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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 754268 5 pageshttpdxdoiorg1011552013754268

Research ArticleFinite-Time Boundedness Control of Time-VaryingDescriptor Systems

Xiaoming Su1 Yali Zhi1 and Qingling Zhang2

1 School of Science Shenyang University of Technology Shenyang 110870 China2 School of Science Northeastern University Shenyang 110004 China

Correspondence should be addressed to Xiaoming Su suxmsuteducn

Received 29 August 2013 Accepted 2 December 2013

Academic Editor Yijing Wang

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

This paper mainly studies a control problem of finite-time boundedness of time-varying descriptor systems Firstly a sufficientand necessary condition of finite-time stability is given then a sufficient condition of finite-time boundedness for time-varyingdescriptor systems is given Secondly we analyze the finite-time boundedness control problem and design the finite-time statefeedback controller the controller is given based on LMIs for time-varying descriptor systems and time-varying uncertaindescriptor systems respectively Finally a numerical example is given to prove the effectiveness of the method

1 Introduction

Recently descriptor systems (which are also known as singu-lar systems semistate systems systems of differential-algebraicequations or generalized state space systems) theory has beenwell studied since they are very important from the engineer-ing point of view Let the time-varying descriptor systems bedescribed as 119864(119905)(119905) = 119860(119905)119909(119905) + 119861(119905)119906(119905) + 119866(119905)120596(119905) Theyarise naturally inmany physical applications such as electricalnetworks aircraft and robot dynamics neutral delay andlarge-scale systems economics and optimization problembiology constrainedmechanics as result of partial discretiza-tion of partial differential equation and so forth The time-varying descriptor system has the common properties ofgeneral descriptor system and some special properties Incontrol theory the time-varying descriptor system stabilityis mainly Lyapunov stability however Lyapunov stabilitydeals with the whole state performance of the system butit does not reflect a transient performance of the systemThe transient performance refers to the system stability ina short period but it is different from Lyapunov stability Inengineering a whole stability of the system may have a verybad transient performance which would cause a bad effect inthe engineeringTherefore people tend to bemore concerned

about the transient performance of the system [1ndash5] than theoverall steady state performance of the system

In order to study the transient performance of the systemscholars have given the concept of finite-time stability Finite-time stability of time-varying descriptor system is a new fieldscholars have mainly studied the finite-time stability of linearsystem for years and also made some corresponding resultsFor example [6ndash10] proposed the definition of finite-timestability and finite-time boundedness [5ndash8] proposed thesufficient conditions for the finite-time stability of generallinear system [1 11] studied the input and output finite-timestability for time-varying descriptor system

In addition a finite-time stability problem with externaldisturbance is called finite-time boundedness the conceptof finite-time boundedness came from finite-time stabilityWe have some preliminary research results about the finite-time boundedness For example [12ndash14] studied the finite-time bounded problemof the linear time-varying systemwithimpulse [15ndash18] proposed the design methods of dynamiccompensators [19ndash21] discussed the finite-time control prob-lems of uncertain system with disturbance Scholars intro-duced the definition of finite-time stability and have given thenecessary and sufficient conditions for finite-time stability ofthe descriptor system but the finite-time stability problems

2 Mathematical Problems in Engineering

of time-varying descriptor system have no available researchresults especially when119864(119905) is time-variant So it is necessaryto study finite-time stability of the time-varying uncertaindescriptor system

The paper is divided into two parts firstly we deals withthe problem of finite-time stability of time-varying descriptorsystem We give the necessary and sufficient condition offinite-time stability and finite-time boundedness of systemwith time-varying function matrix 119864(119905) the unstable systemcan be controlled by the state feedback controllers We alsomake the corresponding research for time-varying uncertaindescriptor system Secondly we design a finite-time bound-edness state feedback controller for time-varying uncertaindescriptor system to make the close-loop system finite-timeboundedness

The paper is organized as follows In Section 2 we givesome results of finite-time stability and boundedness fortime-varying descriptor system In Section 3 some results oftime-varying uncertain descriptor system are provided InSection 4 a numerical example is presented to illustrate theefficiency of the proposed result

2 Finite-Time Control of Time-VaryingDescriptor Systems

21 Finite-Time Stable of Time-Varying Descriptor Systems

Definition 1 Time-varying matrix 119864(119905) is singular on timeinterval [0 119879] if there exists a 119905 isin [0 119879] such that rank119864(119905) lt

119899Consider the following time-varying descriptor systems

119864 (119905) (119905) = 119860 (119905) 119909 (119905) (1)

where 119909(119905) isin 119877119899 is the state 119860(sdot) is given continuous matrix-

valued function 119864(119905) isin 119877119899 is singular on time interval [0 119879]

Definition 2 The time-varying descriptor system (1) is saidto be finite-time stable with respect to (119888

1 1198882 119879 119877(119905)) with

positive definite matrix 119877(119905) and given three positive scalars1198881 1198882 119879 with 119888

1lt 1198882 if 119909119879(0)119864119879(0)119877(0)119864(0)119909(0) le 119888

1 then

119909119879

(119905)119864119879

(119905)119877(119905)119864(119905)119909(119905) lt 1198882 for all 119905 isin [0 119879]

Theorem 3 The following statements are equivalent

(i) system (1) is FTS respect to (1198881 1198882 119879 119877(119905))

