- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Journal of Applied Mathematics

Volume 2012 (2012), Article ID 693163, 12 pages

http://dx.doi.org/10.1155/2012/693163

## Exponential Stability for a Class of Stochastic Reaction-Diffusion Hopfield Neural Networks with Delays

^{1}College of Information Science and Engineering, Ocean University of China, Qingdao 266100, China^{2}Department of Mathematics, Ocean University of China, Qingdao 266100, China

Received 7 August 2011; Accepted 28 November 2011

Academic Editor: Jitao Sun

Copyright © 2012 Xiao Liang and Linshan Wang. 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 studies the asymptotic behavior for a class of delayed reaction-diffusion Hopfield neural networks driven by finite-dimensional Wiener processes. Some new sufficient conditions are established to guarantee the mean square exponential stability of this system by using Poincaré’s inequality and stochastic analysis technique. The proof of the almost surely exponential stability for this system is carried out by using the Burkholder-Davis-Gundy inequality, the Chebyshev inequality and the Borel-Cantelli lemma. Finally, an example is given to illustrate the effectiveness of the proposed approach, and the simulation is also given by using the Matlab.

#### 1. Introduction

Recently, the dynamics of Hopfield neural networks with reaction-diffusion terms have been deeply investigated because their various generations have been widely used in some practical engineering problems such as pattern recognition, associate memory, and combinatorial optimization (see [1–3]). However, under closer scrutiny, that a more realistic model would include some of the past states of the system, and theory of functional differential equations systems has been extensively developed [4, 5], meanwhile many authors have considered the asymptotic behavior of the neural networks with delays [6–9]. In fact random perturbation is unavoidable in any situation [3, 10]; if we include some environment noise in these systems, we can obtain a more perfect model of this situation [3, 11–16]. So, this paper is devoted to the exponential stability of the following delayed reaction-diffusion Hopfield neural networks driven by finite-dimensional Wiener processes:

There are neural network units in this system and denote the potential of the cell at and . are positive constants and denote the rate with which the th unit will reset its potential to the resting state in isolation when it is disconnected from the network and external inputs at , and are the output connection weights from the th neuron to the th neuron. are the active functions of the neural network. is the time delay of a neuron. denotes an open bounded and connected subset of with a sufficient regular boundary , is the unit outward normal on , , and are noise intensities. Initial data are -measurable and bounded functions, almost surely.

We denote a complete probability space with filtration satisfying the usual conditions (see [10]). , are scale standard Brownian motions defined on .

For convenience, we rewrite system (1.1) in the vector form: where , , , , , , , , , and is the Hadamard product of matrix and ; for the definition of divergence operator , we refer to [2, 3].

#### 2. Preliminaries and Notations

In this paper, we introduce the following Hilbert spaces , , according to [17–19], , where denote the dual of the space , respectively, the injection is continuous, and the embedding is compact. represent the norm in , respectively.

is the space of vector-valued Lebesgue measurable functions on , which is a Banach space under the norm

is the Banach space of all continuous functions from to , when equipped with the sup-norm

With any continuous adapted valued stochastic process , , we associate a continuous -adapted -valued stochastic process , by setting , .

denote the space of all bounded continuous processes such that is -measurable for each and .

is the set of all linear bounded operators from into ; when equipped with the operator norm, it becomes a Banach space.

In this paper, we assume the following.**H1** and are Lipschitz continuous with positive Lipschitz constants such that and , and , .**H2** There exists such that .**H3** Let , .

*Remark 2.1. *We can infer from H1 that system (1.1) has an equilibrium .

Let us define the linear operator as follows:
and .

Lemma 2.2 (Poincaré’s inequality). *Let be a bounded domain in and belong to a collection of twice differentiable functions defined on into ; then
**
where the constant depends on the size of .*

Lemma 2.3. *Let us consider the equation
**
For every , let denote the solution of (2.3); then is a contraction map in .*

*Proof. *Now we take the inner product of (2.3) with in ; by employing the Gaussian theorem and condition H2, we get that , is the inner product in , denote the norm of (see [3]), which means
Thanks to the Poincaré inequality, one obtains
Multiplying in both sides of the inequality, we have
Integrating the above inequality from 0 to , we obtain
By the definition of , we have .

