# [IEEE 2010 Third International Workshop on Advanced Computational Intelligence (IWACI) - Suzhou, China (2010.08.25-2010.08.27)] Third International Workshop on Advanced Computational Intelligence - Exponential stability of singular impulsive systems with time-varying delays

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1

Exponential Stability of Singular Impulsive Systems withTime-varying Delays

Zhiguo Yang and Zhichun Yang

Abstract In this article, a model of singular systems involv-ing time-varying delays and impulses is considered. By estab-lishing a singular delay differential inequality with impulsiveinitial condition and using the property of -matrix, some newsufficient conditions ensuring the global exponential stability ofthe zero solution of singular systems are obtained. The resultscan extend and improve those of the earlier publications. Anexample is given to illustrate the theory.

I. INTRODUCTION

S INGULAR systems (known as semistate systems, dif-ferential algebraic systems, generalized state-space sys-tems, etc.) have been of interest in the literature since theyhave many important applications in, for example, circuitsystems, robotics, aircraft modelling, social, biological, andmultisector economic systems, dynamics of thermal nuclearreactors singular perturbation systems, and so on. Manyinteresting results in dealing with singular systems have beenreported [1][2]. Furthermore, singular systems with delayshave been extensively studied in the past years due to thefact that delayed singular systems can even accurately depictthe evolutionary processes [3][4][5].

However, besides delay effects, impulsive effects likewiseexist in a wide variety of evolutionary processes in whichstates are changed abruptly at certain moments of time,involving such fields as medicine and biology, economics,mechanics, electronics and telecommunications, etc.. Aspointed out by [6], many sudden and sharp changes occurinstantaneously in singular systems, in the form of impulses.Therefore, it is very important, and indeed necessary, to studysingular impulsive systems.

The stability analysis plays an important role in the studyof singular system theory. Various stability properties ofsingular systems have been investigated in papers [4][5][6].But stability investigation to singular systems with impulsesand delays have not yet been fully developed [7]. Especially,there are few papers dealing with the exponential stability ofsingular impulsive systems with time-varying delays. Thisprompted us to discuss this problem.

Zhiguo Yang is with the College of Mathematics and SoftwareScience, Sichuan Normal University, Chengdu, 610068, China (email:zhiguoyang@126.com). Zhichun Yang is with the Department of Mathe-matics, Chongqing Normal University, Chongqing 400047, China (email:zhichy@yahoo.com.cn).

This work was supported by National Natural Science Foundation ofChina under the grant No. 10926033, 10971147 and 10971240, A ProjectSupported by Scientific Reserch Fund of SiChuan Provincial EducationDepartment (08zb026), Key Research Project of Sichuan Normal University,Natural Science Foundation of Chongqing under Grant CSTC2008BB2364.

More specifically, in this paper, we will obtain some newsufficient conditions ensuring the global exponential stabilityof the zero solution of a singular impulsive system with time-varying delays by establishing a singular delay differentialinequality with impulsive initial condition and using theproperty of -matrix. The results extend and improve thoseof the earlier publications.

II. MODEL AND PRELIMINARIES

To begin with, we introduce some notations and recallsome basic definitions. Let = {1, 2, . . . , }. For , or , , ( , > , < )means that each pair of corresponding elements of and satisfies the inequality (, >,

2

() and () denote the distributional derivatives [8]of the functions (), (), () and () respectively.Without loss of generality, we may assume that

() = 1 +

=1

( ),

() = 1 +=1

( ),

() = 1 +

=1

( ), ,

where () is the Dirac impulse function. , and are constants. The impulsive moments ( = 1, 2, . . . )satisfy 1 < 2 < . . . and lim+ = +.

Throughout this paper, we assume that for any [[, 0], ], the system (1) has at least one solutiondenoted by (, 0, ) or (). Moreover, we assume that(0) = (0) = 0, for the stability purpose of thispaper. Then the system (1) admits the zero solution () 0.

