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This paper considers the dynamics of switched cellular neural networks (CNNs) with mixed delays. With the help of the Lyapnnov function combined with the average dwell time method and linear matrix inequalities (LMIs) technique, some novel sufficient conditions on the issue of the uniformly ultimate boundedness, the existence of an attractor, and the globally exponential stability for CNN are given. The provided conditions are expressed in terms of LMI, which can be easily checked by the effective LMI toolbox in Matlab in practice.

Cellular neural networks (CNNs) introduced by Chua and Yang in [

Although the use of constant fixed delays in models of delayed feedback provides of a good approximation in simple circuits consisting a small number of cells, recently, it has been well recognized that 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. Therefore, there will be a distribution of conduction velocities along these pathways and a distribution of propagation delays. As the fact that delays in artificial neural networks are usually time varying and sometimes vary violently with time, system (

On the other hand, neural networks are complex and large-scale nonlinear dynamics; during hardware implementation, the connection topology of networks may change very quickly and link failures or new creation in networks often bring about switching connection topology [

Corresponding to the switching signal

Over the past decades, the stability of the unique equilibrium point for switched neural networks has been intensively investigated. There are three basic problems in dealing with the stability of switched systems: (

Just as pointed out in [

In fact, studies on neural dynamical systems involve not only the discussion of stability property but also other dynamics behaviors such as the ultimate boundedness and attractor [

Motivated by the above discussions, in the following, the objective of this paper is to establish a set of sufficient criteria on the attractor and ultimate boundedness for the switched system. The rest of this paper is organized as follows. Section

For the sake of convenience, throughout this paper, two of the standing assumptions are formulated below:

Assume the functions

Assume there exist constants

We shall point out that the constants

Without loss of generality, let

System (

System (

The nonempty closed set

For any switching signal

Assume there is a constant

Choose the following Lyapunov functional:

From assumption

Then we have

Therefore, we obtain

If one chooses

If all of the conditions of Theorem

If one chooses

In addition to all of the conditions of Theorem

If

By (

We now consider the switched cellular neural networks without uncertainties as system (

For a given constant

Define the Lyapunov functional candidate

If one chooses

If all of the conditions of Theorem

If one chooses

In addition to all of the conditions of Theorem

If

Up to now, various dynamical results have been proposed for switched neural networks in the literature. For example, in [

We notice that Lian and Zhang developed an LMI approach to study the stability of switched Cohen-Grossberg neural networks and obtained some novel results in a very recent paper [

When investigating the stability, although the adopted Lyapunov function in this paper is similar to those used in [

When the uncertainties appear in the system (

In this section, we present an example to illustrate the effectiveness of the proposed results. Consider the switched cellular neural networks with two subsystems.

Consider the switched cellular neural networks system (

From assumptions

Choosing

In this paper, the dynamics of switched cellular neural networks with mixed delays (interval time-varying delays and distributed-time varying delays) are investigated. Novel multiple Lyapunov-Krasovkii functional methods are designed to establish new sufficient conditions guaranteeing the uniformly ultimate boundedness, the existence of an attractor, and the globally exponential stability. The derived conditions are expressed in terms of LMIs, which are more relaxed than algebraic formulation and can be easily checked by the effective LMI toolbox in Matlab in practice.

The authors are extremely grateful to Professor Jinde Cao and the anonymous reviewers for their constructive and valuable comments, which have contributed much to the improvement of this paper. This work was jointly supported by the National Natural Science Foundation of China under Grants nos. 11101053, 70921001, and 71171024, the Key Project of Chinese Ministry of Education under Grant no. 211118, and the Excellent Youth Foundation of Educational Committee of Hunan Provincial no. 10B002, the Scientific Research Funds of Hunan Provincial Science and Technology Department of China.