Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 429568, 13 pages

doi:10.1155/2012/429568

Research Article

Stability of Stochastic Reaction-Diffusion Systems with Markovian Switching and Impulsive Perturbations

Yanbo Li1,2 and Yonggui Kao¹

¹Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China

²School of Mathematical Sciences, Guangxi Teachers Education University, Guangxi 530001, China

Received 4 September 2012; Accepted 23 September 2012

Academic Editor: Alexei Mailybaev

Copyright © 2012 Yanbo Li and Yonggui Kao. 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.

This paper is devoted to investigating mean square stability of a class of stochastic reaction-diffusion systems with Markovian switching and impulsive perturbations. Based on Lyapunov functions and stochastic analysis method, some new criteria are established. Moreover, a class of semilinear stochastic impulsive reaction-diffusion differential equations with Markovian switching is discussed and a numerical example is presented to show the effectiveness of the obtained results.

1. Introduction

Markovian jump systems, introduced by Krasovskiĭ and Lidskiĭ [1] in 1961, have received increasing attention, see [2–15] and references therein. Shi and Boukas [3] have probed control for Markovian jumping linear systems with parametric uncertainty. Zhang et al. [4–6] have discussed markovian jump linear systems with partly unknown transition probability. Mao et al. [7–13] have established a number of stability criteria for stochastic differential equations with Markovian switching. However, impulsive perturbations have not been included in the above results.

In fact, impulsive effects widely exist in many fields, such as medicine and biology, economics, mechanics, electronics, and telecommunications [16–19]. Recently, impulsive stochastic differential equations have attracted more and more researchers [20–27]. L. Xu and D. Xu [20] have investigated mean square exponential stability of impulsive control stochastic systems with time-varying delay. Li [23] has obtained the attracting set for impulsive stochastic difference equations with continuous time. Pan and Cao [24] have considered exponential stability of impulsive stochastic functional differential equations. Zhang et al. [25] have studied stability of impulsive stochastic differential equations in terms of two measures via perturbing Lyapunov functions. Moreover, Markovian Jump Systems with impulsive perturbations have been investigated [28–31]. Zhang et al. [28] have established several criteria for stochastic stability analysis of neutral-type impulsive neural networks with mixed time-varying delays and Markovian jumping. Zhu and Cao [30] have obtained several sufficient conditions on stability of Markovian jump neural networks with impulse control and time-varying delays.

Besides impulsive and stochastic effects, reaction diffusion phenomena cannot be ignored in real systems [32–42]. Kao et al. [34] have discussed exponential stability of impulsive stochastic fuzzy reaction-diffusion Cohen-Grossberg neural networks with mixed delays. Wang et al. [40] have probed stochastic exponential stability of the delayed reaction-diffusion recurrent neural networks with Markovian jumping parameters. However, to the best of our knowledge, there are few considering the Markovian jump systems with impulsive perturbations and reaction-diffusion effects.

Motivated by the above discussions, in this paper, we consider mean square stability of a class of impulsive stochastic reaction-diffusion differential systems with Markovian switching. In Section 2, model description and preliminaries are presented. In Section 3, by utilizing Lyapunov function and stochastic analysis, we obtain some new conditions ensuring mean square stability of impulsive stochastic reaction-diffusion differential equations with Markovian switching. Moreover, mean square stability of a class of semilinear stochastic impulsive reaction-diffusion systems has also been discussed. In Section 4, an example is provided. Section 5 is conclusions.

2. Model Description and Preliminaries

In this section, we investigate the impulsive stochastic reaction diffusion equations with Markovian switching described by

with boundary condition

and initial condition

where is a bounded set with smooth boundary is the outward normal derivative. is the impulsive moment satisfying , and and denote the right-hand limit and left-hand limit of at , respectively. is continuous for all but points and exist, furthermore, . is a matrix, and are continuous, in addition, = , = . = represents the impulsive perturbation of at time . is a one-dimensional standard Brownian motion on a complete probability space with a natural filtration . is a left-continuous Markov process on the probability space and takes values in the finite space with generator given by

where and , is the transition rate from to if and . We suppose that the Markov chain is independent of the Brownian motion . Moreover, we assume that , then system (2.1) admits a trivial solution . For , we define where . For simplicity, we denote by throughout this paper.