Definition 1: The zero solution of the singular system(1) is said to be globally exponentially stable if there existconstants > 0 and > 0 such that for any solution(, 0, ) with the initial condition [[, 0], ],

(, 0, ) (0), 0, (3)where (, 0, ) = max1{(, 0, )}, =max1{sup0 ()}.

For an -matrix [9], we define

()= { > 0, > 0}.

Lemma 1: [9] For an -matrix , () is nonemptyand satisfies,

11 + 22 (),for any 1, 2 > 0, 1, 2 ().

III. SINGULAR DELAY DIFFERENTIAL INEQUALITY

For the singular impulsive system (1), we need to estimateevery part on [, +1) with its initial function on [, ]for = 1, 2, . . . . It is therefore difficult to obtain the estimate(3). To overcome these difficulties we establish the followingsingular delay differential inequality with impulsive initialcondition.

Theorem 1: Let = () and 0 for = , = () 0 and = ( + ) be an -matrix. For (0,+), let () = (1(), . . . , ()) [[0, ),

] be a solution of the following delay differen-tial inequality with the initial condition () [[0 , 0],

],

+() ()(() +[()] ), (0, ), (4)where = diag{1, . . . , } 0, +() is the upper rightderivative of (). () > 0 satisfies sup0

()

< . Then()

0

(), [0, ), (5)

provided that the initial condition satisfies

() 0

(), 0 0, (6)

where = (1, . . . , ) () and the positive constant satisfies the following inequality

[ + + ] < 0. (7)

Proof: Since is an -matrix, there exists a vector () such that

> 0 or [ +] < 0.

By using continuity, we obtain that there must exist a positiveconstant satisfying the inequality (7), that is,

=1

[ + ] < , . (8)

We at first shall prove that for any positive constant

() (1 + ) 0

() = (),

[0, ), . (9)If inequality (9) is not true, by (6), there must be constant

1 (0, ) and some integer such that(1) = (1),

+(1) (1), (10)

() (), [0 , 1], . (11)By using (4), (8), (9), (10), (11) and 0 ( = ),

0, we obtain that

+(1) (1)

=1

[(1) + [(1)] ]

(1)

=1

[(1 + ) 10 ()

+(1 + ) 10 ()]

(1)(1 + ) 10

()

=1

[ + ]

< (1)(1 + ) 10

(). (12)

If > 0, the inequality (12) yields that

+(1) < (1),

which contradicts the inequality in (10). If = 0, theinequality (12) yields that 0 < 0. It is a contradiction.

Thus (9) holds. Therefore, letting 0, we have

() 0

(), [0, ), .

The proof is completed.

631

3

IV. GLOBAL EXPONENTIAL STABILIT

In this section, we will obtain the sufficient conditionensuring global exponential stability of the zero solution ofthe singular system (1) by using the property of -Matrixand employing Theorem 1. Here, we firstly introduce thefollowing assumptions.(1) There exist nonnegative constants and such

that continuous functions () and () satisfy

() , () , , .

(2) Let = ( +) be an -matrix, where

= + []+, = []+, = (),

= diag{1, . . . , } > 0, = (),

= diag{1, . . . , }, = diag{1, . . . , }.

(3) Let = [+]+ []+ []+, 1 exist and1 0, = 1, 2, . . . , where

= diag{1, . . . , } 0,

= diag{1, . . . , },

= diag{1, . . . , }.

(4) There exists a positive constant such that

ln 1 < , = 1, 2, . . . . (13)

Where the positive constant satisfies

[ + + ] < 0, (14)

for a given () and 1 satisfies

(1 + 1 []+ []+ ),

= 1, 2, . . . , (15)

where = diag{1, . . . , }.Theorem 2: Assume that (1) - (4) hold. Then the zero

solution of the singular system (1) is globally exponentiallystable and the exponential convergence rate is equal to .

Proof: By the property of the Dirac impulse function (),we see that the derivative () of () exists on (1, ).Thus the system (1) becomes

() = () +

=1

(())

+

=1

(( ())),

, (1, ). (16)

From (16) and (1), it is easy to obtain

+() ()+

=1

(())

+

=1

(( ()))

()+

=1

()

+

=1

( ()),

, (1, ).