(ii) For all 119905 isin [0 119879] Φ119879

(119905 0)119864119879

(119905)119877(119905)119864(119905) lt

(11988821198881)119864119879

(0)119877(0)119864(0) where Φ(119905 0) is the state tran-sition matrix and 119877(119905) is positive definite matrix

(iii) For all 119905 isin [0 119879] the differential Lyapunov inequalitywith terminal and initial conditions

(a) 119872(119905) lt 0 where

119872(119905) = 119860119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905) 119860 (119905)

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(2)

(b) 119877(119905) le 119875(119905) le 119875(0) lt (11988821198881)119877(0) 119877(119905) is positive

definite matrix admits a piecewise continuouslydifferentiable symmetric solution 119901(sdot)

Proof (ii) rArr (i) Let 119909119879(0)119864119879(0)119877(0)119864(0)119909(0) le 1198881 assume

119909(119905) = Φ(119905 0)119909(0) have

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

= 119909119879

(0)Φ119879

(119905 0) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (0)

lt1198882

1198881

119864119879

(0) 119877 (0) 119864 (0)

lt 1198882

(3)

Therefore system (1) is FTS(i) rArr (ii) by contradiction Let us assume exist119905 119909

119909119879

(119905)Φ119879

(119905 0) 119864119879

(119905) 119877 (119905) 119864 (119905)Φ (119905 0) 119909 (119905)

ge1198882

1198881

119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0)

(4)

Let 119909119879(0)119864119879(0)119877(0)119864(0)119909(0) = 1198881 exist120582 such that 119909(119905) = 120582119909(0)

then (4) implies that

119909119879

(0)Φ119879

(119905 0) 119864119879

(119905) 119877 (119905) 119864 (119905)Φ (119905 0) 119909 (0) ge 1198882 (5)

therefore

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

= 119909119879

(0)Φ119879

(119905 0) 119864119879

(119905) 119877 (119905) 119864 (119905)Φ (119905 0) 119909 (0)

ge 1198882

(6)

Obviously it contradicts the initial assumption that system (1)is FTS

(iii) rArr (i) Let119881(119905 119909) = 119909119879

(119905)119864119879

(119905)119875(119905)119864(119905)119909(119905) (119905 119909) =

119909119879

(119905)119872(119905)119909(119905)Then (a) implies that (119905 119909) is negative definite along the

trajectories of system (1)Now 119909

119879

(0)119864119879

(0)119877(0)119864(0)119909(0) le 1198881 then for a generic 119905

such that

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

le 119909119879

(119905) 119864119879

(119905) 119875 (119905) 119864 (119905) 119909 (119905)

lt 119909119879

(0) 119864119879

(0) 119875 (0) 119864 (0) 119909 (0)

le1198882

1198881

119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0)

le 1198882

(7)

(i) rArr (iii) By contradiction Because system (1) is FTS Let119911 = 120598119909 for a small 120598 gt 0 for all 119905 isin [0 119879] such that

119909119879

(0) 119864119879

(0) 119875 (0) 119864 (0) 119909 (0) le 1198881

997904rArr 119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905) + 1199112

2lt 1198882

(8)

Mathematical Problems in Engineering 3

Let 119875(sdot) be the solution of

119872(119905) = 1205982

119868 (9)

119877 (119905) = 119875 (119905) (10)

And assume that exist119909 such that

119909119879

(119905) 119864119879

(119905) 119875 (0) 119864 (119905) 119909 (119905)ge1198882

1198881

119909119879

(119905) 119864119879

(119905) 119877 (0) 119864 (119905) 119909 (119905)

(11)

Now let 119909119879(0)119864119879(0)119877(0)119864(0)119909(0) = 1198881 exist120582 such that 119909(119905) =

120582119909(0)Then (11) implies

119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0) ge 1198882 (12)

from (9) we obtain that

119889

119889119905119909119879

(119905) 119864119879

(119905) 119875 (119905) 119864 (119905) 119909 (119905) = minus1205982

119909119879

(119905) 119909 (119905) (13)

Integrating (13) from 0 to 119905 we have

119909119879

(119905) 119864119879

(119905) 119875 (119905) 119864 (119905) 119909 (119905) minus 119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0)

= minus1205982

1199092

2

(14)

Therefore

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

ge 119909119879

(119905) 119864119879

(119905) 119875 (119905) 119864 (119905) 119909 (119905)

= 119909119879

(0) 119864119879

(0) 119875 (0) 119864 (0) 119909 (0) minus 1205982

1199092

2

ge 1198882minus 1199112

2

(15)

which contradicts (8)

22 Finite-Time Boundedness of Time-Varying Descriptor Sys-tems Consider the following time-varying descriptor sys-tems

119864 (119905) (119905) = 119860 (119905) 119909 (119905) + 119866 (119905) 120596 (119905) (16)

where 119909(119905) isin 119877119899 is the state and 120596(119905) isin 119877

119897 is exogenous input119860(119905) isin 119877

119899times119899 119866(119905) isin 119877119899times119897 are given constant matrices 119864(119905) isin

119877119899times119899 is a singular function matrix and rank 119864(119905) = 119902 lt 119899The exogenous disturbance 120596(119905) is time varying and

satisfies the constraint

int

119879

0

120596119879

(119905) 120596 (119905) 119889119905 le 119889 119889 ge 0 (17)

Definition 4 The system (16) subject to an exogenous dis-turbance 120596(119905) satisfies (17) and is said to be finite-timebounded with respect to (119888

