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This paper investigates the synchronization problem for neural networks with leakage delay and both discrete and distributed time-varying delays under sampled-data control. By employing the Lyapunov functional method and using the matrix inequality techniques, a delay-dependent LMIs criterion is given to ensure that the master systems and the slave systems are synchronous. An example with simulations is given to show the effectiveness of the proposed criterion.

Since the pioneering works of Pecora and Carroll [

In the past decades, some works dealing with the synchronization of neural networks have also appeared; for example, see [

On the other hand, with the development of networked control systems, sampled-data control in the presence of a constant input delay has been an important research area in recent years, because networked control systems are usually modeled as sampled-data systems under variable sampling with an additional network-induced delay [

Motivated by the previous discussions, the objective of this paper is to study the synchronization for neural networks with leakage delay and both discrete and distributed time-varying delays by using sampled-data control approach. The obtained sufficient conditions do not require the differentiability of time-varying delays and are expressed in terms of linear matrix inequalities, which can be checked numerically using the effective LMI toolbox in MATLAB. An example is given to show the effectiveness and less conservatism of the proposed criterion.

Consider the following neural networks with leakage delay and mixed time-varying delays:

In this paper, system (

Throughout this paper, we make the following assumptions.

For any

for all

The leakage delay

where

By defining the error signal as

In many real-world applications, it is difficult to guarantee that the state variables transmitted to controllers are continuous. In addition, in order to make full use of modern computer technique, the sampled-data feedback control is applied to synchronize delayed neural networks [

Substituting control law (

Clearly, it is difficult to analyze the synchronization of neural networks based on error system (

By substituting (

The main purpose of this paper is to design controller with the form (

To prove our result, the following lemmas that can be found in [

For any constant matrix

Given constant matrices

Suppose that (H1) and (H2) hold. If there exist seven symmetric positive definite matrices

in which

From assumption (H1), we know that

From model (

with

By using Lemma

To verify the effectiveness of the theoretical result of this paper, consider the following example.

Consider master system (

The chaotic behaviors of master system (

The chaotic behavior of master system (

The chaotic behavior of slave system (

It can be verified that assumptions (H1) and (H2) are satisfied, and

Further, the sampling period is taken as

State trajectory of

In this paper, we have dealt with the synchronization problems for chaotic neural networks with leakage delay and both discrete and distributed time-varying delays. Based on the sampled-data control techniques, Lyapunov stability theory, and the matrix inequality techniques, a delay-dependent criterion sufficient condition has been developed to guarantee synchronization of the considered coupled neural networks. An example has been provided to demonstrate the effectiveness of the proposed criterion since the feasible solutions to the given LMIs criterion in this paper have been found.

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

The authors would like to thank the editor and the anonymous reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the National Natural Science Foundation of China under Grants 61273021, 60974132, 11172247, and 51208538 and in part by the Natural Science Foundation Project of CQ cstc2013jjB40008.