Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 176375, 16 pages

doi:10.1155/2012/176375

Research Article

Stability and Stabilization of Impulsive Stochastic Delay Differential Equations

Kaining Wu and Xiaohua Ding

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

Received 6 February 2012; Revised 27 March 2012; Accepted 27 March 2012

Academic Editor: Zhijian Ji

Copyright © 2012 Kaining Wu and Xiaohua Ding. 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.

We consider the stability and stabilization of impulsive stochastic delay differential equations (ISDDEs). Using the Lyapunov-Razumikhin method, we obtain the sufficient conditions to guarantee the pth moment exponential stability of ISDDEs. Then the almost sure exponential stability is considered and the sufficient conditions of the almost sure exponential stability are obtained. Moreover, the stabilization problem of ISDDEs is studied and the criterion on impulsive stabilization of ISDDEs is established. At last, examples are presented to illustrate the correctness of our results.

1. Introduction

In recent years, the systems with stochastic or impulsive effects were studied by many authors due to their importance in many branches of science and industry, see [1–10] and references therein. In practice, a given system may be with stochastic, delay and impulsive effects simultaneously, so it is necessary to investigate the properties of impulsive stochastic delay differential equations.

There are a lot of papers discussing ISDDEs, for example, see [11–17] and the references therein. In [11, 13], the authors studied the stability of a class of impulsive delay differential equations where the impulsive effects are nonlinear. In [12], the stability of a nonlinear ISDDE was studied and the equivalent relation between the stability of the nonlinear stochastic differential delay system under impulsive control and that of a corresponding nonlinear stochastic differential delay system without impulses was established. In [14], the authors studied the stability of nonlinear impulsive stochastic differential equations in terms of two measures and the concept of perturbing Lyapunov functions is introduced to discuss stability properties. In [15], the th exponential stability and almost sure exponential stability were studied by the Lyapunov-Krasovskii functional method. In [16], the authors considered the th moment exponential stability by using an inequality and the propertied of M-cone. In [17], the authors studied the mean square exponential stability of ISDDEs by the formula for the variation of parameters and Cauchy matrix.

From the existing lectures, we can see the stability of ISDDEs is a main research direction. For ISDDEs, there are at least two questions on stability that need be answered, one is that if the stochastic delay differential equation (SDDE) without impulse is stable, what kinds of the impulses can the system tolerate so that it remains stable? The other is if a SDDE without impulse is unstable, what kind of impulsive effects should we take to make the ISDDEs be stable? The first one we call the question of stability, the second one we call the question of stabilization.

As we all know, the Lyapunov-Razumikhin method is a powerful tool to investigate the stability; however, to our best knowledge, there is few work on ISDDEs by using Lyapunov-Razumikhin method.

In this paper, we use the Lyapunov-Razumikhin method to answer the questions of stability and stabilization of ISDDEs, give the sufficient conditions ensuring the th moment exponential stability of ISDDEs, and present the criteria of almost sure exponential stability of ISDDEs. The Lyapunov-Razumikhin method does not require that the formal derivative of the Lyapunov function falls into a restriction in all time; it just need to satisfy the restriction under some situation; therefore, our results relax the restrictions in some existing lectures. At last, examples are given to illustrate the efficiency of our results.

2. Preliminaries

Let be a complete probability space with a natural filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). denotes the n-dimensional Euclidean space with the Euclidean norm . Let to denote the set of piecewise right continuous functions with the sup-norm . Take to denote the family of all bounded -measurable -valued random variables satisfying , and denote the family of all -measurable -valued random variables satisfying , where denotes the expectation with respect to the given probability measure .

In this paper, we consider the following impulsive stochastic delay differential system:

where , represents the impulsive perturbation of at time and satisfies , . The impulsive moments satisfy and . The functions and are continuous functions; is an -dimensional standard Brownian motion defined on .

The following initial value is imposed on system (2.1)

where .

