Pth Moment Exponential Stability of Impulsive Stochastic Neural Networks with Mixed Delays

This paper investigates the problem of pth moment exponential stability for a class of stochastic neural networks with time-varying delays and distributed delays under nonlinear impulsive perturbations. By means of Lyapunov functionals, stochastic analysis and differential inequality technique, criteria on pth moment exponential stability of this model are derived. The results of this paper are completely new and complement and improve some of the previously known results Stamova and Ilarionov 2010 , Zhang et al. 2005 , Li 2010 , Ahmed and Stamova 2008 , Huang et al. 2008 , Huang et al. 2008 , and Stamova 2009 . An example is employed to illustrate our feasible results.


Introduction
The dynamics of neural networks have drawn considerable attention in recent years due to their extensive applications in many fields such as image processing, associative memories, classification of patters, and optimization.Since the integration and communication delays are unavoidably encountered in biological and artificial neural systems, it may result in oscillation and instability.The stability analysis of delayed neural networks has been extensively investigated by many researchers, for instance, see 1-30 .In real nervous systems, there are many stochastic perturbations that affect the stability of neural networks.The result in Mao 24 suggested that one neural network could be stabilized or destabilized by certain stochastic inputs.It implies that the stability analysis of stochastic neural networks has primary significance in the design and applications of neural networks, such as 7, 12-16, 18, 20, 22-24, 26, 27, 30 .On the other hand, it is noteworthy that the state of electronic networks is often subjected to some phenomenon or other sudden noises.On that account, the electronic networks will experience some abrupt changes at certain instants that in turn affect dynamical behaviors of the systems 5, 6, 17-23, 28, 29 .Therefore, it is necessary to take both stochastic effects and impulsive perturbations into account on dynamical behaviors of delayed neural networks 18, 20, 22, 23 .Very recently, Li et al. 22 have employed the properties of M-cone and inequality technique to investigate the mean square exponential stability of impulsive stochastic neural networks with bounded delays.Wu et al. 23 studied the exponential stability of the equilibrium point of bounded discrete-time delayed dynamic systems with linear impulsive effects by using Razumikhin theorems.To the best of authors' knowledge, however, few authors have considered the pth moment exponential stability of impulsive stochastic neural networks with mixed delays.
Motivated by the discussions above, our object in this paper is to present the sufficient conditions ensuring pth moment exponential stability for a class of stochastic neural networks with time-varying delays and distributed delays under nonlinear impulsive perturbations by virtue of Lyapunov method, inequality technique and It ô formula.The results obtained in this paper generalize and improve some of the existing results 5, 8, 18, 19, 26-28 .The effectiveness and feasibility of the developed results have been shown by a numerical example.

Model Description and Preliminaries
Let R denote the set of real numbers, R n the n-dimensional real space equipped with the Euclidean norm | • |, Z the set of nonnegative integral numbers.E • stands for the mathematical expectation operator.L denotes the well-known L-operator given by the It ô formula.w t w 1 t , . . ., w m t is m-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 .Let σ t, x, y σ il t, x i , y i n×m ∈ R n×m , and σ i t, x i , y i be ith row vector of σ t, x, y .
In 5, 6 , the researchers investigated the following impulsive neural networks with time-varying delays:

2.1
The authors in 7, 26, 27 studied the stochastic recurrent neural networks with time-varying delays:

2.2
In this paper, we will study the generalized stochastically perturbed neural network model with time-varying delays and distributed delays under nonlinear impulses defined by the state equations: where Λ {1, 2, . . ., n}, the time sequence . ., x n t T and x i t corresponds to the state of the ith unit at time t; b ij , c ij , and d ij denote the constant connection weight; τ j t is the timevarying transmission delay and satisfies 0 ≤ τ j t ≤ τ j , 0 < η j inf t∈R {1 − τj t }, for j ∈ Λ. f j • , g j • , h j • denote the activation functions of the jth neuron; the delay kernel K ij • is the real-valued nonnegative piecewise continuous functions defined on 0, ∞ ; n corresponds to the numbers of units in a neural network; I i denotes the external bias on the ith unit; p ik x t k represents the abrupt change of the state x i t at the impulsive moment t k .System 2.3 is supplemented with initial condition given by where The norms are defined by the following norms, respectively: Throughout this paper, the following standard hypothesis are needed.H1 Functions a i • : R → R are continuous and monotone increasing, that is, there exist real numbers a i > 0 such that

Mathematical Problems in Engineering
H2 Functions f j , g j , and h j are Lipschitz-continuous on R with Lipschitz constants L f j > 0, L g j > 0, and L h j > 0, respectively.That is, H3 The delay kernels where K s : 0, ∞ → R is continuous and integrable, and the constant μ 0 denotes some positive number.H4 There exist nonnegative constants e i , l i such that We end this section by introducing three definitions.
where it is assumed that impulse functions 3 is said to be pth moment exponentially stable if there exist λ > 0 and M ≥ 1 such that where x t is an any solution of system 2.
Definition 2. 3 Forti and Tesi, 1995 25 .A map H : R n → R n is a homeomorphism of R n onto itself if H is continuous and one-to-one, and its inverse map H −1 is also continuous.

