Stability And Synchronization Of Several Kinds Of Fractional-order Differential Systems

Fractional-order differential equations,as a natural generalization of the case of integer-order,can accurately characterize some complex phenomena in nature.It has been proved that there are many unique advantages in the fields of engineering,chemistry,signal processing,physics and so on,which has become a hot research topic of common concern in the world.Stability,as one of the most important performance indexes of control system,is the precondition to ensure the system to operate normally.Many do-mestic and foreign experts and scholars have invested in the research of the stability and synchronization of the fractional order complex system,and have obtained a series of meaningful research results.The stability of fractional-order differential systems with de-lay has good application value and practical significance because the time-delay is widely existed in various practical systems and has important influence on the stability of the system.However,due to the complexity of fractional derivative,the research results of stability of fractional-order time-delay systems are still rare.Motivated by above,this pa-per will discuss the stability of fractional-order nonlinear differential systems,the stability of fractional-order neutral systems,and study the global synchronization problem for a class of fractional-order complex networks with non-delayed and delayed couplings.The main research contents of this paper are as follows:In the first chapter,we introduce the relative background,research progress,basic concepts and necessary lemmas of fractional-order system.Then,two important inequal-ities of fractional derivatives of Reimann-Liouville are presented.In the second chapter,we mainly study the stability of Reimann-liouville fractional-order nonlinear(delayed)systems.By using our proposed fractional derivative inequality and Lyapunov Direct method,we obtain the linear matrix inequality criterion on asymp-totic stability of these two kinds of systems.Finally,concrete examples are presented to verify the validity of the theoretical results.In the third chapter,by using the Lyapunov direct method and the linear matrix in-equality,two algebraic criteria on asymptotic stability of the Reimann-Liouville fractional-order neutral linear system are obtained.The main advantage of our proposed method is to judge the stability of the system by calculating the integer-order derivative of the Lyapunov function,and the numerical example further verifies the validity and simplicity of the theoretical results.In the fourth chapter,the synchronization problem of a class of Caputo fractional-oeder complex networks with non-delayed and delayed coupling is studied.By constructing a simple quadratic Lyapunov function and using fractional-order Razumikhin theorem,three linear matrix inequality criteria on global synchronization of complex network are obtained,and numerical simulations are given to show the efficiency of the obtained results.