Exponential Stability Of Several Kinds Of Stochastic Differential Equations With Delays

It is well known that the study of stability on system,which provides the control system with theoretical support,is one of the basic problems to analyze system.In 1892 the distinguished Russian mathematician and dynamicist Lyapunov established the Lyapunov second method as a power tool for analyzing certain systems,and bring the possibility of analyzing the stability of stochastic systems into being.In the course of the practice,stochastic factors exist objectively,and the system described by certain method may lose some properties that lead to errors.Therefore, the stochastic factor must be taken into account in the description of the system.In addition,the effects of delay on the states of the system,to which development trend is related either at present or in history,are considered.Thus,it is necessity to study the stochastic system with delay.At the same time,the phenomenon of impulsive is an unavoidable in practical application,so the study of impulsive stochastic system with delay has important theories meaning and practical value.Neural network,a dynamic system with special structure,which is developed by elicitation of the function of brain,has been applied to many fields.The impact of stochastic factor and delay on stability of the neural network is significant to investigate the stability of stochastic neural network with delay.As concerned above,based on the Lyapunov stability theory on the stochastic differential system,basic theory on functional differential equation,It(?) formula,the principle of stochastic analysis,and Lyapunov-Krasovskii functional(function) etc. are employed to study the exponential stability of the stochastic differential equation with time-varying delay,the impulsive stochastic differential equation with delays and the stochastic neural network with multi-delay.Some significant results are obtain.Details are as follows: In introduction,we briefly outline the significance,background,progress,application prospects of the stochastic system with delay of the achievements of previous studies on this.In chapter 2,exponential stability for a class of stochastic differential equations with time-varying delay is analyzed by constructing suitable Lyapunov function and using the nonnegative semi-martingale convergence theorem.New algebraic criteria are given for the almost surely exponential stability,which extend main conclusions in existing literature.An example is also given for illustration.In chapter 3,we study the exponential p-stability of It(?) impulsive stochastic differential equation with delays.Based on Lyapunov-Krasovskii functional method and stochastic analysis theory,we obtain some new results ensuring the exponential p-stability of equilibrium solution of this kind of system.In chapter 4,we consider exponential stability for a class of stochastic neural network with multi-delay.Similarly,by constructing suitable Lyapunov function and using the nonnegative semi-martingale convergence theorem,we get some new criterions ensuring the almost surely exponential stability of the system.