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The synchronization of coupled networks with mixed delays is investigated by employing Lyapunov functional method and intermittent control. A sufficient condition is derived to ensure the global synchronization of coupled networks, which is controlled by the designed intermittent controller. Finally, a numerical simulation is constructed to justify the theoretical analysis.

Various large-scale and complicated systems can be modelled by complex networks, such as the Internet, genetic networks, ecosystems, electrical power grids, and the social networks. A complex network is a large set of interconnected nodes, which can be described by the graph with the nodes representing individuals in the graph and the edges representing the connections among them. The most remarkable recent advances in study of complex networks are the developments of the small-world network model [

The dynamical behaviors of complex networks have become a focal topic of great interest, particularly the synchronization phenomena, which is observed in natural, social, physical, and biological systems and has been widely applied in a variety of fields, such as secure communication, image processing, and harmonic oscillation generation. It is noted that the dynamical behavior of a complex network is determined not only by the dynamical rules governing the isolated nodes, referred to as self-dynamics, but also by information flow along the edges, which depends on the topology of the network. Synchronization in an array of linearly coupled dynamical systems was investigated in [

In the case that the whole network cannot synchronize by itself, some controllers should be designed and applied to force the network to synchronize. Recently, another interesting intermittent control was introduced and studied, that is, the control time is periodic, and in any period the time is composed of work time and rest time. It is a straightforward engineering approach to process control of any typelan approach that has been used for a variety of purposes in such engineering fields as manufacturing, transportation, and communication. Intermittent control has been introduced to control nonlinear dynamical systems [

Another type of time delays, namely, distributed delays, has begun to receive research attention. The main reason is that a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths, and it is desirable to model them by introducing continuously distributed delays over a certain duration of time, such that the distant past has less influence compared to the recent behavior of the state [

Motivated by the above discussion, the intermittent controller will be designed to achieve the synchronization for coupled networks with mixed delay. The rest of the paper is organized as follows. In Section

Consider a dynamical network consisting of

Note that a solution to an isolated node satisfies

We assume that

For any positive integers

By the definition of Kronecker product, the following properties hold:

For any vectors

For any constant matrix

Let

Let

Suppose that assumption

Consider the following Lyapunov function:

For

From Lemma

From Lemma

Suppose that

First, we have

For any

For given control period

In Theorem

Furthermore, let

Therefore, (a)–(f) hold if

Corollary

Consider the following coupled networks:

Error state

Error state

In this paper, synchronization of coupled networks with mixed time delay has been investigated via intermittent control. Some criteria for ensuring coupled networks synchronization have been derived, and some analytical techniques have been proposed to obtain appropriate control period

This work was jointly supported by the National Natural Science Foundation of China under Grant 61004043 and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 2009092120066.