As a standing hypothesis, we suppose that system (2.1) has a unique solution for any given initial value , and there exists an for any such that . Suppose is left continuous and right limitable. Moreover, we assume that , and for any positive integer , then system (2.1) admits a trivial solution.

Let denote the set of nonnegative functions on , which are twice continuously differential in and once in on .

If , define an operator associated with system (2.1) as follows:

where

Definition 2.1 (see [15, 16]). The trivial solution of system (2.1) is said to be th moment exponentially stable if there exist positive constants and such that for any initial value ,

Definition 2.2 (see [15]). The trivial solution of system (2.1) is said to be almost surely exponentially stable if there exists a positive constant such that for any initial value and ,

If the trivial solution of system (2.1) is th moment exponentially stable or almost surely exponentially stable, we also say the system (2.1) is th moment exponentially stable or almost surely exponentially stable.

3. Stability and Stabilization of ISDDEs

In this section, we establish the criteria of th moment exponential stability for system (2.1) by using the Lyapunov-Razumikhin technique, and the almost sure exponential stability is also considered. Moreover, the stabilization theorem is presented for system (2.1). The results show that impulses may change the stability of a given system. Some techniques used in the proof are motivated by the paper [5].

Theorem 3.1. Assume that there exist a function and positive constants , , and is any positive integer, such that the following conditions hold:

(1) for any and ;(2) for all in whenever for any , ;(3);(4), , and .

Then, for any and , the solution of system (2.1) satisfies

where , that is, system (2.1) is th moment exponentially stable.

Proof. For a given , let and write for the simplicity.

We claim

for any , where .

Let

It is easy to check is continuous and differentiable in , and

for .

To verify (3.2), we just need to show that for all .

We first show that for .

For , we have ; using condition 1, we can get . Let be an arbitrary positive constant; we claim

for .

If (3.5) is not true, then there must exist a , such that , which implies that there exists a such that and for being the continuity of in . Noting

using the fact , , for , we have, for any ,

By virtue of condition 2, we can obtain ; then

which contradicts the definition of . So we get for all . Let ; we obtain for .

Now, assume for . In view of condition 3, we have

Next, we will show, for arbitrary ,

For the sake of contradiction, suppose (3.10) is not true. Define

From (3.9), we have . The continuity of in yields and for .

Since , then, for ,

In view of condition 2, we obtain , then

which contradicts the definition of . Therefore, for all . Let ; we have for . Thus, by induction, we obtain holds for ; hence

Then by condition 1, we have

This completes the proof.

The following theorem states the almost sure exponential stability of system (2.1). In the proof, the classical method used in [4] is borrowed and this method was also adopted in paper [15].

Theorem 3.2. Suppose all of the conditions of Theorem 3.1 are satisfied and in addition . If there exist positive constants and such that and for all

then for all ,

Proof. From system (2.1), we can get, for ,

Using Hölder’s inequality, condition (3.16), and Theorem 3.1, we derive that

In virtue of Burkholder-Davis-Gundy inequality, we have

where is a positive constant dependent on only. One can then show in the same way as in (3.19) that

Substituting (3.15), (3.19), and (3.21) into (3.18) yields

where

When , keeping and (3.15) in mind, we can get

Taking , we have

We now show that (3.25) implies the required (3.17).

Let be an arbitrary constant satisfying . By virtue of (3.25) and Markovian inequality, we have

In view of Borel-Cantelli lemma, we can obtain that for almost all ,

holds for all but finitely many . Hence there exists an , for all but a -null set, such that (3.27) holds when . Then we have, for almost all , if , ,

that is,

and (3.17) follows by letting .

In the following, we give two corollaries.

Corollary 3.3. Assume there exist positive constants and a function such that

(1) for all ;(2) for all , where ;(3);(4), , and .

Then, the trivial solution of system (2.1) is th moment exponentially stable.

Furthermore, if there exists a positive constant such that and the following inequality holds for any :

then the trivial solution of system (2.1) is almost surely exponentially stable.

