Study On The Existence And Control Of Solutions For Fractional Stochastic Differential Equations

This paper is devoted to existence,controllability and stability of mild solutions for some fractional stochastic differential equations and random impulsive differential equations.The following problems are discussed:(1)the existence and approximate controllability of mild solutions for Riemann-Liouville's fractional stochastic differential equations with order of 1<?<2;(2)the reasonable expression of solutions for impulsive fractional differential evolution equations,then the results are applied to the existence and related control problems of mild solutions to a class of fractional impulsive stochastic functional differential equations;(3)the existence of extremal solutions for boundary value problems of nonlinear fractional random impulsive differential equations;(4)the solvability and stability of second order stochastic differential equations with random impulses.This dissertation is divided into six chapters,the specific contents are as follows:In the first chapter,the background and significance of fractional stochastic differential equation,random impulsive system and the development of controllability theory are introduced briefly.The present progress of our main problems are analyzed.The main work of this paper are summarized as well.In the second chapter,the basic knowledge of fractional calculus theory,stochastic differential equation and non-compact measure theory,some important properties,conclusions,lemma and theorem are introduced.In the third chapter,using Laplace transformation and the semigroup theory of fractional order operators,the mild solution of a class of Riemann-liouville's fractional stochastic evolution equations with nonlocal initial value conditions of order 1<?<2 is given.The module of the solution operators are estimated,then the properties of their solution operators are derived.On this basis,with the aid of Hausdorff non-compactness measure and Monch fixed point theorem,the existence of mild solution is studies under the condition of non-compactness.According to the definition of Hausdorff non-compact measure and using the distance formula of corresponding space,a new criterion of non-compactness measurement about Ito integral is derived.Finally,the approximate controllability of the system is proved by using the dominated convergence theorem under the assumption that the associated linear system is approximately controllable.In the fourth chapter,considering a general form of Caputo's fractional impulsive evolution equation of order 0<?<1 as an example,we show the problems about the solution form for this class of equation in the existing literature by giving counter examples.Then,based on the definition and characteristics of fractional derivative operators and their solution operators,a more appropriate new definition of mild solutions for the impulsive fractional evolution equation is proposed,which improves the existing results.We further study the existence and optimal control of the mild solutions for a class of impulsive fractional stochastic functional differential equations with infinite delay base on the above researches.In the fifth chapter,we develop the method of upper and lower solutions for the boundary value problem of nonlinear fractional ordinary differential equations with random impulses.We first deduce the solution form of the corresponding linear system with the aid of Mittag-Lefller function.A comparison result for the boundary value problem of random impulsive differential equation is established in term of the properties of its Green function and the solution operator,and the theory of cone and partial order in random systems.By a comparison result and using the method of upper and lower solutions combined with monotone iterative technique,several suitable sufficient conditions for the existence of extremal mild solutions of the considered problem are established.Finally,we give an example to illustrate our results.In the sixth chapter,The existence,the exponential stability in mean square and Hyers-Ulam stability of mild solutions to the second order neutral stochastic functional differential equation with random pulses are considered.Without the need to assume that the system generates a compact semigroup,we deal with the existence problems of mild solution to this system by using fixed point via measure of non-compactness.Then,by establishing an integral inequality that applied to second-order functional differential equations with random impulse,the exponential stability in mean square of the mild solution to the equation is discussed.Next we extend the Ulam stability results to random impulsive stochastic functional differential equations.Our work may generalize some existing results of impulsive stochastic differential equations to more general random impulses cases,which has some theoretical significance and application values.