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In this paper the globally exponential stability criteria of delayed Hopfield neural networks with variable-time impulses are established. The proposed criteria can also be applied in Hopfield neural networks with fixed-time impulses. A numerical example is presented to illustrate the effectiveness of our theoretical results.

Hopfield neural networks [

It is well known that time delay is unavoidable due to finite switching speeds of the amplifiers and it may cause oscillations or instability of dynamic systems. The effects of time delay on the dynamical behavior of neural networks are nonnegligible. Some stability criteria for delayed Hopfield neural networks have been proposed in [

However, up to now, the vast majority of stability results for impulsive Hopfield neural networks are focused on the case of fixed-time impulses. As we know, variable-time impulses arise naturally in biological and physiological systems. The primary difference between neural network with fixed-time impulses and neural network with variable-time impulses is the impulsive instant. In the neural network with fixed-time impulses, the impulsive instant is completely fixed and not about the state of system. But in neural network with variable-time impulses, the impulsive instant is not fixed and determined by state of system. In [

This paper is organized as follows. In the coming section we introduce some notations, definition, and lemmas. In Section

In this paper, we consider the following Hopfield neural networks with variable-time impulses:

there is

Throughout this paper, it is always assumed that there is at least one equilibrium point of (

In the sequel, we introduce some notations, basic definition, and lemmas:

The equilibrium point

Let

Consider the following differential inequality:

See the Appendix section.

In Lemma

The solutions of system (

Suppose that

for any

for any

From Theorem

In this section, we establish some sufficient criteria for the exponential stability of system (

Assume that, in addition to condition (A1), the following conditions are satisfied:

there are a symmetric positive definite matrix

Then the equilibrium point

Based on (A2), we know that

Because

As mentioned in [

Assume that (A1), (A2) hold, and

there are a symmetric positive definite matrix

Then the equilibrium point

In this section, we consider one example to illustrate the effectiveness of theoretical results.

Consider the following system:

Now we verify that there is no beating phenomenon in system (

It is obvious that, for

Based on [

Let

It is easy to see that

For convenience, choose

The time response curves of system (

Let

First, for

Now suppose that, for

We can find

Now we show that (

By mathematical induction, it is easy to illustrate that (

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is funded by the Research Foundation of The Natural Foundation of Chongqing City (cstc2014jcyjA40024, cstc2012jjA1459), Teaching & Research Program of Chongqing Education Committee (KJ1401307, KJ131401), and Research Project of Chongqing University of Science and Technology (CK2013B15).