Syhchronize Different Coupled Systems Via Different Controllers Based On Graph-Theoretic Approach

The mathematical description of the network is a directed graph consisting of vertices and directed arcs connecting them.At each vertex,the local dynamics are given by a system of differential equations called vertex systems,and dynamic behavior cannot be directly analyzed,and many applied dynamic properties cannot be modified.Therefore,controller is required to achieve stabilization or synchronization.The purpose of this paper is to analyze the dynamic behavior of such systems under different controllers via graph theory,considering the stability conditions and synchronization conditions of time-delay stochastic coupling networks under the action of three types of controllers.After proposed in last century,Lyapunov functional method was widely used to analysis the dynamical behavior of differential equations.However,it is difficult to give out a global Lyapunov function of a given system directly.A method combing graph-theoretical method and Lyapunov theory which can avoid the difficulty of construct Lyapunov for any given system was put forward by Michael Li and his co-works` work.In Chapter 2 the stability of the Lyapunov form for the stability of the trivial solution of a coupled network with coupled delays based on a strongly connected directed graph under periodic intermittent controllers will be derived.Theorem and stability criteria based on coefficients of the entire system will be given.What’s more,the numerical simulation of the stochastic coupled oscillator with time-varying delay and the corresponding application stability conditions are given.The stability theorem of Lyapunov form under pinning aperiodical intermittent control and the stability criterion of coefficients based on the whole system are given in the form of two corollaries.Chapter 3 of this article will focus on the synchronization problem under adaptive control instead of pinning aperoidically intermittent control in Chapter 2.The stability theorem of Lyapunov form and the stability criterion of coefficient type will be given by using of the method combing graph-theoretical method and Lyapunov theory.Furtherlmore,in order to verify the validity of the obtained results,an numerical example will be proposed based on Chua`s circuit is going to be given.