1 1198882 119879 119877(119905) 119889) with positive def-

inite matrices 119877(119905) and given three positive scalars 1198881 1198882 119879

with 1198881

lt 1198882 if 119909

119879

(0)119864119879

(0)119877(0)119864(0)119909(0) le 1198881 then

119909119879

(119905)119864119879

(119905)119877(119905)119864(119905)119909(119905) lt 1198882

Theorem 5 The time-varying descriptor system (16) is finite-time bounded if there exist two positive definite nonsingularmatrices 119875(119905) isin 119877

119899times119899 119876 isin 119877119897times119897 such that

[119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) 119876] lt 0 (18)

119877 (119905) le 119875 (119905) le 119875 (0) le 119877 (0) (19)

1198881+ 119889120582max (119876) lt 119888

2(20)

hold where 120582max(sdot) denote the maximum eigenvalue of theargument

119872(119905) = 119860119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905) 119860 (119905)

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(21)

Proof Let 119881(119905 119909) = 119909119879

(119905)119864119879

(119905)119875(119905)119864(119905)119909(119905)We have

(119905 119909) = 119909119879

(119905)119872 (119905) 119909 (119905) + 120596119879

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) 119909 (119905)

+ 119909119879

(119905) 119864119879

(119905) 119875 (119905) 119866 (119905) 120596 (119905)

= (119909119879

(119905) 120596119879

(119905))

times [119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) 119876]

times (119909 (119905)

120596 (119905)) minus 120596

119879

(119905) 119876120596 (119905)

lt 0

(22)

Therefore

(119905 119909) lt 120596119879

(119905) 119876120596 (119905) (23)

integrating from 0 to 119905 with 119905 isin [0 119879] we have

119881 (119905 119909) lt 1198810+ 119889120582max (119876) (24)

From (19) we have

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

le 119881 (119905 119909) lt 119881 (0 119909) + 119889120582max (119876)

lt 119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0) + 119889120582max (119876)

lt 1198881+ 119889120582max (119876)

lt 1198882

(25)

Therefore system (16) is FTB The proof is completed


23 Finite-Time Bounded Control of Time-Varying DescriptorSystems Consider the following time-varying descriptor sys-tems

119864 (119905) (119905) = 119860 (119905) 119909 (119905) + 119861 (119905) 119906 (119905) + 119866 (119905) 120596 (119905) (26)

where 119909(119905) isin 119877119899 is the state 119906(119905) isin 119877

119898 is control input 120596(119905) isin

119877119897 is exogenous input 119860(119905) isin 119877

119899times119899

119861(119905)) isin 119877119899times119898

119866(119905) isin 119877119899times119897

are given continuous matrix-valued functions 119864(119905) isin 119877119899times119899 is

a singular function matrix and rank 119864(119905) lt 119899Find a state feedback control law

119906 (119905) = 119870 (119905) 119909 (119905) (27)

where 119870(119905) isin 119877119898times119899 Let 119860

119888(119905) = (119860(119905) + 119861(119905)119870(119905)) Then the

closed-loop system is given by

119864 (119905) (119905) = 119860119888(119905) 119909 (119905) + 119866 (119905) 120596 (119905) (28)

Theorem 6 The time-varying descriptor system (19) is finite-time bounded if there exist a symmetric positive definitematrix119875(119905) isin 119877

119899times119899 and a nonsingular matrix 119876 isin 119877119897times119897 such that

inequality (19) (20) and the following condition hold

[119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) minus119876119879

] lt 0 (29)

Moreover the state feedback control law is given by

119870 (119905) = 119861minus1

(119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) (30)

where 120582max(sdot) denote the maximum eigenvalue of the argu-ment

119872(119905) = 119860119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905) 119860 (119905)

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(31)

Proof From Theorem 5 the conditions for FTB is that thereexist a nonsingular matrix 119876 such that (19) (20) and thefollowing matrix inequality hold

[119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) 119876] lt 0 (32)

where

119872(119905) = (119860 (119905) + 119861 (119905)119870 (119905))119879

119875 (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119860 (119905) + 119861 (119905)119870 (119905))

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(33)

According to Schurrsquos theorem it is easy to see that (32) can berewritten as

119872(119905) minus 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) lt 0

(34)

Using (30) (34) we have

119872(119905) + 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus119879

(119905) 119866119879

(119905) 119875 (119905) 119864 (119905) lt 0

(35)

Since 119875(119905) is symmetric then

119872(119905) + 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus119879

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) lt 0

(36)

By Schurrsquos theorem (36) is equivalent to (29) The proof iscompleted

24 Finite-Time Bounded Control of Time-Varying UncertainDescriptor Systems Consider the following time-varyingdescriptor systems

119864 (119905) (119905) = (119860 (119905) + Δ119860 (119905) 119909 (119905))

+ (119861 (119905) + Δ119861 (119905)) 119906 (119905) + 119866 (119905) 120596 (119905)

(37)




119899times119899 119861(119905) isin 119877119899times119898 119866(119905) isin 119877

119899times119897


a singular function matrix and rank 119864(119905) lt 119899

[Δ119860 (119905) Δ119861 (119905)] = 119867119865 [1198641(119905) 119864

2(119905)] (38)

where 119865 isin 119877119902times119904 is unknown and satisfies

119865119879

119865 le 119868 (39)



inequality (19) (20) and (32) hold and the state feedbackcontrol law is given by

119870 (119905) = (119861 (119905) + 1198671198651198642(119905))minus1

times (119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) minus 1198671198651198641(119905))

