmic scale used for ordinates)

0 20 40 60 80

10minus2

10minus1

10minus3

u

T

120572 = 07120572 = 09

120572 = 05

120572 = 03 120572 = 01

Figure 4 The dependencies for norm of control from 119879 Logarith-mic scale used for ordinates

Consider now the minimal control time 119879lowast dependencyfrom differentiation index 120572 In case of 1199040 = 0 and 119902119879 = 0 (16)subject to formula (32) can be solved explicitly and have onlyone real root

119879lowast= (

10038161003816100381610038161003816119904010038161003816100381610038161003816

119897)

1199011015840

(1199011015840(120572 minus 1) + 1) (56)

In Figure 5 the computational results are shown for theobtained formula used at different 1199011015840 at 119897 = 1 and1199040= 1 (dash-dotted lines) Also the similar dependencies

(calculated by formulas from [8]) are represented for Caputointegrator at 119897 = 1 and 1199020 = 1 (solid lines) As can be seenfrom Figure 5 the dependency at 1199011015840 = 1 is linear for both ofintegrator types In case of 1199011015840 gt 1 the dependency for Caputointegrator becomes nonlinear

52 One-Dimensional System of General Type Consider (asabove) the task of system transfer from any state with 1199040 = 0to the state 119902119879 = 0

Figures 6 and 7 represent the dependencies of controlnorm from differentiation index at 1199011015840 = 1 and 1199011015840 = 2calculated by formulas (24) and (26) correspondingly (dash-doted lines) The analogical dependencies for Caputo one-dimensional system are also shown (solid lines) calculated byformulas from [5 8] The curves were calculated for differentvalues of parameter 119886 and the following values of otherparameters 119879 = 2 119887 = 1 1199040 = 1 (RL-system) and 1199020 = 1

0 02 04 06 080

02

04

06

08

120572

Tlowast

p998400 = 1

p998400 = 32

p998400 = 2

p998400 = 3

Figure 5 The dependencies for minimal control time from 120572

0 02 04 06 080

05

1

120572

a = 1

a = 01

a = 05

u

Figure 6 The dependencies for control norm from 120572 at 1199011015840 = 1 anddifferent values of parameter 119886

06 07 08 090

05

1

120572

a = 1

a = 01a = 05

u

Figure 7 The dependencies for control norm from 120572 at 1199011015840 = 2 anddifferent values of parameter 119886

(C-system) It is seen from Figures 6 and 7 that differencein derivative type has qualitatively effect on behaviour ofdependencies This behaviour changes at 119886 lt 1 and 119886 = 1In case of 1199011015840 = 1 the curves for C-system decrease andcurves for RL-system increase at 119886 lt 1 (analogically to singleintegrator) At 119886 = 1 the curves for both of systems increaseFor1199011015840 = 2 the dependencies increase for both of system types

The dependencies of control norm from control time aresimilar in general for single integrator and are not shownhere

The dependencies of minimal control time from 120572 atdifferent values of parameter 119886 are shown in Figures 8 and 9for 1199011015840 = 1 and 1199011015840 = 2 correspondingly The following valuesof other parameters were chosen 119897 = 2 1199040 = 1 (RL-system)

Journal of Control Science and Engineering 9

0 02 04 06 080

02

04

06

120572

a = 1

a = 05

a = 01

Tlowast

Figure 8The dependencies forminimal control time from120572 at1199011015840 =1 and different values of parameter 119886

06 065 07 075 08 085 09 0950

01

02

03

120572

a = 1

a = 05a = 01

Tlowast

Figure 9The dependencies forminimal control time from120572 at1199011015840 =2 and different values of parameter 119886

and 1199020 = 1 (C-system) Dash-dotted lines correspond to RL-system and solid lines correspond to C-system It is seen thatin case of RL-system the curves have the quasi-linear trendand differ qualitatively from the curves for C-system

53 Double Integrator As in case of one-dimensional sys-tems we will consider the task of system transfer from anystate with nonzero values of phase coordinates into the originat phase plane wewill suppose that 1199040

1= 0 and 119902119879

1= 119902119879

2= 1199040

2=

0The dependencies of control norm from one of differ-

entiation indices for Riemann-Liouville double integrator atseveral fixed values of other indexes are shown in Figures10 and 11 (dash-dotted lines) The curves were calculated byexpression (40) for 1199011015840 = 1 119879 = 100 and 1199040

1= 1 Also the

similar dependencies for Caputo double integrator are shownat these figures calculated for the same values of parameters1199011015840 and 119879 and for 1199020

