【作者】 黄海；

【导师】 傅显隆；

【作者基本信息】 华东师范大学 ， 应用数学， 2021， 博士

【摘要】 积分微分发展系统理论是无穷维发展系统理论的重要分支.许多情形下,相较于一般的微分方程,积分微分方程可以更准确地描述科学领域中的自然现象.因此,对这类系统的各种动力学行为的研究具有重要的理论和应用意义.本文综合考虑了随机现象,脉冲现象,非局部条件和时滞对系统的影响,通过利用算子半群理论,预解算子理论,分数幂算子理论,基本解理论,随机分析理论及不动点定理,研究了几类半线性积分微分发展系统解的渐近性质,近似可控性与最优控制问题.本文的工作推广了这一领域已有的一些结论.全文共分五章.第一章介绍了积分微分发展方程的研究背景和研究意义,综述了近年来关于积分微分发展方程的研究现状,并概述了本文的主要工作.第二章研究一类脉冲中立型随机泛函积分微分系统解的渐近性质.利用预解算子理论,Banach不动点原理和随机分析理论分别研究了该方程全局温和解的存在唯一性,全局吸引集和拟不变集.另外,还得到了温和解的-阶矩指数稳定和几乎必然指数稳定的充分条件.第三章证明了具有非局部条件的半线性中立型积分微分系统的近似可控性.由于引进了分数幂算子和-范数,本章的结论能够应用到非线性项包含空间变量偏导数的系统.值得一提的是,这里不需要非局部函数2)满足紧性或满足Lipschitz条件.在第四章,首先建立了带有无穷时滞的线性积分微分发展系统的基本解理论,之后利用Laplace变换及基本解得到了一类带有无穷时滞的半线性随机积分微分系统的温和解的表达式,再结合预解算子型条件证明了所讨论系统的近似可控性.特别地,由于这里利用了基本解理论,部分克服了系统的非线性项需要一致有界的约束.第五章利用最近建立的线性中立型积分微分发展系统的预解算子,首先讨论了带有无穷时滞的半线性中立型积分微分系统温和解的存在唯一性并证明了解算子的紧性.然后在适当条件下,研究了该控制系统的最优控制与时间最优控制问题.更多还原

【Abstract】 The theory of integro-differential evolution systems is an important branch of the theory of infinite dimensional evolution systems.Compared with the general differential equations,in many cases integro-differential evolution systems can more accurately describe the natural phenomena arising from many fields of science.Thus,the research of various dynamic behaviors for this kind of systems has vital theoretical and practical significance.In this dissertation,the effects of stochastic phenomena,impulsive phenomena,nonlocal conditions and time delay on the system are considered.By using theory of semigroups,resolvent operators,fractional powers of operators,fundamental solutions,stochastic analysis as well as fixed point theorems,we study some asymptotic properties,approximate controllability and optimal control problems of solutions for several kinds of integro-differential evolution systems.The work in this paper generalizes some existing conclusions in this field.The whole dissertation consists of five chapters.In Chapter 1,we first introduce some backgrounds and research status of integrodifferential evolution equations and present some recent relevant works on integrodifferential evolution equations.We then state briefly the main work of this dissertation.In Chapter 2,we study the asymptotic properties of solutions for a class of impulsive neutral stochastic functional integro-differential systems.By applying the theory of resolvent operators,Banach fixed point principle and results on stochastic analysis,we study respectively the existence,uniqueness,global attracting and quasi-invariant sets of mild solutions for the considered equation.Also,we derive some sufficient conditions on -th moment exponential stability and almost surely exponential stability of the mild solutions.In Chapter 3 we prove the approximate controllability of semi-linear neutral integrodifferential systems with nonlocal conditions.Since fractional power operators and -norm are used to discuss the problems,the obtain results in this chapter can apply to the systems involving partial derivatives of spatial variables.It is worth mentioning that the compactness condition or Lipschitz condition for the function 2)in the nonlocal condition appearing in literature is not required here.In Chapter 4,we first construct the fundamental solutions of linear integro-differential evolution systems,from which the expressions of mild solutions for a class of semi-linear stochastic integro-differential systems with infinite delay are obtained via the fundamental solution based on Laplace transform arguments.From this we show the approximate controllability of the considered systems through the so-called resolvent conditions.Due to the fundamental solution theory,the uniform boundedness for the nonlinear terms is partly overcame and the results in this chapter generalize the results in literature.By virtue of the theory of resolvent operators for linear neutral integro-differential evolution systems constructed recently in literature,Chapter 5 first discusses the existence and uniqueness of mild solutions and shows the compactness of solution operators for the neutral integro-differential system with infinite delay.Then optimal control and time optimal control problems for the considered control system are investigated under some assumptions.更多还原