(40)

where

119872(119905) = [119860 (119905) + Δ119860 (119905) + (119861 (119905) + Δ119861 (119905))119870 (119905)]119879

times 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905)

times [119860 (119905) + Δ119860 (119905) + (119861 (119905) + Δ119861 (119905))119870 (119905)]

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(41)

Proof Applying (40) to (34) and 119875(119905) is symmetric then theproof is completed


3 Numerical Example

Consider a time-varying descriptor system (26) with

119864 = [119905 0

0 0] 119860 = [

119905 0

0 minus1] 119861 = [

1 0

0 119905]

119866 = [1

minus119905]

(42)

Let 119877(119905) = 119875(119905) 119876 = 119905 119879 = 1 Exist 119875(119905) = [minus119905 0

0 119905] such that

119872(119905) = [minus21199053

minus 31199052

0

0 0] 119864

119879

(119905) 119875 (119905) 119866 (119905) = [minus1199052

0]

119866119879

(119905) 119875119879

(119905) 119864 (119905) = [minus1199052

0]

(43)

So condition (29) (19) (20) hold the system (32) is finite-time bounded and the state feedback controller with

119870 (119905) = 119861minus1

(119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) = [minus119905 0

1 0]

(44)

4 Conclusions

In this paper we have studied the finite-time stability andgiven a sufficient and necessary condition of the finite-timestability the state feedback controller was designed thenwe studied the finite-time boundedness of time-varyingdescriptor systems and a sufficient and necessary conditionof the finite-time boundedness is given and a state feedbackcontroller was designed In the end a numerical example isgiven to prove the effectiveness of the method

Acknowledgment

This project was supported by the National Nature ScienceFoundation of China (no 61074005) the Talent Project of theHighEducation of Liaoning province China underGrant noLR2012005

References

[1] J Yao Z-H Guan G Chen and D W C Ho ldquoStability robuststabilization and 119867

infincontrol of singular-impulsive systems via

switching controlrdquo Systems amp Control Letters vol 55 no 11 pp879ndash886 2006

[2] X Lin H Du and S Li ldquoFinite-time boundedness and 1198712-

gain analysis for switched delay systems with norm-boundeddisturbancerdquo Applied Mathematics and Computation vol 217no 12 pp 5982ndash5993 2011

[3] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009

[4] Z-H Guan C W Chan A Y T Leung and G Chen ldquoRobuststabilization of singular-impulsive-delayed systems with non-linear perturbationsrdquo IEEE Transactions on Circuits and Sys-tems vol 48 no 8 pp 1011ndash1019 2001

[5] Y Tong BWu and F Huang ldquoFinite-time bound and LL2-gain

analysis for linear time-varying singular impulsive systemsrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo12) no 24 2012

[6] E Moulay M Dambrine N Yeganefar and W PerruquettildquoFinite-time stability and stabilization of time-delay systemsrdquoSystems amp Control Letters vol 57 no 7 pp 561ndash566 2008

[7] J Yao J-e Feng L Sun and Y Zheng ldquoInput-output finite-time stability of time-varying linear singular systemsrdquo Journalof Control Theory and Applications vol 10 no 3 pp 287ndash2912012

[8] F Amato R Ambrosino C Cosentino and G DeTommasildquoInput-output finite-time stability of linear systemsrdquo in Pro-ceedings of the 17th Mediterranean Conference on Control andAutomation pp 342ndash346 Hessaloniki Greece June 2009

[9] F Amato M Ariola C Cosentino C T Abdallah and PDorato ldquoNecessary and sufficient conditions for finite-timestability of linear systemsrdquo in Proceedings of the AmericanControl Conference pp 4452ndash4456 Denver Colo USA June2003

[10] N A Kablar and D L Debeljkovic ldquoFinite-time stability oftime-varying linear singular systemsrdquo in Proceedings of the 37thIEEE Conference on Decision and Control (CDC rsquo98) pp 3831ndash3836 Tampa Fla USA December 1998

[11] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[12] S Zhao J Sun and L Liu ldquoFinite-time stability of linear time-varying singular systems with impulsive effectsrdquo InternationalJournal of Control vol 81 no 11 pp 1824ndash1829 2008

[13] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoIEEETransactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[14] F Amato R Ambrosino M Ariola and C Cosentino ldquoFinite-time stability of linear time-varying systems with jumpsrdquoAutomatica A vol 45 no 5 pp 1354ndash1358 2009

[15] F Amato M Ariola and C Cosentino ldquoFinite-time stabiliza-tion via dynamic output feedbackrdquo Automatica A vol 42 no 2pp 337ndash342 2006

[16] F Amato M Ariola M Carbone and C Cosentino ldquoFinite-time output feedback control of linear systems via differentiallinear matrix conditionsrdquo in Proceedings of the 45th IEEEConference on Decision and Control (CDC rsquo06) pp 5371ndash5375San Diego Calif USA December 2006

[17] P Dorato ldquoShort time stability in linear time-varying systemsrdquoin Proceedings of the IRE International Convention Record pp83ndash87 1961

[18] J Sun and Z Cheng ldquoFinite-time control for one kind ofuncertain linear singular systems subject to norm boundeduncertaintiesrdquo in Proceedings of the 5th World Congress onIntelligent Control and Automation pp 980ndash984 HangzhouChin June 2004