1= 1 (solid lines) The dependencies for

1199011015840= 2 are similar and not demonstrated here It is seen

from Figure 10 that curves corresponding to different systems(RL- and C-systems) at identical values of 120572

2converge into

the same point at 1205721= 1 In Figure 11 the distance between

different curves for C- and RL-systems decreases when 1205721

increasesThe dependencies of control norm from control time for

double integrator are analogous generally to that in case ofone-dimensional systems and are not demonstrated here

Let us consider the dependency of minimal control timefrom differentiation indices 120572

1and 120572

2 In case of 1199011015840 = 1 this

0 02 04 06 08

100

10minus2

10minus4

u

1205721

1205722 = 09

1205722 = 011205722 = 05

Figure 10 The dependencies for control norm from 1205721at 1199011015840 = 1

and several values of 1205722

0 02 04 06 08

10minus2

10minus4

u

1205722

1205721 = 091205721 = 01 1205721 = 05

Figure 11The dependencies for control norm from 1205722at 1199011015840 = 1 and

several values of 1205721

dependency is given by formula (43) In case of 1199011015840 = 2 withadditional assumption 119902119879

1= 119902119879

2= 1199040

2= 0 the explicit formula

also can be found for such dependency (as solution of (16)subject to (40))

119879lowast= (radic21205721+ 21205722+ 1 (120572

1+ 21205722+ 1) Γ (120572

1+ 1205722)

Γ (1205721+ 1)

sdot

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897)

2(21205722+1)

(57)

In Figures 12ndash15 the dependencies are shown for minimalcontrol time from one of differentiation indices at fixedvalue of other indices The curves are calculated at 119897 = 2for 1199011015840 = 1 (Figures 12 and 13) and 1199011015840 = 2 (Figures 14and 15) Dash-dotted lines correspond to Riemann-Liouvilledouble integrator and are calculated for 1199040

1= 1 Solid lines

correspond toCaputo double integrator and are calculated for1199020

1= 1 As seen from these figures the dependencies for RL-

and C-systems differ qualitatively from each otherLet us now analyze the qualitative dynamics of Riemann-

Liouville double integrator its boundary trajectories andphase trajectories in optimal controlmodeNote that last typeof trajectories in case of Riemann-Liouville double integratorwill not have an obvious initial point at phase plane since theinitial condition (4) defines not the system state but the valueof some integral functional at initial timeThe system state in

10 Journal of Control Science and Engineering

0 02 04 06 08

Tlowast

102

100

10minus2

1205721

1205722 = 01

1205722 = 05 1205722 = 09

Figure 12 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

100

10minus2

1205722

1205721 = 01

1205721 = 05

1205721 = 09

Figure 13 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

105

100

1205721

1205722 = 071205722 = 09

Figure 14 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 2 Logarithmic scale used for

ordinates

this case can be undefined as follows from solutions (35) andexpressions (55)

The boundary trajectories for studied system are repre-sented in Figure 16 for Riemann-Liouville double integrator(dash-dotted lines) and for analogous Caputo system (solidlines) For the last system calculations were performed usingformulas from [5] It is clear that trajectories for both ofsystems are very close to each other In Figure 17 the fragmentof trajectories is shown at enlarged scale

The phase trajectories of Riemann-Liouville double inte-grator in optimal control mode were calculated using expres-sions (55) In Figure 18 the results are shown for different

055 06 065 07 075 08 085 09 0950

1

2

Tlowast

1205722

1205721 = 03 1205721 = 09

1205721 = 07

Figure 15 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 2 Logarithmic scale used for

ordinates

0 200 400

q2

q1

50

0

minus50

minus600 minus400 minus200

Figure 16 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

q2

q1

minus50

minus48

minus46

minus400 minus380 minus360 minus340

Figure 17 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

0 005

0

q1

1205721 = 071205722 = 04

1205721 = 01 1205722 = 021205721 = 01 1205722 = 1

1205721 = 05 1205722 = 1

1205721 = 03 1205722 = 05

q2

minus0005

minus001

minus0015minus005

Figure 18 The phase trajectories in optimal control mode forRiemann-Liouville double integrator

Journal of Control Science and Engineering 11

0

q2

q1

minus1

minus1

minus2

minus2

minus3

minus3

minus4

minus4minus5

minus5minus6

1205721 = 03 1205722 = 05

1205721 = 07 1205722 = 04

1205721 = 05 1205722 = 1

1205721 = 01 1205722 = 02

1205721 = 01 1205722 = 1

Figure 19 The phase trajectories in optimal control mode forCaputo double integrator