*Definition 2.4 (see [20–22]). *A stochastic process is called a global mild solution of (1.1) if(i) is adapted to (ii) is measurable with almost surely and
for all with probability one.

*Definition 2.5. *Equation (1.1) is said to be almost surely exponentially stable if, for any solution with initial data , there exists a positive constant such that

*Definition 2.6. *System (1.1) is said to be exponentially stable in the mean square sense if there exist positive constants and such that, for any solution with the initial condition , one has

#### 3. Main Result

Theorem 3.1. *Suppose conditions H1–H3 hold; then (1.1) is exponentially stable in the mean square sense.*

*Proof. *Let be the mild solution of (1.1); thanks to the Itô formula, we observe that
where is a positive constant that will be defined below. Then, by integration between 0 and , we find that
Integrating the above equation over , by virtue of Fubini’s theorem, we prove that
Taking the expectation on both sides of the last equation, by means of [3, 10, 16]
Then, by Fubini’s theorem, we have
We observe that
From the Neumann boundary condition, by means of Green’s formula and H2 (see [3, 6, 7]), we know
Then, by using the positiveness of , one gets the relation
where . By using the Young inequality as well as condition H1, we have that
where , and
We infer from (3.6)–(3.11) that
Adding (3.12) from to , we obtain
due to
we induce from the previous equations that
where and ; so we choose such that . By using the classical Gronwall inequality we see that
in other words, we get
So, for , we also have
and we can conclude that

Theorem 3.2. *If the system (1.1) satisfies hypotheses H1–H3, then it is almost surely exponentially stable.*

*Proof. *Let be the mild solution of (1.1). By Definition 2.4 as well as the inequality , we have
Using the contraction of the map and the result of Theorem 3.1, we find
By the Hölder inequality, we obtain
where .

By virtue of Theorem 3.1, Hölder inequality, and H1, we have
Then, by the Burkholder-Davis-Gundy inequality (see [18, 22]), there exists such that
where is an dimensional vector.

We can deduce from (3.21)–(3.24) that
where .

Thus, for any positive constants , thanks to the Chebyshev inequality we have that
Due to the Borel-Cantelli lemma, we see that
This completes the proof of the theorem.

#### 4. Simulation

Consider two-dimensional stochastic reaction-diffusion recurrent neural networks with delay as follows:

is the Laplace operator. We have , , , , , , , and ; by Theorems 3.1 and 3.2, this system is mean square exponentially stable as well as almost surely exponentially stable. The results can be shown in Figures 1, 2 and 3.

We use the forward Euler method to simulate this example [23–25]. We choose the time step and space step , and .

#### Acknowledgment

The authors wish to thank the referees for their suggestions and comments. We are also indebted to the editors for their help. This work was supported by the National Natural Science Foundation of China (no. 11171374), Natural Science Foundation of Shandong Province (no. ZR2011AZ001).