[19] F Amato M Ariola and P Dorato ldquoFinite-time control oflinear systems subject to parametric uncertainties and distur-bancesrdquo Automatica vol 37 no 9 pp 1459ndash1463 2001

[20] J-E Feng Z Wu and J-B Sun ldquoFinite-time control of linearsingular systems with parametric uncertainties and distur-bancesrdquoActaAutomatica Sinica vol 31 no 4 pp 634ndash637 2005

[21] R Ambrosino F Calabrese C Cosentino and G de TommasildquoSufficient conditions for finite-time stability of impulsivedynamical systemsrdquo IEEE Transactions on Automatic Controlvol 54 no 4 pp 861ndash865 2009

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


of time-varying descriptor system have no available researchresults especially when119864(119905) is time-variant So it is necessaryto study finite-time stability of the time-varying uncertaindescriptor system

The paper is divided into two parts firstly we deals withthe problem of finite-time stability of time-varying descriptorsystem We give the necessary and sufficient condition offinite-time stability and finite-time boundedness of systemwith time-varying function matrix 119864(119905) the unstable systemcan be controlled by the state feedback controllers We alsomake the corresponding research for time-varying uncertaindescriptor system Secondly we design a finite-time bound-edness state feedback controller for time-varying uncertaindescriptor system to make the close-loop system finite-timeboundedness

The paper is organized as follows In Section 2 we givesome results of finite-time stability and boundedness fortime-varying descriptor system In Section 3 some results oftime-varying uncertain descriptor system are provided InSection 4 a numerical example is presented to illustrate theefficiency of the proposed result

2 Finite-Time Control of Time-VaryingDescriptor Systems

21 Finite-Time Stable of Time-Varying Descriptor Systems

Definition 1 Time-varying matrix 119864(119905) is singular on timeinterval [0 119879] if there exists a 119905 isin [0 119879] such that rank119864(119905) lt

119899Consider the following time-varying descriptor systems

119864 (119905) (119905) = 119860 (119905) 119909 (119905) (1)

where 119909(119905) isin 119877119899 is the state 119860(sdot) is given continuous matrix-

valued function 119864(119905) isin 119877119899 is singular on time interval [0 119879]

Definition 2 The time-varying descriptor system (1) is saidto be finite-time stable with respect to (119888

1 1198882 119879 119877(119905)) with

positive definite matrix 119877(119905) and given three positive scalars1198881 1198882 119879 with 119888

1lt 1198882 if 119909119879(0)119864119879(0)119877(0)119864(0)119909(0) le 119888

1 then

119909119879

(119905)119864119879

(119905)119877(119905)119864(119905)119909(119905) lt 1198882 for all 119905 isin [0 119879]

Theorem 3 The following statements are equivalent

(i) system (1) is FTS respect to (1198881 1198882 119879 119877(119905))

(ii) For all 119905 isin [0 119879] Φ119879

(119905 0)119864119879

(119905)119877(119905)119864(119905) lt

(11988821198881)119864119879

(0)119877(0)119864(0) where Φ(119905 0) is the state tran-sition matrix and 119877(119905) is positive definite matrix

(iii) For all 119905 isin [0 119879] the differential Lyapunov inequalitywith terminal and initial conditions

(a) 119872(119905) lt 0 where

119872(119905) = 119860119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905) 119860 (119905)

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(2)

(b) 119877(119905) le 119875(119905) le 119875(0) lt (11988821198881)119877(0) 119877(119905) is positive

definite matrix admits a piecewise continuouslydifferentiable symmetric solution 119901(sdot)

Proof (ii) rArr (i) Let 119909119879(0)119864119879(0)119877(0)119864(0)119909(0) le 1198881 assume

119909(119905) = Φ(119905 0)119909(0) have

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

= 119909119879

(0)Φ119879

(119905 0) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (0)

lt1198882

1198881

119864119879

(0) 119877 (0) 119864 (0)

lt 1198882

(3)

Therefore system (1) is FTS(i) rArr (ii) by contradiction Let us assume exist119905 119909

119909119879

(119905)Φ119879

(119905 0) 119864119879

(119905) 119877 (119905) 119864 (119905)Φ (119905 0) 119909 (119905)

ge1198882

1198881

119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0)

(4)

Let 119909119879(0)119864119879(0)119877(0)119864(0)119909(0) = 1198881 exist120582 such that 119909(119905) = 120582119909(0)

then (4) implies that

119909119879

(0)Φ119879

(119905 0) 119864119879

(119905) 119877 (119905) 119864 (119905)Φ (119905 0) 119909 (0) ge 1198882 (5)

therefore

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

= 119909119879

(0)Φ119879

(119905 0) 119864119879

(119905) 119877 (119905) 119864 (119905)Φ (119905 0) 119909 (0)

ge 1198882

(6)

Obviously it contradicts the initial assumption that system (1)is FTS

(iii) rArr (i) Let119881(119905 119909) = 119909119879

(119905)119864119879

(119905)119875(119905)119864(119905)119909(119905) (119905 119909) =

119909119879

(119905)119872(119905)119909(119905)Then (a) implies that (119905 119909) is negative definite along the

trajectories of system (1)Now 119909

119879

(0)119864119879

(0)119877(0)119864(0)119909(0) le 1198881 then for a generic 119905

such that

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

le 119909119879

(119905) 119864119879

(119905) 119875 (119905) 119864 (119905) 119909 (119905)

lt 119909119879

(0) 119864119879

(0) 119875 (0) 119864 (0) 119909 (0)

le1198882

1198881

119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0)

le 1198882

(7)