That is,

+[()]+ [()]+ +[()]+ , (1, ), = 1, 2, . . . . (17)

Since is an -matrix, from Lemma 1 we may choosea vector = (1, . . . , ) () and =(1, . . . , 1) such that

> 0 or [ +] < 0.

By using continuity, we obtain that there must exist a positiveconstant satisfying the inequality (14).

For the initial condition (0 + ) = (), [, 0],where [[, 0], ] and 0 (without loss ofgenerality, we assume 0 < 1), we can get

[()]+ (0), 0 0. (18)

Then, all conditions of Theorem 1 are satisfied by (17),(18), Condition (2), () 1 and = . So

[()]+ (0), 0 < 1. (19)

Suppose that for all = 1, . . . , , the inequalities

[()]+ 0 . . . 1(0),1 < , (20)

hold, where 0 = 1.On the other hand, (1) implies that

[() ( )]=

[()()

+

=1

(())()

+

=1

(( ()))()], , (21)

632

4

where > 0 is sufficiently small, as 0+, which yields[() ( )]

= () +

=1

(())

+

=1

(( ())), . (22)

This, together with (1), yields that

( + )()

= ( ) +

=1

(())

+

=1

(( ()))

( )+

=1

()

+

=1

( ())

( )+

=1

()

+

=1

[()] , . (23)

That is,

[ + ]+[()]

+

[( )]+ + []+ []+[()]++[]+ []

+[()]+ . (24)

Then, we have

([ + ]+ []+ []+)[()]+

[( )]+ + []+ []+[()]+ (25)yielding, together with (3), that

[()]+ 1 [( )]+ + 1 []+ []+[()]+ . (26)

Then, from (15), (20) and (26)

[()]+ (1 + 1 []+ []+ )

0 . . . 1(0) 0 . . . 1(0). (27)

This, together with (20) and 1, = 1, 2, . . . , leads to[()]+ 0 . . . 1(0)

= 0 . . . 1(0)(), [ , ]. (28)

By Lemma 1 again, the vector

0 . . . 1(0) ().

Then, all conditions of Theorem 1 are satisfied again by (17),(28), Condition (2), () 1 and = . So[()]+ 0 . . . 1(0)()

= 0 . . . 1(0), < +1. (29)

By the mathematical induction, we can conclude that

[()]+ 0 . . . 1(0),1 < , = 1, 2, . . . . (30)

Noticing that (1) by (13), we can use (30) toconclude that

[()]+ (10) . . . (12)(0) (0)(0)= ()(0),

[1, ), = 1, 2, . . . .So,

[()]+ ()(0), 0.This implies that the conclusion of the theorem holds.

Remark 1: Condition (3) must hold when > 0 and is sufficiently small. In fact, the positive diagonal matrix[ + ]

+ is an -matrix when > 0. So, = [ +]

+ []+ []+ is also an -matrix for sufficientlysmall . Then,

1 exists and

1 0.

Remark 2: When () and = , some globalexponential stability criteria for the singular impulsive system(1) have been established in [7]. However, from assumption1) in Theorem 2 and Corollary 4.1 of [7], the impulsivecoefficient max1

5

with () [11]. In [11], () and () must satisfy() 0 and () 0, , . However, inCorollary 1 above, these restrictive conditions are removed.

V. ILLUSTRATIVE EXAMPLE

The following illustrative example will demonstrate theeffectiveness of our results.

Example 1: Consider the following impulsive singularsystem with time-varying delays

1() = 61()1() + 31( 11())1() 22( 12())2(),

0 = 52()2() 1( 21())1()+3 cos(2( 22()))2(),

(32)

where () = sin((+ )) 1 = and

() = 1+=1

( ), () = 1+=1

( ),

for , = 1, 2, where () is the Dirac impulse function andthe impulsive moments ( = 1, 2, . . . ) satisfy: 1 = 0.3,1 < 2 < . . . and lim+ = +.

