This work addresses the asymptotic stability for a class of impulsive cellular neural networks with time-varying delays and reaction-diffusion. By using the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality as well as piecewise continuous Lyapunov functions, we summarize some new and concise sufficient conditions ensuring the global exponential asymptotic stability of the equilibrium point. The provided stability criteria are applicable to Dirichlet boundary condition and showed to be dependent on all of the reaction-diffusion coefficients, the dimension of the space, the delay, and the boundary of the spatial variables. Two examples are finally illustrated to demonstrate the effectiveness of our obtained results.

Cellular neural networks (CNNs), proposed by Chua and Yang in 1988 [

With reference to neural networks, however, it is noteworthy that the state of electronic networks is actually subject to instantaneous perturbations more often than not. On this account, the networks experience abrupt change at certain instants which may be caused by a switching phenomenon, frequency change, or other sudden noise; that is, the networks often exhibit impulsive effects [

In reality, besides impulsive effects, diffusion effects are also nonignorable since diffusion is unavoidable when electrons are moving in asymmetric electromagnetic fields. As such, the model of neural networks with both impulses and reaction-diffusion should be more accurate to describe the evolutionary process of the systems in question, and it is necessary to consider the effects of both diffusion and impulses on the stability of CNNs and DCNNs.

In the past years, there have been a few theoretical contributions to the stability of CNNs and DCNNs with impulses and diffusion. For instance, Qiu [

In this paper, unlike the methods of impulsive differential inequalities and Poincaré inequality used in [

The rest of the paper is organized as follows. In Section

Let

We consider the following impulsive neural networks with time delays and reaction-diffusion terms:

Denote by

The solution

Throughout this paper, the norm of

Before moving on, we introduce two hypotheses as follows.

Activation function

The functions

According to (H1) and (H2), it is easy to see that problems ((

The equilibrium point

Assume that

(A1) the sequence

(A2)

(A3)

Let

If

Provided that

for

there exists a constant

Multiplying both sides of (

Regarding the right-hand part of (

Moreover, we derive from (H1) that

We define a Lyapunov function

Construct

Set

Choose small enough

which yields after letting

We now proceed to estimate the value of

If we let

Note that

holds for

This, together with (

Recalling assumptions that

By induction argument, we reach

Therefore,

According to Lemma

According to the conditions of Theorem

Providing that

for

there exists constant

Define a Lyapunov function

Construct another Lyapunov function defined by

Set

The relations (

By induction argument, we arrive at

Introducing

It then results from Lemma

On the other hand, since

And (

Theorem

In the sequel, we follow the same procedures as in Theorems

Provided that

for

there exist constants

Assume that

for

there exist constants

Then, the equilibrium point

Further, on the condition that

Identical with the proof of Theorem

Assume that

for

there exist constants

Then, the equilibrium point

Different from Theorems

Consider the following impulsive reaction-diffusion delayed neural network:

By selecting

According to Theorem

Consider the following impulsive reaction-diffusion delayed neural network:

Select

It is then concluded from Theorem

The work is supported by National Natural Science Foundation of China under Grant 60904028.