Main Result
For convenience, we denote that where , and η l,ij are real numbers and satisfy

3.2
Lemma 3.1.If a i i 1, 2, . . ., p denote p nonnegative real numbers, then where p ≥ 1 denotes an integer.A particular form of 3.3 , namely, If H : R n → R n is a continuous function and satisfies the following conditions.
Then, H x is homeomorphism of R n .

Theorem 3.3. System 2.3 exists a unique equilibrium x * under the assumptions (H1)-(H3) if the following condition is also satisfied: (H6)
In the following, we will prove that H x is a homeomorphism.Firstly, we claim that H x is an injective map on R n .Otherwise, there exist x T , y T ∈ R n , and x T / y T such that H x H y , then

3.6
It follows from H1 -H3 that

3.8
From H6 , it leads to a contradiction with our assumption.Therefore, H x is an injective map on R n .
To demonstrate the property H x → ∞ as x → ∞, we have 3.9 where

3.10
Using the H ölder inequality, we obtain which leads to From 3.12 , we see that H x → ∞ as x → ∞.Thus, the map H x is a homeomorphism on R n under the sufficient condition H6 , and hence it has a unique fixed point x * .This fixed point is the unique solution of the system 2.3 .The proof is now complete.
To establish some sufficient conditions ensuring the pth moment exponential stability of the equilibrium point of x * of system 2.3 , we transform x * to the origin by using the transformation y i t x i t − x * for i ∈ Λ.Then system 2.3 can be rewritten as the following form:

3.13
where g j y j t − τ j t g j y j t − τ j t x * j − g j x * j , h j y j s h j y j s x * j − h j x * j , σ ij t, y j t , y j t − τ j t σ ij t, y j t x * j , y j t − τ j t x * j − σ ij t, x * j , x * j , p ik y t k p ik y t k x * − p ik x * .

3.14
In order to obtain our results, the following assumptions are necessary.
Mathematical Problems in Engineering 9 K s e μ 0 s ds.
Then regardless of cases, H6 and K s e λs u j t − s ds.

3.20
while Lemma 3.1 is used in the second inequality.Let Ψ i e λτ i u i t

3.22
From H8 , we have On the other hand, we have where ρ 0 1.On the other hand, we observe that

3.27
It follows that for t ≥ 0, where Therefore, the equilibrium point of system 2.3 is pth moment exponentially stable.
Remark 3.6.In Corollary 3.5, if p 2, the conditions H9 and H10 are less weak than the following conditions: max i∈Λ e i .
Corollary 3.7.Under the assumptions H2 and H8 , system 2.1 exists a unique equilibrium point x * , and x * is globally exponentially stable if 0 < η i ≤ 1, and the following condition is also satisfied:

3.39
Especially, if p 2, then the equilibrium x * is exponentially stable in mean square if the following condition is also satisfied:

Mathematical Problems in Engineering
This is less conservative than the following inequality: 3.42 while 3.42 was required in Theorem 3.3 in 26 .

3.43
this is less conservative than the following inequality: 3.44 while 3.44 was required in Theoerm 3.3 in 27 .

Illustrative Example
In this section, we will give an example to show that the conditions given in the previous sections are less weak than those given in some of the earlier literatures, such as 18 .
Consider the following stochastic neural networks with mixed time delays

4.2
In the following, we introduce the following nonlinear impulsive controllers:

4.3
In this case, we have L f j L g j L h j 1, K s e −s , e j 0, l j 0.01 for j 1, 2, and μ 0 0.9.For p 2, we can compute that  K s e μ 0 s ds 10.

4.5
Thus, all conditions of Theorem 3.4 in this paper are satisfied; the equilibrium solution is exponentially stable in mean square.From above discussion, it is easy to see that 2min which implies that the condition H 9 in 18 do not hold for this example.So our results are less weaker than some previous results.

Conclusion
In this paper, we investigate the pth moment exponential stability for stochastic neural networks with mixed delays under nonlinear impulsive effects.By means of Lyapunov functionals, stochastic analysis, and differential inequality technique, some sufficient conditions for the pth moment exponential stability of this system are derived.The results of this paper are new, and they supplement and improve some of the previously known results 5, 8, 18, 19, 26, 27 .Moreover, examples are given to illustrate the effectiveness of our results.Furthermore, the method given in this paper may be extended to study other neural networks, such as the model in 29 and stochastic Cohen-Grossberg neural networks in 30 , and we can get improved results too.