Stability Analysis for Impulsive Stochastic Reaction-Diffusion Differential System and Its Application to Neural Networks

This paper is concerned with the stability of impulsive stochastic reaction-diffusion differential systems with mixed time delays. First, an equivalent relation between the solution of a stochastic reaction-diffusion differential system with time delays and impulsive effects and that of corresponding system without impulses is established. Then, some stability criteria for the stochastic reaction-diffusion differential system with time delays and impulsive effects are derived. Finally, the stability criteria are applied to impulsive stochastic reaction-diffusion Cohen-Grossberg neural networks with mixed time delays, and sufficient conditions are obtained for the exponentialp-stability of the zero solution to the neural networks. An example is given to illustrate the effectiveness of our theoretical results. The systems we studied in this paper are more general, and some existing results are improved and extended.


Introduction
In recent years, impulsive dynamical systems have attracted considerable attention due to its wide applications in the areas of economics, physics, population dynamics, engineering, biology, and so on.These systems arise because they are subject to abrupt state changes at certain moments of time, and these changes may be related to such phenomena as shocks, harvesting, or other faults.Meanwhile, time delays are frequently encountered in real world, which can cause instability and oscillations in a system.A large number of stability criteria of impulsive delay systems have been reported (see [1][2][3][4][5][6] and references therein).
They showed the stability results of system (1) by transforming (1) into an equivalent system without impulses.Generally speaking, diffusion effects cannot be avoided in systems modeling many real world phenomena.As a representation example in neural networks, when electrons are moving in an asymmetric electromagnetic field, it inevitably leads to diffusion phenomena.In [5,[9][10][11], the stabilities of the equilibrium points of some types of neural networks with reaction-diffusion terms have been investigated.On the other hand, distributed delay systems can characterize the cumulative effects of the past values of the dynamic and are often used to model the time lag phenomena in thermodynamics, ecology, epidemiology, and neural networks.For example, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths.It is desired to model them by introducing continuously distributed delays over a certain duration of time such that the distant past has less influence compared to the recent behavior of the state.
However, in [8], the authors neglected the effects of diffusion and distributed delays.To the best of our knowledge, there are few results about the stability of impulsive stochastic reaction-diffusion deferential systems (ISRDDSs) with mixed time delays.Motivated by [8] and the previous discussions, we are concerned with the stability of the following ISRDDS with time-varying discrete delays and distributed delays:  The organization of this paper is as follows.In Section 2, some preliminaries are given.In Section 3, by transforming the solutions of the stochastic reaction-diffusion differential system with delay and impulsive effects into that of the corresponding system without impulses, some stability criteria for the stochastic reaction-diffusion differential system with delay and impulsive effects are derived.In Section 4, the stability criteria are applied to impulsive stochastic reaction-diffusion Cohen-Grossberg neural networks (ISRD-CGNNs) with mixed time delays, and sufficient conditions are obtained for the exponential -stability of the zero solution to the neural networks.In Section 5, a numerical example is provided to illustrate the effectiveness of the theoretical results.A concluding remark is given in Section 6 to end this work.

Preliminaries
For convenience, we introduce several notations.Let Definition 2. The zero solution of system ( 2) is said to be as follows.
(ii) Exponentially -stable if there is a pair of positive constants  and  such that, for any initial condition  ∈    0 ([−, 0] × Ω, R  ), there holds ‖(, )‖  ≤ ‖‖   − ,  ≥ 0. Here,  is called the exponential convergence rate.When  = 2 especially, it is said to be exponentially stable in mean square.

