The Properties Of Solutions And Approximate Controllability To Some Impulsive Differential Equations

Impulsive phenomenon is the common problems in various fields of modern science and technology,the mathematical model can often be attributed to impul-sive differential system. In recent years,with the development of fractional differ-ential equations and stochastic differential equations,the study on the properties and approximate controllability of impulsive fractional differential equations and impulsive stochastic differential equations has aroused great interest of scholars at home and abroad. This paper study the properties of solutions and approxi-mate controllability to some impulsive differential equations by using the theory of functional differential equation、fractional differential equation、stochastic d-ifferential equation and semigroup theory. This Ph.D thesis is composed of six chapters,which mainly study the existence of positive periodic solutions for higher-dimension functional differential equation with impulses、exponential stability for stochastic delay neural network with impulses and approximate controllability of impulsive fractional differential systems.In chapter one,the research background、significance and the main work done in this paper are reviewed.In chapter two,some preliminaries are briefly introduced,such as the basic knowledge of stochastic differential equation,the basic concept and properties of the fractional calculus and the theorems、lemmas used in this paper.In chapter three,we study the existence of positive periodic solutions for higher-dimension functional differential equation with impulses. Firstly,by using Leray-Schauder theorem,we discuss the existence of positive periodic solutions for a class of higher-dimension impulsive functional differential equations,some sufficient con-ditions for the existence of positive periodic solutions are obtained,which improve the results of literature. Secondly,by employing Krasnoselskii fixed point theo-rem,we establish some criteria for existence of positive periodic solutions of a class of n-dimension periodic functional differential equations with impulses.In chapter four,exponential stability for stochastic delay neural network with impulses is discussed. By employing fixed point theorem and some analysis tech-niques,sufficient conditions about the exponential stability for a stochastic neural network with impulses are derived.In chapter five,approximate controllability of impulsive fractional differential systems are investigated.Firstly,we study the approximate controllability for semi-linear fractional impulsive differential systems,by using Krasnoselskii’s fixed point theorem,semigroup theory and fractional power theory,sufficient conditions for ap-proximate controllability of impulsive fractional semilinear differential systems are established.Secondly,we study the approximate controllability for fractional impul-sive neutral functional differential evolution equations with infinite delay,sufficient conditions for the existence of mild solutions and approximate controllability about the systems are established.In chapter six,some concluding remarks are given.