Mathematical Problems in Engineering

Volume 2012, Article ID 429568, 13 pages

http://dx.doi.org/10.1155/2012/429568

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

^{1}Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China^{2}School 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).

#### References

- N. N. Krasovskiĭ and È. A. Lidskiĭ, “Analytical design of controllers in systems with random attributes. I. Statement of the problem, method of solving,”
*Automation and Remote Control*, vol. 22, pp. 1021–1025, 1961. View at Google Scholar - P. Balasubramaniam, A. Manivannan, and R. Rakkiyappan, “Exponential stability results for uncertain neutral systems with interval time-varying delays and Markovian jumping parameters,”
*Applied Mathematics and Computation*, vol. 216, no. 11, pp. 3396–3407, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Shi and E. K. Boukas, “${H}_{\infty}$-control for Markovian jumping linear systems with parametric uncertainty,”
*Journal of Optimization Theory and Applications*, vol. 95, no. 1, pp. 75–99, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Zhang and E.-K. Boukas, “Mode-dependent ${H}_{\infty}$ filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities,”
*Automatica*, vol. 45, no. 6, pp. 1462–1467, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Zhang and J. Lam, “Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions,”
*IEEE Transactions on Automatic Control*, vol. 55, no. 7, pp. 1695–1701, 2010. View at Publisher · View at Google Scholar - L. Zhang, E.-K. Boukas, and J. Lam, “Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities,”
*IEEE Transactions on Automatic Control*, vol. 53, no. 10, pp. 2458–2464, 2008. View at Publisher · View at Google Scholar - X. Mao and C. Yuan,
*Stochastic Differential Equations with Markovian Switching*, Imperial College Press, London, UK, 2006. - X. Mao, Y. Shen, and C. Yuan, “Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching,”
*Stochastic Processes and their Applications*, vol. 118, no. 8, pp. 1385–1406, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Mao, A. Matasov, and A. B. Piunovskiy, “Stochastic differential delay equations with Markovian switching,”
*Bernoulli*, vol. 6, no. 1, pp. 73–90, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Mao and L. Shaikhet, “Delay-dependent stability criteria for stochastic differential delay equations with Markovian switching,”
*Stability and Control*, vol. 3, no. 2, pp. 88–102, 2000. View at Google Scholar - X. Mao, “Exponential stability of stochastic delay interval systems with Markovian switching,”
*IEEE Transactions on Automatic Control*, vol. 47, no. 10, pp. 1604–1612, 2002. View at Publisher · View at Google Scholar - X. Mao, “Stochastic functional differential equations with Markovian switching,”
*Functional Differential Equations*, vol. 6, no. 3-4, pp. 375–396, 1999. View at Google Scholar · View at Zentralblatt MATH - L. Huang and X. Mao, “On input-to-state stability of stochastic retarded systems with Markovian switching,”
*IEEE Transactions on Automatic Control*, vol. 54, no. 8, pp. 1898–1902, 2009. View at Publisher · View at Google Scholar - S. Wu, D. Han, and X. Meng, “$p$-moment stability of stochastic differential equations with jumps,”
*Applied Mathematics and Computation*, vol. 152, no. 2, pp. 505–519, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Wu and J. Sun, “$p$-moment stability of stochastic differential equations with impulsive jump and Markovian switching,”
*Automatica*, vol. 42, no. 10, pp. 1753–1759, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Lalshmikantham, D. Bainov, and P. Simeonov,
*Theory of Impulsive Diffusin Equation*, World Scientific, Singapore, 1989. - Q. Wang and X. Liu, “Exponential stability for impulsive delay differential equations by Razumikhin method,”
*Journal of Mathematical Analysis and Applications*, vol. 309, no. 2, pp. 462–473, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Zhang and J. Sun, “Stability of impulsive infinite delay differential equations,”
*Applied Mathematics Letters*, vol. 19, no. 10, pp. 1100–1106, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Xu and Z. Yang, “Impulsive delay differential inequality and stability of neural networks,”
*Journal of Mathematical Analysis and Applications*, vol. 305, no. 1, pp. 107–120, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Xu and D. Xu, “Mean square exponential stability of impulsive control stochastic systems with time-varying delay,”
*Physics Letters A*, vol. 373, no. 3, pp. 328–333, 2009. View at Publisher · View at Google Scholar · View at Scopus - Z. Yang, D. Xu, and L. Xiang, “Exponential $p$-stability of impulsive stochastic differential equations with delays,”
*Physics Letters A*, vol. 359, no. 2, pp. 129–137, 2006. View at Publisher · View at Google Scholar - J. Yang, S. Zhong, and W. Luo, “Mean square stability analysis of impulsive stochastic differential equations with delays,”
