Journal of Applied MathematicsVolume 2012 (2012), Article ID 693163, 12 pagesdoi:10.1155/2012/693163

Research Article

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

Xiao Liang¹ and Linshan Wang²

¹College of Information Science and Engineering, Ocean University of China, Qingdao 266100, China²Department 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.

(a)

(b)

Figure 1

Figure 2

Figure 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).

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