Mean square exponential stability of stochastic delay cellular neural networks

The dynamical behaviors of stochastic neural networks have appeared as a novel subject of research and applications, such as optimization, control, and image processing(see [1-12]). Obviously, finding stability criteria for these neural networks becomes an attractive research problem of importance. Some well results have just appeared, for example, in [1-5], for stochastic delayed Hopfield neural networks and stochastic Cohen-Grossberg neural networks, the linear matrix inequality approach is utilized to establish the sufficient conditions on global stability for the neural networks. In particular, in [2], by using the method of variation parameter and stochastic analysis, the sufficient conditions are given to guarantee the exponential stability of an equilibrium solution. However, there are few results about stochastic effects to the stability property of cellular neural networks with delays in the literature today. In this paper, exponential stability of equilibrium point of stochastic cellular neural networks with delays(SDCNNs) is investigated. Following [13], that activation functions require Lipschitz conditions and boundedness, by utilizing general Lyapunov function, stochastic analysis, Young inequality method and Poincare contraction theory are utilized to derive the conditions guaranteeing the existence of periodic solutions of SDCNNs and the stability of periodic solutions. Different from the LMI (linear matrix inequality) approach [13], [15] and variation parameter method, the Young inequality method is firstly developed to investigate the stability of SDCNN. These sufficient conditions improve and extend the early works in Refs. [18,19], and they include those governing parameters of SDCNNs, so they can be easily checked by simple algebraic methods, comparing with the results of [13-17]. Furthermore, one example is given to demonstrate the usefulness of the results in this paper.


Introduction
The dynamical behaviors of stochastic neural networks have appeared as a novel subject of research and applications, such as optimization, control, and image processing(see [1][2][3][4][5][6][7][8][9][10][11][12]).Obviously, finding stability criteria for these neural networks becomes an attractive research problem of importance.Some well results have just appeared, for example, in [1][2][3][4][5], for stochastic delayed Hopfield neural networks and stochastic Cohen-Grossberg neural networks, the linear matrix inequality approach is utilized to establish the sufficient conditions on global stability for the neural networks.In particular, in [2], by using the method of variation parameter and stochastic analysis, the sufficient conditions are given to guarantee the exponential stability of an equilibrium solution.However, there are few results about stochastic effects to the stability property of cellular neural networks with delays in the literature today.
In this paper, exponential stability of equilibrium point of stochastic cellular neural networks with delays(SDCNNs) is investigated.Following [13], that activation functions require Lipschitz conditions and boundedness, by utilizing general Lyapunov function, stochastic analysis, Young inequality method and Poincare contraction theory are utilized to derive the conditions guaranteeing the existence of periodic solutions of SDCNNs and the stability of periodic solutions.Different from the LMI (linear matrix inequality) approach [13], [15] and variation parameter method, the Young inequality method is firstly developed to investigate the stability of SDCNN.These sufficient conditions improve and extend the early works in Refs.[18,19], and they include those governing parameters of SDCNNs, so they can be easily checked by simple algebraic methods, comparing with the results of [13][14][15][16][17]].Furthermore, one example is given to demonstrate the usefulness of the results in this paper.
The organization of this paper is as follows.In Section 2, problem formulation and preliminaries are given.In Section 3, some new results are given to ascertain the exponential stability of the neural networks with time-varying delays based on Lyapunov method.Section 4 gives an example to illustrate the effectiveness of our results.

Preliminaries and lemmas
This paper, we are concerned with the model of continuous-time neural networks described by the following integro-differential systems: or equivalently where n denotes the number of the neurons in the network, x i (t) is the state of the ith neuron at time t, x(t n×n are the feedback matrix and the delayed feedback matrix, respectively, J = (J 1 , J 2 , . . ., J n ) T ∈ R n be a constant external input vector, the kernels k j : [0, +∞) → [0, +∞) are piece continuous functions with the time delay τ j (t) is any nonnegative continuous function with 0 ≤ τ j (t) ≤ τ, where τ is a constant, τ (t) = [τ 1 (t), τ 2 (t), . . ., τ n (t)].
In our analysis, we will employ that each f i , i = 1, 2, . . ., n is bounded and satisfying the following condition: (H)There exist constant scalars This class of functions is clearly more general than both the usual sigmoid activation functions and the piecewise linear function: , which is used in [11].The initial conditions associated with system (1) are of the form T is an equilibrium of Eq. ( 1), one can derive from (1) that the transformation y i = x i − x * i transforms system (1) or (2) into the following system: EJQTDE, 2013 No. 34, p. 2 where Note that since each function f j (•) satisfies the hypothesis (H), hence, each g j (•) satisfies To prove the stability of x * of Eq. ( 1), it is sufficient to prove the stability of the trivial solution of Eq. ( 3) or ( 4).Consider the following stochastic delayed recurrent neural networks with time varying delay σ ij (t, y j (t), y j (t − τ j (t)))dw j (t) ( or equivalently where i = 1, 2, . . ., n; w(t) = (w 1 (t), w 2 (t), . . ., w n (t)) T is an n-dimensional Brownian motion defined on a complete probability space (Ω, F , P ) with a natural filtration {F t } t≥0 generated by {w(s) : 0 ≤ s ≤ t}, where we associate Ω with the canonical space generated by w(t), and denote by F the associated σ-algebra generated by w(t) with the probability measure P.
Definition 1.The solution y(t; φ) of system ( 5) is said to be pth moment exponentially stable if there exists a pair of positive constants λ and c such that E y(t; φ) p ≤ cE φ p e −λt , t ≥ 0 holds for any φ, where E stands for the mathematical expectation operator.In this case The right-hand side of ( 7) is called the pth moment Lyapunov exponent of the solution.It is usually called the exponential stability in the mean square when p = 2.
Let C 2,1 (R n × R + ; R + ) denote the family of all non-negative functions V (y, t) on R n × R + which are continuously twice differentiable in y and once differentiable in t.For each V ∈ C 2,1 (R n × R + , R + ), define an operator LV associated with stochastic delayed neural networks (5) In the following, we will use the notation A > 0 (or A < 0) to denote a symmetric and positive definite (or negative definite) matrix.The notation A T and A −1 means the transpose of and the inverse of a square matrix A. If A, B are symmetric matrices, A > B(A ≥ B) means that A − B is positive definite (positive semi-definite).
In order to obtain our result, we need the following lemma In this section, we present and prove our main results.
Consider the following positive definite Lyapunov function defined by: From Lemma 2, we have EJQTDE, 2013 No. 34, p. 6 2y (13) From (10-13), we have We can easily obtain the following corollary When the feedback matrix A = 0 in Theorem 1, we can easily obtain the following corollary Corollary 2. Assume that there exist positive diagonal matrices , then the equilibrium point of system ( 5) is exponentially stable in the mean square if there exist a positive diagonal matrix P = diag(p 1 , p 2 , . . ., p n ) such that When the delayed feedback matrix C = 0, the feedback matrix A = 0, in Theorem 1, we can easily obtain the following corollary Corollary 3. Assume that there exist positive diagonal matrices , then the equilibrium point of system ( 5) is exponentially stable in the mean square if there exist a positive diagonal matrix P = diag(p 1 , p 2 , . . ., p n ) such that Remark.Obviously, the results in Corollary 1,2,3 are more simple than Theorem 2 in [5] and Theorem 1 in [15,16].Thus, Theorem 1 above generalizes the result in [5,15,16].

An example
In this section, an example is used to demonstrate that the method presented in this paper is effective.
Choose P = I, then we have −0.17 < 0, Therefore, by theorem 1, the equilibrium point of Eq. ( 1) is exponentially stable in the mean square.