0 02 04 06 08 1

0

q1

q2 1205722 = 03 1205722 = 05

1205722 = 08

minus001

minus002

Figure 20 The phase trajectories in optimal control mode forRiemann-Liouville and Caputo double integrator in case of 120572

1= 1

values of differentiation indices 119879 = 100 and 11990401= 1 In

Figure 19 the phase trajectories of Caputo double integrator inoptimal control mode are represented which were calculatedby formulas from [5] at 119879 = 100 and 1199020

1= 1 It is clear

from Figures 18 and 19 that both of systems demonstrate theovercontrolling effect In case of 120572

1= 1 this effect does not

appear and phase trajectories coincide for both of systemsstudied (Figure 20)

6 Summary

In this paper the optimal control problem is investigated forlinear dynamical systems of fractional order described byequations with Riemann-Liouville derivative The problemreduced to themoment problem For the last problem the cor-rectness and solvability conditions derived Some examplesof one- and two-dimensional systems were considered andexplicit solutions of corresponding optimal control problemswere found The properties of these solutions were analyzedand compared with analogous integer-order systems andfractional-order systems described by equationswithCaputoderivative

Some computational results are demonstrated that showthe dependency of control norm from differentiation indicesant control time and the dependency ofminimal control timefrom differentiation indices Qualitative dynamics of doubleintegrator are analyzed and show that boundary trajectoriesof Riemann-Liouville system of this type are very close to

the same trajectories for corresponding Caputo system It isshown also that the overcontrolling effect appears in optimalcontrol mode of system studied

Obtained results can be useful in investigation of optimalcontrol problems for fractional-order systems and in designof particular control systems

Competing Interests

The authors declare that they have no competing interests

References

[1] A G Butkovskii S S Postnov and E A Postnova ldquoFractionalintegro-differential calculus and its control-theoretical applica-tions II Fractional dynamic systems modeling and hardwareimplementationrdquo Automation and Remote Control vol 74 no5 pp 725ndash749 2013

[2] R Kamocki ldquoPontryagin maximum principle for fractionalordinary optimal control problemsrdquo Mathematical Methods inthe Applied Sciences vol 37 no 11 pp 1668ndash1686 2014

[3] R Kamocki and M Majewski ldquoFractional linear control sys-tems with Caputo derivative and their optimizationrdquo OptimalControl Applications and Methods vol 36 no 6 pp 953ndash9672015

[4] A G Butkovskiy Distributed Control Systems American Else-vier New York NY USA 1969

[5] V A Kubyshkin and S S Postnov ldquoOptimal control problemfor a linear stationary fractional order system in the form of aproblem of moments problem setting and a studyrdquoAutomationand Remote Control vol 75 no 5 pp 805ndash817 2014

[6] V A Kubyshkin and S S Postnov ldquoOptimal control prob-lem for linear fractional-order systemsrdquo in Proceedings of theInternational Conference on Fractional Differentiation and ItsApplications (ICFDA rsquo14) 6 pages IEEE Catania Italy June2014 Paper ID 3189701

[7] V A Kubyshkin and S S Postnov ldquoAnalysis of two optimalcontrol problems for a fractional-order pendulum by themethod of momentsrdquo Automation and Remote Control vol 76no 7 pp 1302ndash1314 2015

[8] V A Kubyshkin and S S Postnov ldquoThe optimal controlproblem for linear systems of non-integer order with lumpedand distributed parameters Discontinuityrdquo Nonlinearity andComplexity vol 4 no 4 pp 429ndash443 2015

[9] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 ElsevierAmsterdam The Netherlands 2006

[10] R Almeida and D F M Torres ldquoCalculus of variations withfractional derivatives and fractional integralsrdquo Applied Mathe-matics Letters vol 22 no 12 pp 1816ndash1820 2009

[11] N Heymans and I Podlubny ldquoPhysical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivativesrdquo Rheologica Acta vol 45 no 5pp 765ndash771 2006

[12] V E Tarasov Fractional Dynamics Springer Berlin Germany2010

[13] A N Kolmogorov and S V Fomin Elements of the Theory ofFunctions and Functional Analysis Dover Publications Mine-ola NY USA 1999

[14] N I AkhiezerTheClassical Moment Problem and Some RelatedQuestions in Analysis Hafner New York NY USA 1965

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 9

0 02 04 06 080

02

04

06

120572

a = 1

a = 05

a = 01

Tlowast

Figure 8The dependencies forminimal control time from120572 at1199011015840 =1 and different values of parameter 119886

