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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 592036, 15 pages
http://dx.doi.org/10.1155/2010/592036
Research Article

Stability and Stabilization of Impulsive Stochastic Delay Difference Equations

Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China

Received 7 December 2009; Accepted 16 January 2010

Academic Editor: Leonid Shaikhet

Copyright © 2010 Kaining Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

When an impulsive control is adopted for a stochastic delay difference system (SDDS), there are at least two situations that should be contemplated. If the SDDS is stable, then what kind of impulse can the original system tolerate to keep stable? If the SDDS is unstable, then what kind of impulsive strategy should be taken to make the system stable? Using the Lyapunov-Razumikhin technique, we establish criteria for the stability of impulsive stochastic delay difference equations and these criteria answer those questions. As for applications, we consider a kind of impulsive stochastic delay difference equation and present some corollaries to our main results.

1. Introduction

In recent years, stochastic delay difference equations (SDDEs) have been studied by many researchers; a number of results have been reported [17]. In these literatures, stability analysis stays on the focus of attention; see [1, 2, 46] and the references therein. As we all know, when we adopt an impulsive strategy to an SDDE, the stability of the SDDE may be destroyed or strengthen. Impulsive phenomena exist widely in the real world; therefore, it is important to study the stability problem for SDDEs with impulsive effects [810], that is to say, the stability problem for impulsive stochastic delay difference equations (ISDDEs).

For SDDEs, when we take impulsive effects into account, we have at least two problems to deal with. Problem 1. When a SDDE is stable, what kind of impulsive effect can the system tolerate so that it remain stable? Problem 2. If the SDDE is unstable, then what kind of impulsive effect should be taken to make the system stable? Problems 1 and 2 are called the problem of impulsive stability and the problem of impulsive stabilization, respectively.

As well known, Lyapunov-Razumikhin technique is one of main methods to investigate the stability of delay systems [11, 12]. There are little papers on the stability of ISDDEs [13, 14], and up to our knowledge, there is no paper on the stability of ISDDEs using Lyapunov-Razumikhin technique. In this paper, we study the stability of ISDDEs by Lyapunov-Razumikhin technique. We establish criteria for the -moment exponential stability; these criteria present the answers to Problems 1 and 2. As for applications, we consider a kind of ISDDE and present some corollaries to our main theorems.

2. Preliminaries

In the sequel, denotes the field of real numbers, and represents the natural numbers. For some positive integer and , let and . Given a matrix , denotes the norm of induced by the Euclidean vector norm. Let . Given a positive integer , we define for any . Let be a complete probability space and let be a nondecreasing family of sub--algebra of , that is, for .

Consider the impulsive stochastic delay difference equations of the form

where , , and represents the delay in system (2.1), . is defined by for any . are -adapted mutually independent random variables and satisfy , , where denotes the mathematical expectation. . Impulsive moment satisfies: , and as . Let .

Assume that , , and , then system (2.1) admits the trivial solution. We also assume there exists a unique solution of system (2.1), denoted by , for any given initial data .

Definition 2.1. One calls the trivial solution of system (2.1) -moment exponentially stable if for any initial data there exist two positive constants and , such that for all , , the following inequality holds:

If the trivial solution of system (2.1) is -moment exponentially stable, then we also call the system (2.1) -moment exponentially stable.

3. Main Results

In this section, we will establish two theorems on -moment exponential stability of system (2.1); these theorems give the answers to Problems 1 and 2.

First, we present the theorem on impulsive stability. The technique adopted in the proof is motivated by [15].

Theorem 3.1. Assume that there exist a positive function for system (2.1) and positive constants , where , , such that.() for any and .() For , any , whenever .() For , some , whenever , where .(), where and .(), .Then for any initial data , That is the trivial solution of system (2.1) that is moment exponentially stable.

Proof. Let . For any , , , define then .
Next, we will show that, for any ,
For a given , we have two situations to contemplate: and .Case 1 (). Under this situation, we have , then Using condition () and noticing , we obtain Multiplying both sides by and rearranging yield Then we get which implies that Case 2 (). Making use of the definition of and , noticing that for any , we have Under condition (), the above inequality implies that Multiplying both sides by , we have Thus which is the desired assertion.
When , under condition () and using the definition of , we get By induction and taking (3.3) into account, when , for all , we have which yields By virtue of condition (), The desired result follows when we take condition into account.

Now, we are in position to state the theorem on impulsive stabilization. The method used in the proof is motivated by [16].

Theorem 3.2. Assume that there exist a function for system (2.1) and constants , , , , and natural number , such that the following conditions hold. () for any and .() For , any , whenever , where .(), where .(), .Then for any initial data there exists positive constant ; for any , the following inequality holds: that is, the trivial solution of system (2.1) is -moment exponentially stable.

