Pinning Adaptive Synchronization of Delayed Coupled Dynamical Networks via Periodically Intermittent Control

This paper investigates the exponential synchronization problem of delayed coupled dynamical networks by using adaptive pinning periodically intermittent control. Based on the Lyapunov method, by designing adaptive feedback controller, some sufficient conditions are presented to ensure the exponential synchronization of coupled dynamical networks with delayed coupling. Furthermore, a numerical example is given to demonstrate the validity of the theoretical results.


Introduction
Complex networks have received a great deal of attention due to their many potential practical applications [1,2].A family of dynamically interacting units composes a kind of complex networks which can exhibit a number of emerging phenomena.Among various dynamical behaviors of complex networks, synchronization is a significant and interesting phenomenon, such as synchronization phenomena on the Internet, synchronization transfer of digital or analog signals in communication network, and synchronization related to biological neural networks.Recently, much works have been devoted to research the synchronization problem of complex networks [3][4][5].
In the case where the network cannot synchronize by itself, in order to drive the network to synchronize, many effective control techniques have been reported, such as feedback control [6], sampled-data control [7], adaptive control [8,9], pinning control [10], impulsive control [11], and intermittent control [12].In [9], the synchronization of a class of complex network by adding an adaptive controller to all nodes has been discussed.But in practice, it is too costly and impractical to add controllers to all nodes in a largescale network.To reduce the number of controlled nodes, pinning control is introduced [10], in which controllers are only applied to partial nodes.This case of control techniques has been earlier reported in paper [11][12][13][14].In addition, the adaptive pinning control method, which is utilized to get the appropriate control gains effectively, has received considerable research attention.An adaptive pinning control method is proposed in [15] to synchronize for a delayed complex dynamical network with free coupling matrix.Besides these, there are many literatures to study adaptive pinning control problems of networks [16][17][18].
One the other hand, intermittent control has been widely used in engineering fields due to its practical and easy implementation in engineering control.In recent years, many important and interesting results on stabilization and synchronization of delayed dynamical networks by using intermittent control have been obtained.Based on ∞-norm, authors in [19] investigated a class of Cohen-Grossberg neural networks with time-varying delays by designing a periodically intermittent controller.In [20], by using periodically intermittent control, Gan studied the stochastic neural networks with leakage delay and reaction-diffusion terms; some new and less conservative synchronization conditions based on -norm were derived.The pinning periodically intermittent control is used to achieve the synchronization of delayed complex network [21,22].To the best of our knowledge, the problem of adaptive pinning synchronization 2 Mathematical Problems in Engineering for delayed coupled dynamical networks has received very little research attention.
In this paper, we aim to further investigate adaptive pinning synchronization of delayed coupled dynamical network via periodically intermittent control.By using Lyapunov stability theory and designing adaptive feedback control gains, several criteria are given to guarantee synchronization of delayed coupled dynamical networks.A numerical simulation is also presented to show the effectiveness of the proposed method.

Model and Preliminaries
Consider the complex network consisting of  nodes and the th node described by the following state equation: where , where  2 is the transmittal delay. = (  ) × and Ĝ = (ĝ  ) × are the configuration matrices; if there is a link from node  to node  at time  (at time − 2 ), then   > 0 ( ĝ > 0), where  ̸ = .Otherwise,   = 0 ( ĝ = 0).It is assumed that  and Ĝ satisfy the diffusive coupling connection, ∑  =1   = 0 and ∑  =1 ĝ = 0.   ∈   are the control inputs.Note that the coupling configuration matrix  and matrices , , , Γ and Γ  are not assumed to be symmetric.
In order to derive the main results, the following definitions and lemmas are needed in this paper.

Main Result
In order to realize synchronization of the couple network by pinning periodically intermittent control, some controllers are added to selected partial nodes, and the controllers   (1 ≤  ≤ ) can be described by where  > 0 denotes the control period, 0 <  < 1,  ∈ N: and   () is the adaptive feedback strength for which the update law is to be designed.When  ∈ [, ( + )) the error system (4) can be rewritten as When  ∈ [( + ), ( + 1)), the error system (4) can be rewritten as Our objective is to design suitable  and  such that the delayed couple network can realize synchronization.The main results are stated as follows.
When Γ = 0, only delayed coupling exists in the networks.One has the following corollary.

Numerical Simulation
In this section, we present a numerical simulation to illustrate the feasibility and effectiveness of our results.
Consider the coupled network (1) consisting of 6 identical Chua oscillators with time delayed nonlinearity.The dynamics of the Chua oscillator is given by where   () ∈ The dynamical behavior of the synchronization manifold () is shown in Figure 1.

Conclusion
In this paper, we have investigated the exponential synchronization problem for neural networks by pinning periodically intermittent control.Based on Lyapunov stability theory and periodically intermittent control method, some novel conditions for synchronization are derived.Furthermore, numerical simulations have verified the effectiveness of the presented method.