06 065 07 075 08 085 09 0950

01

02

03

120572

a = 1

a = 05a = 01

Tlowast

Figure 9The dependencies forminimal control time from120572 at1199011015840 =2 and different values of parameter 119886

and 1199020 = 1 (C-system) Dash-dotted lines correspond to RL-system and solid lines correspond to C-system It is seen thatin case of RL-system the curves have the quasi-linear trendand differ qualitatively from the curves for C-system

53 Double Integrator As in case of one-dimensional sys-tems we will consider the task of system transfer from anystate with nonzero values of phase coordinates into the originat phase plane wewill suppose that 1199040

1= 0 and 119902119879

1= 119902119879

2= 1199040

2=

0The dependencies of control norm from one of differ-

entiation indices for Riemann-Liouville double integrator atseveral fixed values of other indexes are shown in Figures10 and 11 (dash-dotted lines) The curves were calculated byexpression (40) for 1199011015840 = 1 119879 = 100 and 1199040

1= 1 Also the

similar dependencies for Caputo double integrator are shownat these figures calculated for the same values of parameters1199011015840 and 119879 and for 1199020

1= 1 (solid lines) The dependencies for

1199011015840= 2 are similar and not demonstrated here It is seen

from Figure 10 that curves corresponding to different systems(RL- and C-systems) at identical values of 120572

2converge into

the same point at 1205721= 1 In Figure 11 the distance between

different curves for C- and RL-systems decreases when 1205721

increasesThe dependencies of control norm from control time for

double integrator are analogous generally to that in case ofone-dimensional systems and are not demonstrated here

Let us consider the dependency of minimal control timefrom differentiation indices 120572

1and 120572

2 In case of 1199011015840 = 1 this

0 02 04 06 08

100

10minus2

10minus4

u

1205721

1205722 = 09

1205722 = 011205722 = 05

Figure 10 The dependencies for control norm from 1205721at 1199011015840 = 1

and several values of 1205722

0 02 04 06 08

10minus2

10minus4

u

1205722

1205721 = 091205721 = 01 1205721 = 05

Figure 11The dependencies for control norm from 1205722at 1199011015840 = 1 and

several values of 1205721

dependency is given by formula (43) In case of 1199011015840 = 2 withadditional assumption 119902119879

1= 119902119879

2= 1199040

2= 0 the explicit formula

also can be found for such dependency (as solution of (16)subject to (40))

119879lowast= (radic21205721+ 21205722+ 1 (120572

1+ 21205722+ 1) Γ (120572

1+ 1205722)

Γ (1205721+ 1)

sdot

100381610038161003816100381610038161199040

1

10038161003816100381610038161003816

119897)

2(21205722+1)

(57)

In Figures 12ndash15 the dependencies are shown for minimalcontrol time from one of differentiation indices at fixedvalue of other indices The curves are calculated at 119897 = 2for 1199011015840 = 1 (Figures 12 and 13) and 1199011015840 = 2 (Figures 14and 15) Dash-dotted lines correspond to Riemann-Liouvilledouble integrator and are calculated for 1199040

1= 1 Solid lines

correspond toCaputo double integrator and are calculated for1199020

1= 1 As seen from these figures the dependencies for RL-

and C-systems differ qualitatively from each otherLet us now analyze the qualitative dynamics of Riemann-

Liouville double integrator its boundary trajectories andphase trajectories in optimal controlmodeNote that last typeof trajectories in case of Riemann-Liouville double integratorwill not have an obvious initial point at phase plane since theinitial condition (4) defines not the system state but the valueof some integral functional at initial timeThe system state in

10 Journal of Control Science and Engineering

0 02 04 06 08

Tlowast

102

100

10minus2

1205721

1205722 = 01

1205722 = 05 1205722 = 09

Figure 12 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

100

10minus2

1205722

1205721 = 01

1205721 = 05

1205721 = 09

Figure 13 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

105

100

1205721

1205722 = 071205722 = 09

Figure 14 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 2 Logarithmic scale used for

ordinates

this case can be undefined as follows from solutions (35) andexpressions (55)

The boundary trajectories for studied system are repre-sented in Figure 16 for Riemann-Liouville double integrator(dash-dotted lines) and for analogous Caputo system (solidlines) For the last system calculations were performed usingformulas from [5] It is clear that trajectories for both ofsystems are very close to each other In Figure 17 the fragmentof trajectories is shown at enlarged scale