Proof. Choose such that We will show that, for any , ,
Write for the sake of brevity.
First we will show that, for any , Obviously, when , If (3.20) is not true, then there exists such that And when , At the same time there exists such that and when ,
Note that there may not exist the natural number that satisfies such that (3.22) holds. However, we claim that there must be a natural number satisfing such that (3.22) holds. If not, we have ; then we get Obviously, Under condition () we get That is which contradicts with (3.23). Then there must be an satisfing such that (3.22) holds.
For any , By virtue of (), for any , and for , we have Making use of (3.28), we get Taking (3.3) into account, the above inequality yields which implies that It contradicts with (3.18); then (3.20) holds, that is, (3.19) holds for .
Assume that (3.19) holds for , that is, when , , Under conditions () and (), we have
Now we will show that, when ,
If (3.35) is not true, then there must be an , such that and for At the same time, there exists an such that And, when , If there does not exist an satisfing such that (3.39) holds, then . Obviously, for any , . Using condition () yields , that is, which contradicts with the definition of ; then there exists at least one number satisfing such that (3.39) holds.
For and , we have which implies that, under condition (), Obviously, . Using condition () again, we get since , , we have, under condition () Then which conflicts with the definition of . Then (3.19) holds for .
By induction, we know that (3.19) holds for , .
Using condition (), for any , , we have That is the desired result.

4. Applications

In this section, we consider a kind of impulsive stochastic delay difference equation as follows:

Using the obtained results, we present three corollaries for system (4.1).

Corollary 4.1. Assume that conditions , , and of Theorem 3.1 hold, but conditions and are replaced with the following conditions:() There exist constants and , , , such that If , then the trivial solution of system (4.1) is -moment exponentially stable.

Proof. Let be a solution of system (4.1). Take . It is easy to see that, under the conditions , and , If for any , it follows from the condition () that Then condition () of Theorem 3.1 follows under (4.3).
Let ; using inequality (4.3), we get in Theorem 3.1: .
Now we assume that for some ; by virtue of () and inequality (4.3), Then condition () of Theorem 3.1 follows, which completes the proof.

From the above proof, we know that constant in Theorem 3.1 equals to

Corollary 4.2. Assume that conditions (), (), and () of Theorem 3.1 hold, but conditions () and () are replaced with the following condition.() There exists a constant such that Then the trivial solution of system (4.1) is -moment exponentially stable.

Proof. Let be a solution of system (4.1). Take Since , we have and For any , assume that ; by virtue of condition (), we get that is, condition () of Theorem 3.1.
Since we have . Under condition (), for any , we get that is condition () of Theorem 3.1.

From the above proof, we know that constant in Theorem 3.1 equals to

Then constant in Theorem 3.1 equals

Now, we present a corollary of Theorem 3.2 which establishes a criterion of mean square exponential stability for system (4.1).

Corollary 4.3. Assume that there exist positive constants , and where is a natural number and , such that system (4.1) satisfies the following.(1) where are positive constants, and (2), for any , and . The impulsive moments satisfy Then, for any initial data , the solution of system (4.1) satisfies That is to say, the trivial solution of system (4.1) is mean square exponentially stable.

Proof. Let , then, Assume that holds for any , then The other conditions of Theorem 3.2 are easy to be verified and the conclusion of this corollary now follows.

5. Examples

Now we study some examples to illustrate our results.

We consider a linear impulsive stochastic delay difference equation as following:

First we take , , , , , , and . By virtue of Corollary 4.1, taking , we can get the mean square exponential stability of (5.1). The stability is shown in Figure 1.

592036.fig.001
Figure 1: Mean square exponential stability of (5.1): , , , , , .

Now we take , , , and in (5.1) without impulsive effects. It is easy to see that equation is unstable. This property is shown in Figure 2. Then we take an impulsive strategy: , . In light of Corollary 4.3, we can see that equation is mean square exponentially stable. The stability is shown in Figure 3.

592036.fig.002
Figure 2: Instability without impulsive effects of (5.1): , , , .
592036.fig.003
Figure 3: Mean square exponential stability of (5.1): , , , , , .

It should be pointed that the conditions of Corollary 4.3 are sufficient but not necessary. If we take , , , , and and , then it is not difficult to show that the conditions of Corollary 4.3 are not satisfied again, but under this situation, the equation is still stable. The stability is shown in Figure 4.

592036.fig.004
Figure 4: Stability of (5.1): , , , , , .

6. Conclusions

In this paper, we considered the -moment exponential stability for impulsive stochastic delay difference equations. Using the Lyapunov-Razumikhin method, we established criteria of -moment exponential stability and these criteria presented the answers for the problem of impulsive stability and the problem of impulsive stabilization. As for applications, we considered a kind of impulsive stochastic delay difference equation and obtained three corollaries for our main theorems. The results we got may work in the study of stability of numerical method for the impulsive delay differential equations.

Acknowledgments

This work was supported by the Scientific Foundations of Harbin Institute of Technology at Weihai under Grants no.HIT(WH)20080008 and no.HIT(WH)2B200905. The authors would like to give their thanks to Professor Leonid Shaikhet for his valuable comments and to the referees who suggest that the authors give examples to illustrate the results.

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