(i) rArr (iii) By contradiction Because system (1) is FTS Let119911 = 120598119909 for a small 120598 gt 0 for all 119905 isin [0 119879] such that

119909119879

(0) 119864119879

(0) 119875 (0) 119864 (0) 119909 (0) le 1198881

997904rArr 119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905) + 1199112

2lt 1198882

(8)



119872(119905) = 1205982

119868 (9)

119877 (119905) = 119875 (119905) (10)


119909119879

(119905) 119864119879

(119905) 119875 (0) 119864 (119905) 119909 (119905)ge1198882

1198881

119909119879

(119905) 119864119879

(119905) 119877 (0) 119864 (119905) 119909 (119905)

(11)


120582119909(0)Then (11) implies

119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0) ge 1198882 (12)


119889

119889119905119909119879

(119905) 119864119879

(119905) 119875 (119905) 119864 (119905) 119909 (119905) = minus1205982

119909119879

(119905) 119909 (119905) (13)


119909119879

(119905) 119864119879

(119905) 119875 (119905) 119864 (119905) 119909 (119905) minus 119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0)

= minus1205982

1199092

2

(14)

Therefore

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

ge 119909119879

(119905) 119864119879

(119905) 119875 (119905) 119864 (119905) 119909 (119905)

= 119909119879

(0) 119864119879

(0) 119875 (0) 119864 (0) 119909 (0) minus 1205982

1199092

2

ge 1198882minus 1199112

2

(15)



119864 (119905) (119905) = 119860 (119905) 119909 (119905) + 119866 (119905) 120596 (119905) (16)






int

119879

0

120596119879

(119905) 120596 (119905) 119889119905 le 119889 119889 ge 0 (17)




with 1198881

lt 1198882 if 119909

119879

(0)119864119879

(0)119877(0)119864(0)119909(0) le 1198881 then

119909119879

(119905)119864119879

(119905)119877(119905)119864(119905)119909(119905) lt 1198882



[119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) 119876] lt 0 (18)

119877 (119905) le 119875 (119905) le 119875 (0) le 119877 (0) (19)

1198881+ 119889120582max (119876) lt 119888

2(20)


119872(119905) = 119860119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905) 119860 (119905)

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(21)

Proof Let 119881(119905 119909) = 119909119879

(119905)119864119879

(119905)119875(119905)119864(119905)119909(119905)We have

(119905 119909) = 119909119879

(119905)119872 (119905) 119909 (119905) + 120596119879

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) 119909 (119905)

+ 119909119879

(119905) 119864119879

(119905) 119875 (119905) 119866 (119905) 120596 (119905)

= (119909119879

(119905) 120596119879

(119905))

times [119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) 119876]

times (119909 (119905)

120596 (119905)) minus 120596

119879

(119905) 119876120596 (119905)

lt 0

(22)

Therefore

(119905 119909) lt 120596119879

(119905) 119876120596 (119905) (23)


119881 (119905 119909) lt 1198810+ 119889120582max (119876) (24)

From (19) we have

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

le 119881 (119905 119909) lt 119881 (0 119909) + 119889120582max (119876)

lt 119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0) + 119889120582max (119876)

lt 1198881+ 119889120582max (119876)

lt 1198882

(25)




119864 (119905) (119905) = 119860 (119905) 119909 (119905) + 119861 (119905) 119906 (119905) + 119866 (119905) 120596 (119905) (26)




119899times119899

119861(119905)) isin 119877119899times119898

119866(119905) isin 119877119899times119897



119906 (119905) = 119870 (119905) 119909 (119905) (27)


119888(119905) = (119860(119905) + 119861(119905)119870(119905)) Then the


119864 (119905) (119905) = 119860119888(119905) 119909 (119905) + 119866 (119905) 120596 (119905) (28)




[119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) minus119876119879

] lt 0 (29)


119870 (119905) = 119861minus1

(119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) (30)


119872(119905) = 119860119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905) 119860 (119905)

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(31)


[119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) 119876] lt 0 (32)

where

119872(119905) = (119860 (119905) + 119861 (119905)119870 (119905))119879

119875 (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119860 (119905) + 119861 (119905)119870 (119905))

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(33)


119872(119905) minus 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) lt 0

(34)


119872(119905) + 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus119879

(119905) 119866119879

(119905) 119875 (119905) 119864 (119905) lt 0

(35)


119872(119905) + 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus119879

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) lt 0

(36)



119864 (119905) (119905) = (119860 (119905) + Δ119860 (119905) 119909 (119905))

+ (119861 (119905) + Δ119861 (119905)) 119906 (119905) + 119866 (119905) 120596 (119905)

(37)





119899times119897



[Δ119860 (119905) Δ119861 (119905)] = 119867119865 [1198641(119905) 119864

2(119905)] (38)


119865119879

119865 le 119868 (39)




119870 (119905) = (119861 (119905) + 1198671198651198642(119905))minus1

times (119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) minus 1198671198651198641(119905))

(40)

where

119872(119905) = [119860 (119905) + Δ119860 (119905) + (119861 (119905) + Δ119861 (119905))119870 (119905)]119879

times 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905)

times [119860 (119905) + Δ119860 (119905) + (119861 (119905) + Δ119861 (119905))119870 (119905)]