The phase trajectories of Riemann-Liouville double inte-grator in optimal control mode were calculated using expres-sions (55) In Figure 18 the results are shown for different

055 06 065 07 075 08 085 09 0950

1

2

Tlowast

1205722

1205721 = 03 1205721 = 09

1205721 = 07

Figure 15 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 2 Logarithmic scale used for

ordinates

0 200 400

q2

q1

50

0

minus50

minus600 minus400 minus200

Figure 16 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

q2

q1

minus50

minus48

minus46

minus400 minus380 minus360 minus340

Figure 17 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

0 005

0

q1

1205721 = 071205722 = 04

1205721 = 01 1205722 = 021205721 = 01 1205722 = 1

1205721 = 05 1205722 = 1

1205721 = 03 1205722 = 05

q2

minus0005

minus001

minus0015minus005

Figure 18 The phase trajectories in optimal control mode forRiemann-Liouville double integrator

Journal of Control Science and Engineering 11

0

q2

q1

minus1

minus1

minus2

minus2

minus3

minus3

minus4

minus4minus5

minus5minus6

1205721 = 03 1205722 = 05

1205721 = 07 1205722 = 04

1205721 = 05 1205722 = 1

1205721 = 01 1205722 = 02

1205721 = 01 1205722 = 1

Figure 19 The phase trajectories in optimal control mode forCaputo double integrator

0 02 04 06 08 1

0

q1

q2 1205722 = 03 1205722 = 05

1205722 = 08

minus001

minus002

Figure 20 The phase trajectories in optimal control mode forRiemann-Liouville and Caputo double integrator in case of 120572

1= 1

values of differentiation indices 119879 = 100 and 11990401= 1 In

Figure 19 the phase trajectories of Caputo double integrator inoptimal control mode are represented which were calculatedby formulas from [5] at 119879 = 100 and 1199020

1= 1 It is clear

from Figures 18 and 19 that both of systems demonstrate theovercontrolling effect In case of 120572

1= 1 this effect does not

appear and phase trajectories coincide for both of systemsstudied (Figure 20)

6 Summary

In this paper the optimal control problem is investigated forlinear dynamical systems of fractional order described byequations with Riemann-Liouville derivative The problemreduced to themoment problem For the last problem the cor-rectness and solvability conditions derived Some examplesof one- and two-dimensional systems were considered andexplicit solutions of corresponding optimal control problemswere found The properties of these solutions were analyzedand compared with analogous integer-order systems andfractional-order systems described by equationswithCaputoderivative

Some computational results are demonstrated that showthe dependency of control norm from differentiation indicesant control time and the dependency ofminimal control timefrom differentiation indices Qualitative dynamics of doubleintegrator are analyzed and show that boundary trajectoriesof Riemann-Liouville system of this type are very close to

the same trajectories for corresponding Caputo system It isshown also that the overcontrolling effect appears in optimalcontrol mode of system studied

Obtained results can be useful in investigation of optimalcontrol problems for fractional-order systems and in designof particular control systems

Competing Interests

The authors declare that they have no competing interests

References

[1] A G Butkovskii S S Postnov and E A Postnova ldquoFractionalintegro-differential calculus and its control-theoretical applica-tions II Fractional dynamic systems modeling and hardwareimplementationrdquo Automation and Remote Control vol 74 no5 pp 725ndash749 2013

[2] R Kamocki ldquoPontryagin maximum principle for fractionalordinary optimal control problemsrdquo Mathematical Methods inthe Applied Sciences vol 37 no 11 pp 1668ndash1686 2014

[3] R Kamocki and M Majewski ldquoFractional linear control sys-tems with Caputo derivative and their optimizationrdquo OptimalControl Applications and Methods vol 36 no 6 pp 953ndash9672015

[4] A G Butkovskiy Distributed Control Systems American Else-vier New York NY USA 1969

[5] V A Kubyshkin and S S Postnov ldquoOptimal control problemfor a linear stationary fractional order system in the form of aproblem of moments problem setting and a studyrdquoAutomationand Remote Control vol 75 no 5 pp 805ndash817 2014

[6] V A Kubyshkin and S S Postnov ldquoOptimal control prob-lem for linear fractional-order systemsrdquo in Proceedings of theInternational Conference on Fractional Differentiation and ItsApplications (ICFDA rsquo14) 6 pages IEEE Catania Italy June2014 Paper ID 3189701

[7] V A Kubyshkin and S S Postnov ldquoAnalysis of two optimalcontrol problems for a fractional-order pendulum by themethod of momentsrdquo Automation and Remote Control vol 76no 7 pp 1302ndash1314 2015