Stability Criteria
In this section, we first establish an equivalent relation between the solution of system (2) and that of system (6).
Remark 4. Lemma 3 gives the equivalent relation between the solution of a stochastic reaction-diffusion differential delay system with impulsive effects and the solution of a corresponding system without impulses.Based on the "equivalent method, " the existence and uniqueness of the solution of a stochastic reaction-diffusion differential delay system with impulsive effects can be derived by a new way; that is, any conditions that ensure the existence and uniqueness of the solution of system (6) without impulses will also ensure the existence and uniqueness of the system (2) with impulses.
In what follows, we will reduce the stabilities of system (2) to those of corresponding system (6).
Proof.Let (, ) and V(, ) be the solutions of systems ( 2) and ( 6), respectively.Since the zero solution of ( 6) is -stable, we have that, for any  > 0, there exists a scalar  > 0 such that the initial condition Hence, the zero solution of (2) is -stable.Using similar arguments, we can verify that if the zero solution of ( 6) is exponentially -stable (asymptotically stable), then the zero solution of ( 2) is also exponentially -stable (asymptotically stable).This completes the proof.
In a similar way, we can derive the following results.
and the zero solution of (2) is -stable (exponentially -stable, asymptotically stable), then the zero solution of ( 6) is also stable (exponentially -stable, asymptotically stable).

Theorem 7. Assume that (H1)-(H4) hold and inequalities
From the definition of   in Section 2, we have Obviously, the condition (11) in Theorem 5 is less conservative than (14), in which (11) ensures that the stability of the delayed stochastic reactiondiffusion differential system without impulses can be used to judge the stability of the corresponding system with impulses.
Remark 9.In [2,8], the authors dealt with the stochastic differential systems with delay and nonlinear impulsive effects, the stability results of which were showed by transforming the system into a corresponding system without impulses.However, the distributed delays and diffusion effects were not taken into account in the previous systems.In this paper, we incorporated stochastic perturbations, reaction-diffusion effects, and mixed time delays into impulsive differential system and derived the stability criteria of the system.It is readily seen that our results are more general than those reported in [2,8].
Lemma 10 (see [16]).Let  be an × matrix with nonpositive off-diagonal elements; then  is an M-matrix if and only if there exists a vector  > 0 such that  > 0.
Remark 14.The stability of impulsive Cohen-Grossberg neural networks without spacial diffusion or distributed delays or stochastic disturbance, which are special cases of system (15), have been studied in [5,13,[18][19][20].It should be noted that the main result in [13] is a special case of Theorem 13.Further, the stability criteria derived in [5,[18][19][20] are dependent on the intervals of adjoining impulsive moments, while our results are independent of that.Thus, our results are new, and they effectually complement or improve the previously published results.
Remark 15.In [19,[21][22][23][24][25][26], the reaction-diffusion neural networks have been investigated.Nevertheless, the diffusion terms were eliminated by inequality analysis techniques, and the derived conditions for the stability of neural networks are the same as those obtained in the cases when there are no reaction-diffusion terms in the systems.Thus, our results including reaction-diffusion terms are less conservative than those in [19,[21][22][23][24][25][26].
Remark 16.As far as we know, almost all the existing results concerning the stability of neural networks are based on 2norm (e.g., [5,11,19,21,22,[26][27][28]).In this paper, we derived the stability criteria of ISRDCGNNs with mixed time delays in terms of -norm.Hence, our results generalize and improve the existing results reported in the previous literature.

Numerical Example
In this section, we give an example to illustrate the main theoretical results in Sections 3 and 4.
In system (15), let in which  +  is an M-matrix, and all the conditions of Theorem 13 are satisfied.From Theorem 13, we know that the zero solution of system (15) with the parameters and functions above is exponentially 4-stable (see Figure 1).

Concluding Remark
In this paper, we incorporated stochastic perturbations, reaction-diffusion effects, and mixed time delays into impulsive differential systems.First, an equivalent relation between the solution of a stochastic reaction-diffusion differential system with time delays and impulsive effects and that of corresponding system without impulses was established.Second, some stability criteria for the stochastic reactiondiffusion differential system with time delays and impulsive effects were derived by transforming the solutions of the system to those of corresponding one without impulses.Third, the stability criteria were applied to ISRDCGNNs with mixed time delays, and sufficient conditions were obtained for the exponential -stability of the zero solution to the neural networks.Lastly, a numerical example was provided to illustrate the effectiveness of our theoretical results.Our stability results provide a new, convenient, and efficient approach to study the stability of stochastic reaction-diffusion differential systems with time delays and impulsive effects, and some previously published results are generalized and improved.