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This paper investigates the mean-square exponential synchronization issues of delayed stochastic complex dynamical networks with switching topology and impulsive control. By using the Lyapunov functional method, impulsive control theory, and linear matrix inequality (LMI) approaches, some sufficient conditions are derived to guarantee the mean-square exponential synchronization of delay complex dynamical network with switch topology, which are independent of the network size and switch topology. Numerical simulations are given to illustrate the effectiveness of the obtained results in the end.

Complex networks are everywhere in nature and our daily life, such as the Internet, ecosystems, social networks, World Wide Web, and neural networks. A complex network can be described by a set of nodes and edges interconnecting these nodes together. During the last two decades, complex networks have been focused on by scientists from various fields, such as mathematical, engineering, and social and economic science. There are many literatures concerning the collective behaviors of complex networks [

Due to many complex systems may experience abrupt changes in their connection caused by some phenomena such as link failures, component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbance, and so forth. Thus, the synchronization of complex network with switching topology has been studied in [

Uncertainties commonly exist in the real world, such as stochastic forces on the physical systems and noisy measurements caused by environmental uncertainties. Thus, a stochastic behavior should be produced instead of a deterministic one [

In many systems, the impulsive effects are common phenomena due to instantaneous perturbations at certain moments [

Based on the above discussion, studying the synchronization problem of the complex dynamical network with switch topology and impulsive effects is very useful and meaningful. It should be pointed out that exponential synchronization of the coupling delay complex dynamical networks with switch topology and impulsive effects has received very little research attention.

In this paper, we investigate the problem of exponential synchronization of coupling delay complex dynamical network with switching topology and impulsive effects, basing on the Lyapunov theory and impulsive control theory and by assuming that there exists finite connective topology of the complex dynamical network, and the connective topology may be switched (or jumped) from one to another at different moments. For controlling, the synchronization state can be any weighting average of the network states. It means that each node of the dynamical network can contribute to synchronization of the network in its weight. Finally, some sufficient conditions are given to guarantee the synchronization of the complex dynamical network, which are independent of the network size and switch topology.

The switching complex dynamical networks investigated in this paper consist of

The initial conditions associated with (

In general, the synchronization state

The following assumptions will be used throughout this paper for establishing the synchronization conditions.

(H1)

(H2) Let

(H3)

Define error state

The complex network (

A continuous function

The function class QUAD includes almost all the well-known chaotic systems with or without delays such as the Lorenz system, the Rössler system, the Chen system, the delayed Chua’s circuit, the logistic delayed differential system, the delayed Hopfield neural network, and the delayed CNNs. We shall simply write

In order to derive the main results, it is necessary to propose the following lemmas.

The following linear matrix inequality

In this section, we investigate the exponential stability condition of the error system (

Let assumptions (H1) and (H2) be true and let

Define the Lyapunov-Krasovskii function

On the other hand, from the construction of

According to (

Let

Using Condition (

Theorem

If the switching signal

Let assumptions (H1) and (H2) be true and let

Let impulsive gains

In this section, we give two numerical simulations to illustrate the feasibility and effectiveness of the theoretical results presented in the previous sections.

Consider a three-order Chua’s circuit described as follows:

Consider a network model consists of 5 nodes and 2-connected topology. Each node in the network is a three-order Chua’s circuit described by

The coupling matrices are as follows:

If we choose

Time evolution of synchronization errors

Time evolution of synchronization errors

Time evolution of synchronization errors

In this paper, the exponential synchronization of the coupling delay stochastic complex networks with switch topology and impulsive effects has been investigated. Based on the Lyapunov stability theory, LMI, and impulsive control theory, some simple, yet generic, criteria for exponential synchronization have been derived. It has shown that criteria can provide an effective control scheme to synchronize for a given coupled delay, the network size, and switch topology. Furthermore, the effectiveness of the presented method has been verified by numerical simulations.

This work was supported by the National Science Foundation of China under Grant no. 61070087, the Guangdong Education University Industry Cooperation Projects (2009B090300355), the Shenzhen Basic Research Project (JC200903120040A, JC201006010743A), and the Shenzhen Polytechnic Youth Innovation Project (2210k3010020). The authors are very grateful to the reviewers and editors for their valuable comments and suggestions that improved the presentation of this paper.