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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 exponential

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 [

Stochastic effects are common phenomena due to disturbances or uncertainties in a system. A lot of dynamical systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment switching. Hence, considerable attention has been paid to the study of stochastic systems, and various interesting results have been reported in the literatures; for example, see [

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 [

However, in [

The organization of this paper is as follows. In Section

For convenience, we introduce several notations. Let

A function

The zero solution of system (

Exponentially

Asymptotically stable if it is stable, and there exists a

For system (

except for the zero solution of (

Denoting

A function vector

In this section, we first establish an equivalent relation between the solution of system (

Assume that (H1)–(H4) hold. Then

First, we prove the sufficiency. Letting

On the other hand, for any

Further, if

Lemma

In what follows, we will reduce the stabilities of system (

Under assumptions (H1)–(H4), if there exists a constant

Let

In a similar way, we can derive the following results.

Under assumptions (H1)–(H4), if there exists a constant

Combining Theorems

Assume that (H1)–(H4) hold and inequalities (

For impulsive neural networks, many researchers supposed that the impulsive operators are linear (e.g., [

In [

In this section, we apply our previous stability results to analyze the stability of the following ISRDCGNNs with time delays:

The boundary condition and the initial value of system (

Equivalently, we consider the following stochastic reaction-diffusion Cohen-Grossberg neura1 networks without impulses:

Throughout this section, we make the following assumptions:

there exists a constant

there exist positive constants

there exist positive constants

The following lemmas are useful in proving our main results.

Let

Let

Let

Under assumptions (H5)–(H8), if inequalities (

Since

We can choose a sufficiently small constant

By Lemma

Denoting

The stability of impulsive Cohen-Grossberg neural networks without spacial diffusion or distributed delays or stochastic disturbance, which are special cases of system (

In [

As far as we know, almost all the existing results concerning the stability of neural networks are based on 2-norm (e.g., [

In this section, we give an example to illustrate the main theoretical results in Sections

In system (

By direct calculation, we obtain that

Transient behaviors of the state variables

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 reaction-diffusion 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

This work was supported by the National Natural Science Foundation of China (11071254, 11371368) and the Natural Science Foundation of Hebei Province (A2013506012).