[8] V A Kubyshkin and S S Postnov ldquoThe optimal controlproblem for linear systems of non-integer order with lumpedand distributed parameters Discontinuityrdquo Nonlinearity andComplexity vol 4 no 4 pp 429ndash443 2015

[9] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 ElsevierAmsterdam The Netherlands 2006

[10] R Almeida and D F M Torres ldquoCalculus of variations withfractional derivatives and fractional integralsrdquo Applied Mathe-matics Letters vol 22 no 12 pp 1816ndash1820 2009

[11] N Heymans and I Podlubny ldquoPhysical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivativesrdquo Rheologica Acta vol 45 no 5pp 765ndash771 2006

[12] V E Tarasov Fractional Dynamics Springer Berlin Germany2010

[13] A N Kolmogorov and S V Fomin Elements of the Theory ofFunctions and Functional Analysis Dover Publications Mine-ola NY USA 1999

[14] N I AkhiezerTheClassical Moment Problem and Some RelatedQuestions in Analysis Hafner New York NY USA 1965

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Journal of Control Science and Engineering

0 02 04 06 08

Tlowast

102

100

10minus2

1205721

1205722 = 01

1205722 = 05 1205722 = 09

Figure 12 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

100

10minus2

1205722

1205721 = 01

1205721 = 05

1205721 = 09

Figure 13 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 1 Logarithmic scale used for

ordinates

0 02 04 06 08

Tlowast

105

100

1205721

1205722 = 071205722 = 09

Figure 14 The dependencies for minimal control time from 1205721

at different values of 1205722for 1199011015840 = 2 Logarithmic scale used for

ordinates

this case can be undefined as follows from solutions (35) andexpressions (55)

The boundary trajectories for studied system are repre-sented in Figure 16 for Riemann-Liouville double integrator(dash-dotted lines) and for analogous Caputo system (solidlines) For the last system calculations were performed usingformulas from [5] It is clear that trajectories for both ofsystems are very close to each other In Figure 17 the fragmentof trajectories is shown at enlarged scale

The phase trajectories of Riemann-Liouville double inte-grator in optimal control mode were calculated using expres-sions (55) In Figure 18 the results are shown for different

055 06 065 07 075 08 085 09 0950

1

2

Tlowast

1205722

1205721 = 03 1205721 = 09

1205721 = 07

Figure 15 The dependencies for minimal control time from 1205722

at different values of 1205721for 1199011015840 = 2 Logarithmic scale used for

ordinates

0 200 400

q2

q1

50

0

minus50

minus600 minus400 minus200

Figure 16 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

q2

q1

minus50

minus48

minus46

minus400 minus380 minus360 minus340

Figure 17 The boundary trajectories for double integrator offractional order at 120572

1= 1205722= 05 and 119897 = 1

0 005

0

q1

1205721 = 071205722 = 04

1205721 = 01 1205722 = 021205721 = 01 1205722 = 1

1205721 = 05 1205722 = 1

1205721 = 03 1205722 = 05

q2

minus0005

minus001

minus0015minus005

Figure 18 The phase trajectories in optimal control mode forRiemann-Liouville double integrator

Journal of Control Science and Engineering 11

0

q2

q1

minus1

minus1

minus2

minus2

minus3

minus3

minus4

minus4minus5

minus5minus6

1205721 = 03 1205722 = 05

1205721 = 07 1205722 = 04

1205721 = 05 1205722 = 1

1205721 = 01 1205722 = 02

1205721 = 01 1205722 = 1

Figure 19 The phase trajectories in optimal control mode forCaputo double integrator

0 02 04 06 08 1

0

q1

q2 1205722 = 03 1205722 = 05

1205722 = 08

minus001

minus002

Figure 20 The phase trajectories in optimal control mode forRiemann-Liouville and Caputo double integrator in case of 120572

1= 1

values of differentiation indices 119879 = 100 and 11990401= 1 In

Figure 19 the phase trajectories of Caputo double integrator inoptimal control mode are represented which were calculatedby formulas from [5] at 119879 = 100 and 1199020

1= 1 It is clear

from Figures 18 and 19 that both of systems demonstrate theovercontrolling effect In case of 120572

1= 1 this effect does not

appear and phase trajectories coincide for both of systemsstudied (Figure 20)