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(41)



3 Numerical Example


119864 = [119905 0

0 0] 119860 = [

119905 0

0 minus1] 119861 = [

1 0

0 119905]

119866 = [1

minus119905]

(42)

Let 119877(119905) = 119875(119905) 119876 = 119905 119879 = 1 Exist 119875(119905) = [minus119905 0

0 119905] such that

119872(119905) = [minus21199053

minus 31199052

0

0 0] 119864

119879

(119905) 119875 (119905) 119866 (119905) = [minus1199052

0]

119866119879

(119905) 119875119879

(119905) 119864 (119905) = [minus1199052

0]

(43)


119870 (119905) = 119861minus1

(119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) = [minus119905 0

1 0]

(44)

4 Conclusions


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119872(119905) = 1205982

119868 (9)

119877 (119905) = 119875 (119905) (10)


119909119879

(119905) 119864119879

(119905) 119875 (0) 119864 (119905) 119909 (119905)ge1198882

1198881

119909119879

(119905) 119864119879

(119905) 119877 (0) 119864 (119905) 119909 (119905)

(11)


120582119909(0)Then (11) implies

119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0) ge 1198882 (12)


119889

119889119905119909119879

(119905) 119864119879

(119905) 119875 (119905) 119864 (119905) 119909 (119905) = minus1205982

119909119879

(119905) 119909 (119905) (13)


119909119879

(119905) 119864119879

(119905) 119875 (119905) 119864 (119905) 119909 (119905) minus 119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0)

= minus1205982

1199092

2

(14)

Therefore

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

ge 119909119879

(119905) 119864119879

(119905) 119875 (119905) 119864 (119905) 119909 (119905)

= 119909119879

(0) 119864119879

(0) 119875 (0) 119864 (0) 119909 (0) minus 1205982

1199092

2

ge 1198882minus 1199112

2

(15)



119864 (119905) (119905) = 119860 (119905) 119909 (119905) + 119866 (119905) 120596 (119905) (16)






int

119879

0

120596119879

(119905) 120596 (119905) 119889119905 le 119889 119889 ge 0 (17)




with 1198881

lt 1198882 if 119909

119879

(0)119864119879

(0)119877(0)119864(0)119909(0) le 1198881 then

119909119879

(119905)119864119879

(119905)119877(119905)119864(119905)119909(119905) lt 1198882



[119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) 119876] lt 0 (18)

119877 (119905) le 119875 (119905) le 119875 (0) le 119877 (0) (19)

1198881+ 119889120582max (119876) lt 119888

2(20)


119872(119905) = 119860119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905) 119860 (119905)

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(21)

Proof Let 119881(119905 119909) = 119909119879

(119905)119864119879

(119905)119875(119905)119864(119905)119909(119905)We have

(119905 119909) = 119909119879

(119905)119872 (119905) 119909 (119905) + 120596119879

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) 119909 (119905)

+ 119909119879

(119905) 119864119879

(119905) 119875 (119905) 119866 (119905) 120596 (119905)

= (119909119879

(119905) 120596119879

(119905))

times [119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) 119876]

times (119909 (119905)

120596 (119905)) minus 120596

119879

(119905) 119876120596 (119905)

lt 0

(22)

Therefore

(119905 119909) lt 120596119879

(119905) 119876120596 (119905) (23)


119881 (119905 119909) lt 1198810+ 119889120582max (119876) (24)

From (19) we have

119909119879

(119905) 119864119879

(119905) 119877 (119905) 119864 (119905) 119909 (119905)

le 119881 (119905 119909) lt 119881 (0 119909) + 119889120582max (119876)

lt 119909119879

(0) 119864119879

(0) 119877 (0) 119864 (0) 119909 (0) + 119889120582max (119876)

lt 1198881+ 119889120582max (119876)

lt 1198882

(25)




119864 (119905) (119905) = 119860 (119905) 119909 (119905) + 119861 (119905) 119906 (119905) + 119866 (119905) 120596 (119905) (26)




119899times119899

119861(119905)) isin 119877119899times119898

119866(119905) isin 119877119899times119897



119906 (119905) = 119870 (119905) 119909 (119905) (27)


119888(119905) = (119860(119905) + 119861(119905)119870(119905)) Then the


119864 (119905) (119905) = 119860119888(119905) 119909 (119905) + 119866 (119905) 120596 (119905) (28)




[119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) minus119876119879

] lt 0 (29)


119870 (119905) = 119861minus1

(119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) (30)


119872(119905) = 119860119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905) 119860 (119905)

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(31)


[119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) 119876] lt 0 (32)

where

119872(119905) = (119860 (119905) + 119861 (119905)119870 (119905))119879

119875 (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119860 (119905) + 119861 (119905)119870 (119905))

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(33)


119872(119905) minus 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) lt 0

(34)


119872(119905) + 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus119879

(119905) 119866119879

(119905) 119875 (119905) 119864 (119905) lt 0

(35)


119872(119905) + 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus119879

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) lt 0

(36)



119864 (119905) (119905) = (119860 (119905) + Δ119860 (119905) 119909 (119905))

+ (119861 (119905) + Δ119861 (119905)) 119906 (119905) + 119866 (119905) 120596 (119905)

(37)