*Journal of Computational and Applied Mathematics*, vol. 216, no. 2, pp. 474–483, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Li, “The attracting set for impulsive stochastic difference equations with continuous time,”
*Applied Mathematics Letters*, vol. 25, no. 8, pp. 1166–1171, 2012. View at Publisher · View at Google Scholar · View at Scopus - L. Pan and J. Cao, “Exponential stability of impulsive stochastic functional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 382, no. 2, pp. 672–685, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Zhang, J. Sun, and Y. Zhang, “Stability of impulsive stochastic differential equations in terms of two measures via perturbing Lyapunov functions,”
*Applied Mathematics and Computation*, vol. 218, no. 9, pp. 5181–5186, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Cheng and F. Deng, “Global exponential stability of impulsive stochastic functional differential systems,”
*Statistics & Probability Letters*, vol. 80, no. 23-24, pp. 1854–1862, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Li, J. Shi, and J. Sun, “Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks,”
*Nonlinear Analysis A*, vol. 74, no. 10, pp. 3099–3111, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Zhang, M. Dong, Y. Wang, and N. Sun, “Stochastic stability analysis of neutral-type impulsive neural networks with mixed time-varying delays and Markovian jumping,”
*Neurocomputing*, vol. 73, no. 13-15, pp. 2689–2695, 2010. View at Publisher · View at Google Scholar · View at Scopus - H. Bao and J. Cao, “Stochastic global exponential stability for neutral-type impulsive neural networks with mixed time-delays and Markovian jumping parameters,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 9, pp. 3786–3791, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Q. Zhu and J. Cao, “Stability of Markovian jump neural networks with impulse control and time varying delays,”
*Nonlinear Analysis*, vol. 13, no. 5, pp. 2259–2270, 2012. View at Publisher · View at Google Scholar - P. Balasubramaniam, R. Krishnasamy, and R. Rakkiyappan, “Delay-dependent stability criterion for a class of non-linear singular Markovian jump systems with mode-dependent interval time-varying delays,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 9, pp. 3612–3627, 2012. View at Google Scholar - J. Liang and J. Gao, “Global expoential stability of reaction-diffusion recurrent neural networks with time-varying delays,”
*Physics Letters A*, vol. 314, pp. 434–442, 2003. View at Google Scholar - Y. G. Kao, C. C. Gao, and W. Han, “Global exponential robust stability of reaction-diffusion interval neural networks with continuously distributed delays,”
*Neural Computing and Applications*, vol. 19, pp. 867–873, 2010. View at Google Scholar - C. Wang, Y. Kao, and G. Yang, “Exponential stability of impulsive stochastic fuzzy reaction-diffusion Cohen-Grossberg neural networks with mixed delays,”
*Neurocomputing*, vol. 89, no. 15, pp. 55–63, 2012. View at Google Scholar - Y. G. Kao, J. F. Guo, C. H. Wang, and X. Q. Sun, “Delay-dependent robust exponential stability of Marko-vian jumping reaction-diffusion Cohen-Grossberg neural networks with mixed delays,”
*Journal of The Franklin Institute*, vol. 349, no. 6, pp. 1972–1988, 2012. View at Google Scholar - D. Li, D. He, and D. Xu, “Mean square exponential stability of impulsive stochastic reaction-diffusion Cohen-Grossberg neural networks with delays,”
*Mathematics and Computers in Simulation*, vol. 82, pp. 1531–1543, 2012. View at Google Scholar - Q. Luo, F. Deng, J. Bao, B. Zhao, and Y. Fu, “Stabilization of stochastic Hopfield neural network with distributed parameters,”
*Science in China F*, vol. 47, no. 6, pp. 752–762, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Q. Luo and Y. Zhang, “Almost sure exponential stability of stochastic reaction diffusion systems,”
*Nonlinear Analysis A*, vol. 71, no. 12, pp. e487–e493, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y.-T. Zhang, Q. Luo, and X.-H. Lai, “Stability in mean of partial variables for stochastic reaction diffusion systems,”
*Nonlinear Analysis A*, vol. 71, no. 12, pp. e550–e559, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Wang, Z. Zhang, and Y. Wang, “Stochastic exponential stability of the delayed reaction-diffusion recurrent neural networks with Markovian jumping parameters,”
*Physics Letters A*, vol. 372, no. 18, pp. 3201–3209, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Vidhya and P. Balasubramaniam, “On the stability of the stochastic parabolic Itô equation with delay and Markovian jump,”
*Computers & Mathematics with Applications*, vol. 60, no. 7, pp. 1959–1963, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Balasubramaniam and C. Vidhya, “Exponential stability of stochastic reaction-diffusion uncertain fuzzy neural networks with mixed delays and Markovian jumping parameters,”
*Expert Systems with Applications*, vol. 39, no. 3, pp. 3109–3115, 2012. View at Google Scholar