#### References

- X. X. Liao and Y. L. Gao, “Stability of Hopfield neural networks with reactiondiffusion terms,”
*Acta Electronica Sinica*, vol. 28, pp. 78–82, 2000. View at Google Scholar - L. S. Wang and D. Xu, “Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays,”
*Science in China F*, vol. 46, no. 6, pp. 466–474, 2003. View at Publisher · View at Google Scholar - L. S. Wang and Y. F. Wang, “Stochastic exponential stability of the delayed reaction diffusion interval neural networks with Markovian jumpling parameters,”
*Physics Letters A*, vol. 356, pp. 346–352, 2008. View at Google Scholar - J. K. Hale and V. S. M. Lunel,
*Introduction to Functional-Differential Equations*, vol. 99 of*Applied Mathematical Sciences*, Springer, Berlin, Germany, 1993. - S.-E. A. Mohammed,
*Stochastic Functional Differential Equations*, vol. 99 of*Research Notes in Mathematics*, Pitman, London, UK, 1984. - L. S. Wang and Y. Y. Gao, “Global exponential robust stability of reactiondiffusion interval neural networks with time varying delays,”
*Physics Letters A*, vol. 305, pp. 343–348, 2006. View at Google Scholar - L. S. Wang, R. Zhang, and Y. Wang, “Global exponential stability of reaction-diffusion cellular neural networks with S-type distributed time delays,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 2, pp. 1101–1113, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Y. Zhao and G. L. Wang, “Existence of periodic oscillatory solution of reactiondiffusion neural networks with delays,”
*Physics Letters A*, vol. 343, pp. 372–382, 2005. View at Publisher · View at Google Scholar - J. G. Lu and L. J. Lu, “Global exponential stability and periodicity of reaction-diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions,”
*Chaos, Solitons and Fractals*, vol. 39, no. 4, pp. 1538–1549, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Mao,
*Stochastic Differential Equations and Applications*, Horwood, 1997. - J. Sun and L. Wan, “Convergence dynamics of stochastic reaction-diffusion recurrent neural networks with delays,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 15, no. 7, pp. 2131–2144, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Itoh and L. O. Chua, “Complexity of reaction-diffusion CNN,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 16, no. 9, pp. 2499–2527, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Lou and B. Cui, “New criteria on global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms,”
*International Journal of Neural Systems*, vol. 17, pp. 43–52, 2007. View at Publisher · View at Google Scholar - Q. Song and Z. Wang, “Dynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coefficients and delays,”
*Applied Mathematical Modelling*, vol. 33, no. 9, pp. 3533–3545, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Q. Song, J. Cao, and Z. Zhao, “Periodic solutions and its exponential stability of reaction-diffusion reccurent neural networks with distributed time delays,”
*Nonlinear Analysis B*, vol. 8, pp. 345–361, 2007. View at Publisher · View at Google Scholar - K. Liu, “Lyapunov functionals and asymptotic stability of stochastic delay evolution equations,”
*Stochastics and Stochastics Reports*, vol. 63, no. 1-2, pp. 1–26, 1998. View at Google Scholar · View at Zentralblatt MATH - R. Temam,
*Infinite-Dimensional Dynamical Systems in Mechanics and Physics*, vol. 68 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 1988. - G. Da Prato and J. Zabczyk,
*Stochastic Equations in Infinite Dimensions*, vol. 44 of*Encyclopedia of Mathematics and its Applications*, Cambridge University Press, Cambridge, UK, 1992. View at Publisher · View at Google Scholar - I. Chueshov,
*Introduction to the Theory of Infinite-Dimensional Dissipative Systems*, Acta Kharkov, 2002. - R. Jahanipur, “Stochastic functional evolution equations with monotone nonlinearity: existence and stability of the mild solutions,”
*Journal of Differential Equations*, vol. 248, no. 5, pp. 1230–1255, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Taniguchi, “Almost sure exponential stability for stochastic partial functional-differential equations,”
*Stochastic Analysis and Applications*, vol. 16, no. 5, pp. 965–975, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Caraballo, K. Liu, and A. Truman, “Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property,”
*Proceedings the Royal Society of London A*, vol. 456, no. 1999, pp. 1775–1802, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Kamrani and S. M. Hosseini, “The role of coefficients of a general SPDE on the stability and convergence of a finite difference method,”
*Journal of Computational and Applied Mathematics*, vol. 234, no. 5, pp. 1426–1434, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. J. Higham, “Mean-square and asymptotic stability of the stochastic theta method,”
*SIAM Journal on Numerical Analysis*, vol. 38, no. 3, pp. 753–769, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. E. Kloeden and E. Platen,
*Numerical Solution of Stochastic Differential Equations*, vol. 23, Springer, Berlin, Germany, 3rd edition, 1999.