119899times119897



[Δ119860 (119905) Δ119861 (119905)] = 119867119865 [1198641(119905) 119864

2(119905)] (38)


119865119879

119865 le 119868 (39)




119870 (119905) = (119861 (119905) + 1198671198651198642(119905))minus1

times (119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) minus 1198671198651198641(119905))

(40)

where

119872(119905) = [119860 (119905) + Δ119860 (119905) + (119861 (119905) + Δ119861 (119905))119870 (119905)]119879

times 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905)

times [119860 (119905) + Δ119860 (119905) + (119861 (119905) + Δ119861 (119905))119870 (119905)]

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(41)



3 Numerical Example


119864 = [119905 0

0 0] 119860 = [

119905 0

0 minus1] 119861 = [

1 0

0 119905]

119866 = [1

minus119905]

(42)

Let 119877(119905) = 119875(119905) 119876 = 119905 119879 = 1 Exist 119875(119905) = [minus119905 0

0 119905] such that

119872(119905) = [minus21199053

minus 31199052

0

0 0] 119864

119879

(119905) 119875 (119905) 119866 (119905) = [minus1199052

0]

119866119879

(119905) 119875119879

(119905) 119864 (119905) = [minus1199052

0]

(43)


119870 (119905) = 119861minus1

(119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) = [minus119905 0

1 0]

(44)

4 Conclusions


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119864 (119905) (119905) = 119860 (119905) 119909 (119905) + 119861 (119905) 119906 (119905) + 119866 (119905) 120596 (119905) (26)




119899times119899

119861(119905)) isin 119877119899times119898

119866(119905) isin 119877119899times119897



119906 (119905) = 119870 (119905) 119909 (119905) (27)


119888(119905) = (119860(119905) + 119861(119905)119870(119905)) Then the


119864 (119905) (119905) = 119860119888(119905) 119909 (119905) + 119866 (119905) 120596 (119905) (28)




[119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) minus119876119879

] lt 0 (29)


119870 (119905) = 119861minus1

(119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) (30)


119872(119905) = 119860119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905) 119860 (119905)

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(31)


[119872 (119905) 119864

119879

(119905) 119875 (119905) 119866 (119905)

119866119879

(119905) 119875119879

(119905) 119864 (119905) 119876] lt 0 (32)

where

119872(119905) = (119860 (119905) + 119861 (119905)119870 (119905))119879

119875 (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119860 (119905) + 119861 (119905)119870 (119905))

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(33)


119872(119905) minus 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) lt 0

(34)


119872(119905) + 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus119879

(119905) 119866119879

(119905) 119875 (119905) 119864 (119905) lt 0

(35)


119872(119905) + 119864119879

(119905) 119875 (119905) 119866 (119905) 119876minus119879

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) lt 0

(36)



119864 (119905) (119905) = (119860 (119905) + Δ119860 (119905) 119909 (119905))

+ (119861 (119905) + Δ119861 (119905)) 119906 (119905) + 119866 (119905) 120596 (119905)

(37)





119899times119897



[Δ119860 (119905) Δ119861 (119905)] = 119867119865 [1198641(119905) 119864

2(119905)] (38)


119865119879

119865 le 119868 (39)




119870 (119905) = (119861 (119905) + 1198671198651198642(119905))minus1

times (119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) minus 1198671198651198641(119905))

(40)

where

119872(119905) = [119860 (119905) + Δ119860 (119905) + (119861 (119905) + Δ119861 (119905))119870 (119905)]119879

times 119875 (119905) 119864 (119905) + 119864119879

(119905) 119875 (119905)

times [119860 (119905) + Δ119860 (119905) + (119861 (119905) + Δ119861 (119905))119870 (119905)]

+ 119879

(119905) 119875 (119905) 119864 (119905) + 119864119879

(119905) (119905) 119864 (119905)

+ 119864119879

(119905) 119875 (119905) (119905)

(41)



3 Numerical Example


119864 = [119905 0

0 0] 119860 = [

119905 0

0 minus1] 119861 = [

1 0

0 119905]

119866 = [1

minus119905]

(42)

Let 119877(119905) = 119875(119905) 119876 = 119905 119879 = 1 Exist 119875(119905) = [minus119905 0

0 119905] such that

119872(119905) = [minus21199053

minus 31199052

0

0 0] 119864

119879

(119905) 119875 (119905) 119866 (119905) = [minus1199052

0]

119866119879

(119905) 119875119879

(119905) 119864 (119905) = [minus1199052

0]

(43)


119870 (119905) = 119861minus1

(119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) = [minus119905 0

1 0]

(44)

4 Conclusions


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3 Numerical Example


119864 = [119905 0

0 0] 119860 = [

119905 0

0 minus1] 119861 = [

1 0

0 119905]

119866 = [1

minus119905]

(42)

Let 119877(119905) = 119875(119905) 119876 = 119905 119879 = 1 Exist 119875(119905) = [minus119905 0

0 119905] such that

119872(119905) = [minus21199053

minus 31199052

0

0 0] 119864

119879

(119905) 119875 (119905) 119866 (119905) = [minus1199052

0]

119866119879

(119905) 119875119879

(119905) 119864 (119905) = [minus1199052

0]

(43)


119870 (119905) = 119861minus1

(119905) 119866 (119905) 119876minus1

(119905) 119866119879

(119905) 119875119879

(119905) 119864 (119905) = [minus119905 0

1 0]

(44)

4 Conclusions


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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