Proof. Take . Obviously, we just need to verify the condition 2 of Theorem 3.1.

If and , , the following inequality holds

then

Condition 2 of Theorem 3.1 is verified, then the th moment exponential stability for the trivial solution of system (2.1) is obtained. The almost sure exponential stability is followed directly by virtue of Theorem 3.2.

The 2th moment exponential stability; is also called mean square exponential stability, the following corollary presents the criteria of mean square exponential stability of system (2.1).

Corollary 3.4. For system (2.1), assume there exist positive constants such that

(1), and , and ;(2);(3), , .

Then the trivial solution of system (2.1) is mean square exponentially stable.

Proof. Let ; then

The conditions of Corollary 3.3 are easily to be verified, and the required result can be obtained.

Now we are on the position to state the stabilization theorem.

Theorem 3.5. Assume there exist a function and positive constants , , , , , such that

(1), for any and ;(2), for all , whenever , for , where ;(3), where ;(4) and .

Then the trivial solution of system (2.1) is th moment exponentially stable.

Remark 3.6. From condition 2, we can see that the formal derivative of can be positive since is a positive constant; this means that the original system without impulses may be unstable. Therefore, this theorem is called the stabilization theorem.

Proof. Let be a solution of (2.1) with , and write for simplicity. Choose such that

We will show, for any positive integer ,

We first show that

From condition 1 and (3.35), we have, for ,

If (3.37) is not true, then there must be a such that

where .

Then there exists a such that

and there is a such that

Then we have, for any ,

From condition 2, we have

for . Since , we get

that is,

then

which is in conflict with (3.35). We obtain that (3.37) holds, that is, (3.36) holds when .

Now we assume that (3.36) holds when , is a positive integer and , then

especially,

From conditions 3, 4 and (3.48), we have

Now we will show that (3.36) holds when , that is,

If (3.50) is not true, we define

From (3.49), we know ; by the continuity of in , we get

From (3.49), we have

there must be a such that

Since , and , when , we get . From (3.48) and (3.53), when , we have

From condition 2, we get

But

we get

Then (3.36) holds when . By induction, we have that (3.36) holds, and

From condition 1, we have

where . This completes the proof.

4. Applications and Examples

In this section, we consider a nonlinear impulsive stochastic delay differential system. We present the stability criterion and stabilization criterion for this system, then we illustrate the correctness of our results using the numerical experiments.

The following nonlinear impulsive stochastic delay differential system is considered:

By virtue of Corollary 3.4, we can get the following corollary directly.

Corollary 4.1. Assume there exist positive constants and such that the impulsive moments satisfy , and the following inequalities hold:

then the trivial solution of system (4.1) is mean square exponentially stable and almost surely exponentially stable.

Using Theorem 3.5 and taking , we can easily get the following stabilization corollary for system (4.1).

Corollary 4.2. Assume there exist constants satisfing

(1)there exists a constant such that , where ;(2) and .

Then the trivial solution of system (4.1) is th moment exponentially stable.

Now let us illustrate the correctness of Corollaries 4.1 and 4.2.

Let . Take when and . Let and ; it is easy to verify the conditions of Corollary 4.1 are satisfied. Then the trivial solution of system (4.1) is mean square exponentially stable. The mean square exponential stability is showed in Figure 1. This illustrates the correctness of Corollary 4.1.

Figure 1: Mean square exponential stability of system (4.1).

Now we consider the stabilization of system (4.1). Take . Let when . It is easy to see that the trivial solution of system (4.1) without impulsive effects is not mean square stable, see Figure 2. Then we take and ; it can be verified that the conditions of Corollary 4.2 are satisfied; the the trivial solution of system (4.1) is mean square exponentially stable, see Figure 3.

Figure 2: Nonstability of system (4.1) without impulse.

Figure 3: Mean square exponential stability of system (4.1).

Acknowledgments

This paper was supported by the National Natural Science Foundation of China under Grant 11026189 and by the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF.201014).

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