The Asymptotic Analysis Of Neutral Stochastic Functional Differential Equations

It is well known that the study of asymptotic behavior 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 powerful tool for analyzing certain systems,and bring the possibility of analyzing the asymptotic behavior and controlling 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,when the development trend of this system is related with not only present state but also history or future state,the rule of this kind of systems is often described by stochastic functional differential equation.But,with developments of chemical engineering systems and the theroy aeroelasticity etc,description on practical problem is required to be better and better.The research on general stochastic functional equations is not enough.Kolmanovskii and Nosov introuced a class of neutral stochastic functional differential equations firstly.In principle,the neutral stochastic differential equations invole the general stochastic functional differential equations.So the analysis and control of asymptotic behavior is main objective to the design project,this analysis provides a theory basis for the development of practical application.However,the systemic research on neutral stochastic functional differential equations and those of with Markov switching is very little.Therefore,we will not only establish a fundamental theory for such systems but also discuss some important properties of solutions e.g.the existence and uniqueness ,stabilities,boundedness etc.to close this gap.In this paper,we present new concepts of stability and boundedness,that is,the "ψstability" and the "ψboundedness".With It(?) formula,the theory of Lyapunov functional stability,Borel-Cantelli lemma,semi-martingale convergence theorem,H(o|¨)lder inequality,exponential martingle inequality etc.,we establish criterions on stabilities and boundedness of general neutral stochastic functional differential systems for the fisrt time.The conclusions not only extend the existed results about Razumikhin theorem but also establish the Razumikhin-type theorem of the boundedness.We also establish special delay-dependent Laypunov functional for alterable delay equations to get better results.At the same time,we interpret these theroies through practical examples.For the first time,we analyze the boundedness of the neutral stochastic functional differential equations systematically,improve and extend Mao etc.' results about stability theroy.LaSalle theorems is an important tool to investigate the stability of stochastic systems. It cancells the requirement of the positive Lyapunov functional,and so is abroadly adopted in practical engineering problems.This paper has established the LaSalle-type theorems for general neutral stochastic functional differential equations by using It(?) formula,semi-martingale convergence theorem,stochastic integral moment inequality, Kolmogorov-(?)entsov theorem etc.and other inequality technique.The results includes the stochastic-type LaSalle theorem on stochastic differential systems and stochastic functional differential systems in the existing refences.Meanwhile,we support sufficient conditions for the asympotic stability,such as exponential stability and polynomial stability with stochastic-type LaSalle theorem on neutral stochastic functional differential systems.Compared with classical stability theroy,the results in the paper made best use of the beneficial work of stochastic perturbation item.We also improved the exponential stablity theory exeisted with It(?) formula,semi-martingale convergence theorem.The neutral stochastic variable delay equations with Markovian switching was first systemically investigated.The paper established the existence-uniqueness theorem and estimation of the solution of stochastic hybrid systems with Markovian switching by the existence-uniqueness theorem,H(o|¨)der inequality,Gronwall inequality.In the end,sufficient criterions on stability and boundedness of the solution of neutral stochastic delay differential systems are established with It(?) formula,Borel-Cantelli lemma,Layponov function etc,which extends main conclusions in existing literature that the system is constant time delay.