Let stand for the solution of system (2.1)–(2.4) through .

Definition 2.1. The trivial solution is said to be mean square stable if for any , there exists such that for all , we have

when satisfies .

Definition 2.2. The function belongs to class if

(1) for , the function is once continuously differentiable in and twice in on , and, in addition, holds for ;(2) is locally Lipschitzian in ;(3) for each , there exist finite limits

3. Main Results

In this section, we will discuss mean square stability of the trivial solution of system (2.1)–(2.4). Assume satisfies

Theorem 3.1. If there exist constants and a Lyapunov function such that for , we have the following.

(A1). (A2).  Here the operator is defined as

(A3), where (A4)

Then, the trivial solution of system (2.1)–(2.4) is stable in mean square.

Proof. For any , there must exist a scalar such that . Next we will prove that if satisfies .

Let . Multiplying both sides of (2.1) by , we obtain

By integrating the above equality with respect to on , we then have

Namely,

Applying Itô formula, we further compute, when ,

where and

For , integrating (3.5) with respect to from to , one has

Taking the mathematical expectation of both sides of (3.7), we obtain

Choosing small enough such that , it is easy to see that

We thus derive from (3.8) and (3.9) that

So

Next, we will first prove

Obviously,

If inequality (3.12) does not hold, there must exist some such that

Let . Since is continuous on , there exist such that

From , if , we know that there is such that

On the other hand, noticing , we obtain

Integrating both sides of (3.17) on gives

However,

which is a contradiction. Therefore,

Furthermore,

Now we assume that

and then prove

If not, there must exist some such that

Let

Since is continuous in , there exists satisfying

Because of and , there is such that

Noticing , we obtain

Integrating both sides of (3.28) on , we claim that

However,

This leads to a contradiction. Then, we have

Moreover,

Therefore,

which results in

Namely,

This ends the proof of Theorem 3.1.

As an application, we consider a class of semilinear impulsive stochastic reaction-diffusion equations with Markovian switching as follows:

with boundary condition

and initial condition

where with for . , is matrices. The remainder of system (3.36)–(3.39) is the same as that defined in system (2.1)–(2.4).

Theorem 3.2. Assume that

(A5), where ,(A6).

Then, the trivial solution of system (3.36)–(3.39) is stable in mean square.

Proof. Construct a Lyapunov function , and compute the operator that

By Green formula, we get

It follows from boundary condition that

Thus,

Therefore,

According to Theorem 3.1, we find that the trivial solution of system (3.36)–(3.39) is stable in mean square.

4. Example

Consider the following two dimension Markovian jumping impulsive stochastic reaction diffusion systems with two modes. The parameters are given as follows: Let , when , we have

when

where . By simple calculation, we obtain . From Theorem 3.2, the trivial solution of this system is stable in mean square.

5. Conclusion

In this paper, we discuss mean square stability of stochastic reaction diffusion equations with Markovian switching and impulsive perturbations, by means of Lyapunov function and stochastic analysis. As an application, we investigate a class of semilinear impulsive stochastic reaction-diffusion equations with Markovian switching and establish the stability criterion. Finally, we provide an example to demonstrate the effectiveness and efficiency of the obtained results.

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and constructive suggestions. This research is supported by the National Natural Science Foundations of China (60974025, 60939003), National 863 Plan Project (2008AA04Z401, 2009AA043404), the Natural Science Foundation of Shandong Province (no. Y 2007G30), the Natural Science Foundation of Guangxi Autonomous Region (no. 2012GXNSFBA053003), the Scientific and Technological Project of Shandong Province (no. 2007GG3WZ04016), the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF.2001120), the China Postdoctoral Science Foundation (20100481000) and the Shandong Provincial Key Laboratory of Industrial Control Technique (Qingdao University).

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