6 Summary

In this paper the optimal control problem is investigated forlinear dynamical systems of fractional order described byequations with Riemann-Liouville derivative The problemreduced to themoment problem For the last problem the cor-rectness and solvability conditions derived Some examplesof one- and two-dimensional systems were considered andexplicit solutions of corresponding optimal control problemswere found The properties of these solutions were analyzedand compared with analogous integer-order systems andfractional-order systems described by equationswithCaputoderivative

Some computational results are demonstrated that showthe dependency of control norm from differentiation indicesant control time and the dependency ofminimal control timefrom differentiation indices Qualitative dynamics of doubleintegrator are analyzed and show that boundary trajectoriesof Riemann-Liouville system of this type are very close to

the same trajectories for corresponding Caputo system It isshown also that the overcontrolling effect appears in optimalcontrol mode of system studied

Obtained results can be useful in investigation of optimalcontrol problems for fractional-order systems and in designof particular control systems

Competing Interests

The authors declare that they have no competing interests

References

[1] A G Butkovskii S S Postnov and E A Postnova ldquoFractionalintegro-differential calculus and its control-theoretical applica-tions II Fractional dynamic systems modeling and hardwareimplementationrdquo Automation and Remote Control vol 74 no5 pp 725ndash749 2013

[2] R Kamocki ldquoPontryagin maximum principle for fractionalordinary optimal control problemsrdquo Mathematical Methods inthe Applied Sciences vol 37 no 11 pp 1668ndash1686 2014

[3] R Kamocki and M Majewski ldquoFractional linear control sys-tems with Caputo derivative and their optimizationrdquo OptimalControl Applications and Methods vol 36 no 6 pp 953ndash9672015

[4] A G Butkovskiy Distributed Control Systems American Else-vier New York NY USA 1969

[5] V A Kubyshkin and S S Postnov ldquoOptimal control problemfor a linear stationary fractional order system in the form of aproblem of moments problem setting and a studyrdquoAutomationand Remote Control vol 75 no 5 pp 805ndash817 2014

[6] V A Kubyshkin and S S Postnov ldquoOptimal control prob-lem for linear fractional-order systemsrdquo in Proceedings of theInternational Conference on Fractional Differentiation and ItsApplications (ICFDA rsquo14) 6 pages IEEE Catania Italy June2014 Paper ID 3189701

[7] V A Kubyshkin and S S Postnov ldquoAnalysis of two optimalcontrol problems for a fractional-order pendulum by themethod of momentsrdquo Automation and Remote Control vol 76no 7 pp 1302ndash1314 2015

[8] V A Kubyshkin and S S Postnov ldquoThe optimal controlproblem for linear systems of non-integer order with lumpedand distributed parameters Discontinuityrdquo Nonlinearity andComplexity vol 4 no 4 pp 429ndash443 2015

[9] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 ElsevierAmsterdam The Netherlands 2006

[10] R Almeida and D F M Torres ldquoCalculus of variations withfractional derivatives and fractional integralsrdquo Applied Mathe-matics Letters vol 22 no 12 pp 1816ndash1820 2009

[11] N Heymans and I Podlubny ldquoPhysical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivativesrdquo Rheologica Acta vol 45 no 5pp 765ndash771 2006

[12] V E Tarasov Fractional Dynamics Springer Berlin Germany2010

[13] A N Kolmogorov and S V Fomin Elements of the Theory ofFunctions and Functional Analysis Dover Publications Mine-ola NY USA 1999

[14] N I AkhiezerTheClassical Moment Problem and Some RelatedQuestions in Analysis Hafner New York NY USA 1965

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 11

0

q2

q1

minus1

minus1

minus2

minus2

minus3

minus3

minus4

minus4minus5

minus5minus6

1205721 = 03 1205722 = 05

1205721 = 07 1205722 = 04

1205721 = 05 1205722 = 1

1205721 = 01 1205722 = 02

1205721 = 01 1205722 = 1

Figure 19 The phase trajectories in optimal control mode forCaputo double integrator

0 02 04 06 08 1

0

q1

q2 1205722 = 03 1205722 = 05

1205722 = 08

minus001

minus002

Figure 20 The phase trajectories in optimal control mode forRiemann-Liouville and Caputo double integrator in case of 120572

1= 1

values of differentiation indices 119879 = 100 and 11990401= 1 In

Figure 19 the phase trajectories of Caputo double integrator inoptimal control mode are represented which were calculatedby formulas from [5] at 119879 = 100 and 1199020

1= 1 It is clear

from Figures 18 and 19 that both of systems demonstrate theovercontrolling effect In case of 120572

1= 1 this effect does not

appear and phase trajectories coincide for both of systemsstudied (Figure 20)