Thus the parameters of Conditions (1) - (4) are asfollows:

=

(1 00 0

), =

(1 00 1

),

=

(6 00 5

), =

(0 00 0

),

= diag{, }, = diag{, },

=

(3 21 3

), = []+ =

(3 21 3

),

= + []+ = =( 6 0

0 5),

= ( +) =(

3 21 2

),

= [ + ]+ []+ []+

=

( 1 + 6 00 5

), (33)

which yield that is an -matrix, 1 exists when = 0and = 16 . Furthermore,

() = {(1, 2) > 0 232 < 1 < 22},

1 =

(1

1+6 00 15

).

Let = (1, 1) () and = 0.1 which satisfies theinequality ( + + ) < 0. So can satisfy theinequality

(1 + 1 []+ []+ )

= (1 + 1 []

+0.1), = 1, 2, . . . , (34)

provided that

max{1 + 50.1

1 + 6 ,40.15 },

= 1, 2, . . . . (35)

Case 1: Let = 1, = 0.2, = 0.5 and 1 = 6, then satisfies the inequality (35) and 1,

ln 1 =

ln 0.5

6 0.084 = < ,

= 1, 2, . . . .

Clearly, all conditions of Theorem 2 are satisfied. So the zerosolution of the singular system (32) is globally exponentiallystable and the exponential convergence rate is equal to 0.016.

Case 2: Let = 0.1, = 1, = 1 and 1 = (0, ), then satisfies the inequality (35) and 1,

ln 1 = 0 = < , = 1, 2, . . . .

So by Theorem 2, the zero solution of the singular system(32) is globally exponentially stable and the exponentialconvergence rate is equal to 0.1.

Remark 4: Even if () , the conditions in Theorem4.1 and Corollary 4.1 of [7] are not satisfied since in Case 1 =

0.2 , , while max1 1 in [7].

REFERENCES

[1] J.D. Aplevich, Implicit Linear Systems. Springer-Verlag, New York, US,1991.

[2] S.L. Campbell, Singular Systems of Differential Equations. Pitman, NewYork, US, 1982.

[3] Y. Li and Y. Liu, Bifurcation on stability of singular systems withdelay, International Journal of Systems Science, vol. 30, no. 6, pp.643-649, 1999.

[4] S. Xu, P.V. Dooren, R. Stefan and J. Lam, Robust stability andstabilization for singular systems with state delay and parameter un-certainty, IEEE Transactions on Automatic Control, vol. 47, pp. 1122-1128, 2002.

[5] S. Xu, J. Lam and C. Yang, Robust control for uncertain singularsystems with state delay, International Journal Robust NonlinearControl, vol. 13 pp. 1213-1223, 2003.

[6] Z. Guan, J. Yao and D.J. Hill, Robust control of singularimpulsive systems with uncertain perturbations, IEEE Transactions onCircuits and Systems - II: Express Briefs, vol. 52, pp. 293-298, 2005.

[7] Z. Guan, C.W. Chan, Andrew Y. T. Leung and G. Chen, Robust sta-bilization of singular-impulsive-delayed systems with nonlinear pertur-bations, IEEE Transactions on Circuits and Systems - I: FoundmentalTheory and Application, vol. 48, pp. 1011-1019, 2001.

[8] S.G. Deo and S.G. Pandit, Differential Systems Involving Impulses.Springer-Verlag, New York, US, 1982.

[9] Z. Yang, D. Xu and L. Xiang, Exponential p-stability of impulsivestochastic differential equations with delays, Phys Lett A , vol. 359,pp. 129-137, 2006.

[10] Z. Guan, J. Lam and G. Chen, On impulsive autoassociative neuralnetworks, Neural Networks , vol. 13, pp. 63-69, 2000.

[11] Z. Guan and G. Chen, On delayed impulsive Hopfield neural net-works, Neural Networks, vol. 12, pp. 273-280, 1999.

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