Research On Cluster Synchronization Of Lur’e Dynamical Network Based On Pinning Control

Complex network is a new interdisciplinary research field.Synchronization is a typical collective behavior in network,and cluster synchronization is a special synchronization mode,whose goal is to achieve the same states for the individuals in the same cluster.Due to the complexity of the real environment,control strategies are introduced to help achieve synchronization.Pinning control is a kind of control strategies that only applies controllers to some individuals,it is relatively simple and can reduce the control cost.Lur’e system has the strong engineering application background,many common dynamical systems can be studied by regarding as Lur’e system.In this article,cluster synchronization of Lur’e complex networks is studied,by designing reasonable pinning controllers,sufficient criteria are obtained to ensure the synchronization of networks.The main works of this article can be concluded as follows.Firstly,we study the cluster synchronization of a class of nonlinearly coupled Lur’e dynamical networks.The coupled complex networks consisting of not only identical Lur’e systems but also nonidentical Lur’e systems are discussed.Pinning control strategy is introduced to control part of the systems rather than all systems in the networks,and adaptive updating law is designed to obtain appropriate control strength and reduce control costs.Based on the concept of S-procedure and the acceptable nonlinear coupling function(NCF)class,sufficient conditions for realizing cluster synchronization of nonlinearly coupled Lur’e networks are obtained.In addition,numerical simulations are presented to illustrate the effectiveness of the proposed theorems and control schemes.Secondly,we discuss the cluster synchronization of nonidentical Lur’e networks with time-varying delay coupling in a convex domain.The network can achieve cluster synchronization with the designing negative feedback controller,which is beneficial to weaken the relationships among different clusters.Based on Lyapunov stability theorem,the extended Jensen’s inequality and interval division principle of integration,sufficient conditions for the cluster synchronization of the network in convex domain are derived.In addition,we also consider the systems with uncertain parameters in the networks,assuming the uncertain parameters satisfy the boundedness,we obtain the criterion that guarantee the realization of the cluster synchronization of Lur’e networks with time-varying delay and uncertain parameters.The introduction of convex domain makes the synchronization conditions simpler and less conservative.Two models are selected for simulation,and the simulation results verify the correctness of the conclusions.Finally,an adaptive finite-time cluster synchronization problem for a class of complex networks coupled by nonidentical and discontinuous Lur’e systems is discussed.By introducing the Filippov differential inclusion theory and measurable selection theorem,then effective discontinuous controllers are designed with the combination of pinning control and adaptive control.By jointly applying the finite-time stability theory and Lyapunov stability theorem,sufficient cluster synchronization criteria for the time-varying delay and nonlinearly coupled Lur’e networks are obtained in finite-time,the settling time is intelligently estimated as well.Numerical examples and simulation results show the effectiveness of the results.