On Synchronization Control Of Complex Networks With Spatial Characteristics

Complex networks can be found everywhere,from the complicated nature to the intricate human society.Synchronization of network is one of the ideal dynamics behaviors in practical engineering applications.Due to its rich content and wide range of applications,synchronization has an enduring appeal in the study of complex network dynamics.Spatio-temporal networks,as a significant class of complex networks,not only can more accurately simulate the real systems which depend on time and space evolution,such as heat conduction,wave propagation and so on,but also can profoundly reveal the nature of changes caused by diffusion effect.Therefore,the synchronous control of spatio-temporal networks has both a rich practical background and important theoretical value.In view of the above inspiration,combined with the stability theory of differential equations,algebraic graph theory,matrix analysis and LMI technique,the synchronization of three kinds of spatio-temporal networks are investigated based on distinguished control strategies.Different from the limitation of network topology in previous results,the exponential synchronization for a class of directed spatio-temporal networks with arbitrary topology is investigated in the first part.Firstly,based on the decomposition of the topological graph into several maximal strongly connected subgraphs,an intermittent pinning controller is developed.Secondly,by adding the synchronized state as a virtual node to the original network,the exponential pinning synchronization of the underlying network is discussed.Meanwhile,by utilizing the Lyapunov stability theory and inequality technique,several synchronization criteria are derived.Finally,the detailed scheme about pinning nodes and control parameters is given,whose feasibility is substantiated by an illustrative example.Distinguish from the previous continuous coupling mode,the second part of this paper considers the coupling mechanism where node of network only communicate information at discontinuous time periods,and a class of spatio-temporal networks with intermittent coupling is proposed.Then,the distributed adaptive strategy is developed for the coupling weights.Furthermore,based on the weighted average error,the piecewise Lyapunov function is constructed,and the asymptotic synchronization of controlled network is analysed by piecewise analysis method and series theory.Additionally,in order to reduce control consumption,an adaptive pinning strategy based on directed spanning tree is designed.To illustrate the feasibility of the adaptive strategy,a relevant numerical example is finally given.Considering the phenomenon of diffusion in real networks,the spatial diffusion coupling term is introduced into spatio-temporal networks in the third part,and the fixed-time synchronization for a kind of spatio-temporal networks satisfying mixed boundary conditions is researched.Firstly,based on different norms,two fixed-time boundary controllers without linear terms are developed.Secondly,based on the theory of Lyapunov function and fixed time stability,some simple criteria for fixed-time synchronization and the estimation of settling-time are derived.Finally,the effectiveness of the control strategy is substantiated by a numerical example.