| In recent years,complex systems and complex dynamic networks have received exploration on attentions from scholars in various disciplines such as ecological science,engineering science and social science.Synchronization is a typical collective behavior in complex networks,which has become one of the strategic issues in the frontier of science and technology in the 21 st century because of its important practical significance and universality.It is obvious that the cluster phenomenon exists in many networks in the real world.It’s basic characteristics are that the systems belonging to the same cluster are relatively closely connected,while those belonging to different clusters are relatively sparse.In this thesis,the related theory of synchronous control of complex network is systematically expounded.Considering the complexity of the real environment,several complex cluster networks models with time delay and multiple coupling modes are presented and correspondingly,some effective control strategies are proposed.The main research contents are as follows:Firstly,the cluster synchronization problem for a kind of delayed complex dynamical networks with nonlinearly couplings composed of heterogeneous and homogeneous Lur’e systems is investigated.Considering the cluster-tree topology structures of the networks,a kind of pinning feedback controller is cleverly designed,which only imposed on the Lur’e systems in current cluster which have directed paths with those Lur’e systems in the other clusters.Thus,it effectively reduces the number of controllers and reduces the control cost.By constructing the Lyapunov functional reasonably,sufficient conditions for the cluster synchronization of the heterogeneous and homogeneous Lur’e dynamical networks are obtained respectively,according to the Lyapunov stability theorem,the sector condition and the acceptable nonlinear coupling function class.Finally,the effectiveness of the synchronization conditions and the control strategy are explained via a numerical simulation example.Secondly,the globally and exponentially adaptive cluster synchronization for a kind of nonlinearly and nonidentically coupled Lur’e dynamical networks with multiple timevarying delays is investigated.This part considers the case that all the systems in the same cluster have the same dynamic behaviors,while different dynamic behaviors in different clusters in this networks.Considering the complexity of the controller operating environment,the impulsive pinning controller with time-varying delay is proposed.Then,sufficient conditions for the cluster synchronization are obtained according to the Lyapunov stability theorem,mathematical induction method and the concept of average impulsive interval.Furthermore,to decrease the control costs,a kind of adaptive updating law for the pinning feedback control strength is designed.Thus,the optimal control strengths of the networks to achieve cluster synchronization are obtained.Finally,the validity of the theoretical analysis and the control schemes are illustrated by means of a numerical simulation.Finally,in order to build a more realistic networks model,the problem of globally and exponentially adaptive cluster synchronization for a class of Lur’e dynamical networks with multiple time-varying delays and hybrid couplings is discussed.Considering the derivative coupling among systems in this networks,a novel Lyapunov functional is constructed.In addition,based on the clustering topology of networks,a kind of adaptive pinning controller is cleverly designed,which is only applied to a small fraction of systems.In order to improve the synchronization control rate and control efficiency of complex networks,an impulsive control strategy is introduced that only controls the system at discrete time points.In view of the concept of average impulsive interval,the comparison principle,the extended parameter variation methods and the reductio ad absurdum,sufficient conditions are acquired for achieving the cluster synchronization of complex networks.Finally,through three sets of numerical simulations,the validity of the theoretical results and the effectiveness of the control strategy are verified. |
