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

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.


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 2 Journal of Applied Mathematics 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: . . , n.

1.1
There are n neural network units in this system and u i t, x denote the potential of the cell i at t and x. a i are positive constants and denote the rate with which the ith unit will reset its potential to the resting state in isolation when it is disconnected from the network and external inputs at t, and c ij are the output connection weights from the jth neuron to the ith neuron. f j are the active functions of the neural network. r is the time delay of a neuron. O denotes an open bounded and connected subset of R l with a sufficient regular boundary ∂O, ν is the unit outward normal on ∂O, ∂u i /∂ν ∇u i , ν R l , and g ij are noise intensities. Initial data φ i are F 0 -measurable and bounded functions, almost surely.
We denote Ω, F, P a complete probability space with filtration {F t } t≥0 satisfying the usual conditions see 10 . W i t , i 1, 2, . . . , m, are scale standard Brownian motions defined on Ω, F, P .
For convenience, we rewrite system 1.1 in the vector form: . . , f n u n T , A Diag a 1 , a 2 , . . . , a n , φ φ 1 , φ 2 , . . . , φ n T , G u g ij u i n×m , D D ij n×l , and D • ∇u D ij ∂u i /∂x j n×l is the Hadamard product of matrix D and ∇u; for the definition of divergence operator ∇ · u, we refer to 2, 3 .

Preliminaries and Notations
In this paper, we introduce the following Hilbert spaces H where H , V denote the dual of the space H, V , respectively, the injection is continuous, and the embedding is compact. · , ||| · ||| represent the norm in H, V , respectively. With any continuous F t -adapted U-valued stochastic process u t : Ω → U, t ≥ −r, we associate a continuous F t -adapted C-valued stochastic process u t : Ω → C, t > 0, by setting L K is the set of all linear bounded operators from K into K; when equipped with the operator norm, it becomes a Banach space.
In this paper, we assume the following.
H1 f i and G ij are Lipschitz continuous with positive Lipschitz constants k 1 , k 2 such that Remark 2.1. We can infer from H1 that system 1.1 has an equilibrium u t, x, ω 0. Let us define the linear operator as follows:

Lemma 2.2 Poincaré's inequality . Let O be a bounded domain in R l and φ belong to a collection of twice differentiable functions defined on O into R; then
where the constant β depends on the size of O.

2.3
For every φ ∈ U, let u t S t φ denote the solution of 2.3 ; then S t is a contraction map in U.
Proof. Now we take the inner product of 2.3 with u t in U; by employing the Gaussian theorem and condition H2, we get that Au, u ≤ −α |u| 2 Thanks to the Poincaré inequality, one obtains Multiplying e 2αβ 2 t in both sides of the inequality, we have Integrating the above inequality from 0 to t, we obtain By the definition of T t L U , we have T t L U ≤ 1.
for all t ∈ −r, ∞ with probability one.
Definition 2.5. Equation 1.1 is said to be almost surely exponentially stable if, for any solution u t, x, ω with initial data φ ∈ C b F 0 , there exists a positive constant λ such that lim sup t−→∞ ln u t C ≤ −λ, u t ∈ C, almost surely.

2.9
Definition 2.6. System 1.1 is said to be exponentially stable in the mean square sense if there exist positive constants κ and α such that, for any solution u t, x, ω with the initial condition

Main Result
where λ is a positive constant that will be defined below. Then, by integration between 0 and t, we find that 3.5 6

Journal of Applied Mathematics
We observe that From the Neumann boundary condition, by means of Green's formula and H2 see 3, 6, 7 , we know

3.8
Then, by using the positiveness of a i , one gets the relation where k 3 min{a 1 , a 2 , . . . , a n } > 0. By using the Young inequality as well as condition H1, we have that

3.10
where σ max |c ij |, and e λs E u i s − r 2 ds.

3.12
Adding 3.12 from i 1 to i n, we obtain where c 1 2αβ 2 2k 3 − 1 − nk 2 1 σ 2 e λr − mk 2 2 e λr − λ and c 2 1 mk 2 2 re 2λr nk 2 1 σ 2 re 2λr E φ 2 C ; so we choose λ 1 such that c 1 η > 0. By using the classical Gronwall inequality we see that in other words, we get So, for t θ ≥ t/2 ≥ 0, we also have Journal of Applied Mathematics and we can conclude that

3.19
Theorem 3.2. If the system 1.1 satisfies hypotheses H1-H3, then it is almost surely exponentially stable.
Proof. Let u t be the mild solution of 1.1 . By Definition 2.4 as well as the inequality 3.20 Using the contraction of the map S t and the result of Theorem 3.1, we find

3.21
By the Hölder inequality, we obtain
Journal of Applied Mathematics 9 By virtue of Theorem 3.1, Hölder inequality, and H1, we have

3.23
Then, by the Burkholder-Davis-Gundy inequality see 18, 22 , there exists c 3 such that 3.26 This completes the proof of the theorem.