6 Summary

In this paper the optimal control problem is investigated forlinear dynamical systems of fractional order described byequations with Riemann-Liouville derivative The problemreduced to themoment problem For the last problem the cor-rectness and solvability conditions derived Some examplesof one- and two-dimensional systems were considered andexplicit solutions of corresponding optimal control problemswere found The properties of these solutions were analyzedand compared with analogous integer-order systems andfractional-order systems described by equationswithCaputoderivative

Some computational results are demonstrated that showthe dependency of control norm from differentiation indicesant control time and the dependency ofminimal control timefrom differentiation indices Qualitative dynamics of doubleintegrator are analyzed and show that boundary trajectoriesof Riemann-Liouville system of this type are very close to

the same trajectories for corresponding Caputo system It isshown also that the overcontrolling effect appears in optimalcontrol mode of system studied

Obtained results can be useful in investigation of optimalcontrol problems for fractional-order systems and in designof particular control systems

Competing Interests

The authors declare that they have no competing interests

References

[1] A G Butkovskii S S Postnov and E A Postnova ldquoFractionalintegro-differential calculus and its control-theoretical applica-tions II Fractional dynamic systems modeling and hardwareimplementationrdquo Automation and Remote Control vol 74 no5 pp 725ndash749 2013

[2] R Kamocki ldquoPontryagin maximum principle for fractionalordinary optimal control problemsrdquo Mathematical Methods inthe Applied Sciences vol 37 no 11 pp 1668ndash1686 2014

[3] R Kamocki and M Majewski ldquoFractional linear control sys-tems with Caputo derivative and their optimizationrdquo OptimalControl Applications and Methods vol 36 no 6 pp 953ndash9672015

[4] A G Butkovskiy Distributed Control Systems American Else-vier New York NY USA 1969

[5] V A Kubyshkin and S S Postnov ldquoOptimal control problemfor a linear stationary fractional order system in the form of aproblem of moments problem setting and a studyrdquoAutomationand Remote Control vol 75 no 5 pp 805ndash817 2014

[6] V A Kubyshkin and S S Postnov ldquoOptimal control prob-lem for linear fractional-order systemsrdquo in Proceedings of theInternational Conference on Fractional Differentiation and ItsApplications (ICFDA rsquo14) 6 pages IEEE Catania Italy June2014 Paper ID 3189701

[7] V A Kubyshkin and S S Postnov ldquoAnalysis of two optimalcontrol problems for a fractional-order pendulum by themethod of momentsrdquo Automation and Remote Control vol 76no 7 pp 1302ndash1314 2015

[8] V A Kubyshkin and S S Postnov ldquoThe optimal controlproblem for linear systems of non-integer order with lumpedand distributed parameters Discontinuityrdquo Nonlinearity andComplexity vol 4 no 4 pp 429ndash443 2015

[9] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 ElsevierAmsterdam The Netherlands 2006

[10] R Almeida and D F M Torres ldquoCalculus of variations withfractional derivatives and fractional integralsrdquo Applied Mathe-matics Letters vol 22 no 12 pp 1816ndash1820 2009

[11] N Heymans and I Podlubny ldquoPhysical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivativesrdquo Rheologica Acta vol 45 no 5pp 765ndash771 2006

[12] V E Tarasov Fractional Dynamics Springer Berlin Germany2010

[13] A N Kolmogorov and S V Fomin Elements of the Theory ofFunctions and Functional Analysis Dover Publications Mine-ola NY USA 1999

[14] N I AkhiezerTheClassical Moment Problem and Some RelatedQuestions in Analysis Hafner New York NY USA 1965

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Journal of Control Science and Engineering

[15] M G Krein and A A Nudelman ldquoThe Markov momentproblem and extremal problemsrdquo in Ideas and problems of PL Chebyshev and A A Markov and their further developmentvol 50 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 1977

[16] A A FeldbaumOptimal Control Systems Academic Press NewYork NY USA 1965

[17] A G Butkovskiy Phase Portraits of Control Dynamical SystemsKluwer Academic Dordrecht The Netherlands 1991

[18] C Li D Qian and Y Chen ldquoOn Riemann-Liouville and caputoderivativesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 562494 15 pages 2011

[19] Y Li Y Q Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[20] httpwwwmathworkscommatlabcentralfileexchange8738-mittag-leffler-function

[21] L F Shampine ldquoVectorized adaptive quadrature in MatlabrdquoJournal of Computational and Applied Mathematics vol 211 no2 pp 131ndash140 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of