Banach空间中微分方程解的存在性与可控性
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摘要
Banach空间中微分方程理论是非线性泛函分析的一个重要研究分支,近年来已被广泛应用于偏微分方程、工程技术、控制理论等诸多领域中,例如,算子半群可将发展型偏微分方程转化为抽象空间中的微分方程,从而按统一框架研究偏微分方程.现代控制理论与数学各个分支的联系日益密切,可控性则是控制理论中的基本概念.因此在一定条件下,研究Banach空间中微分方程解的存在性与可控性具有重要的理论和现实意义.
本文主要研究Banach空间中的脉冲微分方程和分数阶微分方程,利用算子半群理论、非紧测度、逼近解等方法,先研究几类泛函微分方程解的存在性,然后将其应用到微分系统的控制问题中.本文具体内容由以下五个章节组成.
本文第一章简要地介绍与本论文研究问题有关的背景知识及我们的主要工作.
第二章主要研究如下具有非局部条件的脉冲微分方程解的存在性其中A是一个线性算子(不一定有界),生成强连续半群；9是非局部项；Ii是脉冲函数.
本章主要使用算子半群理论、非紧测度和不动点方法.在2.1节中,我们介绍一些非紧测度的概念和性质,并证明与脉冲函数密切相关的分段连续函数空间中非紧测度的一个重要性质(见引理2.1.5).2.2节给出本章的主要结论,即脉冲微分方程在不同条件下适度解的存在性条件.利用我们证明的非紧测度性质和Darbo不动点定理,我们在算子半群等度连续的条件下,分别得到了紧性条件,Lipschitz条件和混合型条件下,上述非局部脉冲微分方程的适度解.2.3节中给出这些结果在偏微分方程中的应用.
第三章主要讨论脉冲微分包含解的存在性其中A：D(A)(?)X→X是线性算子,生成Banach空间(X,||·||)中强连续半群T(·),F是上半连续的多值函数.
我们将第二章中单值脉冲微分方程的讨论扩展到多值的情形,但方法和主要着眼点与第二章是不同的.我们在算子半群为紧半群条件下,对非局部项g非紧非Lipschitz连续的情况进行研究.主要运用逼近解的技巧和多值分析的方法.
3.1节中回忆多值分析的一些概念及多值映射不动点定理.3.2节,构造脉冲微分包含的逼近问题,证明逼近解解集是相对紧的,进而得到原来脉冲微分包含问题解的存在性.
第四章致力于研究如下半线性脉冲微分系统的可控性其中A(t)是一族线性算子,生成一个发展算子U：△={(t,s)∈[0,b]×[0,b]：0≤s≤t≤b}→L(X),这里X是Banach空间,L(X)是空间X上有界线性算子的全体；B是从Banach空间V到X的有界线性算子,控制函数u(·)∈L2([0,b],V).
本章我们利用非紧测度的方法,在发展系统不具有紧性的条件下,研究脉冲微分系统的精确可控性.具体地,在定理4.2.1中,我们使用Monch不动点定理,讨论脉冲微分系统在发展系统U(t,s)等度连续条件下的可控性；在定理4.2.2中,通过构造一个新的非紧测度,我们只假设生成的发展系统是强连续的,既不需要紧性,甚至也不需要等度连续性,证得脉冲微分系统的可控性,这里我们对第二章中的方法做了实质性改进,那里需要算子半群是等度连续的.
第五章考虑如下半线性分数阶非局部微分方程的近似可控性其中x(·)取值于Hilbert空间X；Dq是q阶的Caputo分数阶导数,0 由于现实中存在系统误差和技术误差,近似可控性的应用更为广泛.本章利用算子半群和分数阶微积分理论,讨论了分数阶微分方程的近似可控性.使用概率密度函数定义了分数阶微分系统的适度解,在假设相应的线性微分系统近似可控的前提下,得到上述半线性分数阶非局部微分方程的近似可控性,其中非局部函数g不需要紧性条件和Lipschitz连续的条件.
The theory of differential equations in Banach spaces is an important branch of nonlinear analysis, which is applied to many fields, such as partial differential equations, engineering, control theory. For example, by means of semigroup of lin-ear operators, evolution equations can be transformed into differential equations in abstract spaces, which implies that we can deal with partial differential equa-tions in a unified way. Modern control theory is closely related to many branches of mathematics and controllability is the fundamental concept of control theory. Therefore, it has vital theoretical and practical significance to study the existence and controllability of differential equations in Banach spaces.
The present dissertation focuses on the impulsive differential equations and fractional differential equations. By using semigroup of linear operators, measure of noncompactness and approximate solutions, we firstly study the existence of solutions to several types of functional differential equations. Then the results are extended to the control problems of differential systems. The present dissertation consists of five chapters.
Chapter1introduces briefly the background and our main work.
In Chapter2, we discuss the existence of mild solutions to the following nonlocal impulsive differential equations where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators{T(t):t≥0}, g is a nonlocal item,Ii are impulsive functions.
The tools used in this chapter are semigroup of linear operators, measure of non-compactness and fixed point theorems. In Section2.1, we introduce some concepts and properties of measure of noncompactness. and prove an important property of measure of noncompactness in PC([0, b]; X), which is closely related to impulsive functions (see Lemma2.1.5). In Section2.2, we give the main results of this chap-ter, i.e., the existence of impulsive differential equations are obtained under various conditions. By supposing that semigroup is equicontinuous, we get the existence results of impulsive differential equations when compactness conditions, Lipschitz conditions and mixed-type conditions are satisfied, respectively. In Section2.3, we give an example applied to partial differential equations.
Chapter3is concerned with the existence of the impulsive differential inclusions where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators{T(t):t≥0}, F is an upper semicontinuous multifunction.
We extend the discussions on impulsive differential equations to the scenario of multifunctions. However, the method and focus used in this chapter are different from Chapter2. By supposing the semigroup is compact, we discuss the existence results when nonlocal item g is not compact and not Lipschitz continuous.
In Section3.1, we recall some concepts and fixed point theorems on multi-valued analysis. In Section3.2, by constructing the approximate problem of the above impulsive differential inclusions, we get the existence of the impulsive differential inclusions by means of the compactness of approximate solutions set.
Chapter4is devoted to the controllability of the following semilinear impulsive differential system where A(t) is a family of linear operators which generates an evolution operator U:Δ={(t, s)∈[0,b] x [0,b]:0 By means of measure of noncompactness, we study the exact controllability of impulsive differential system under the noncompact semigroup. By using Monch's fixed point theorem, we discuss the controllability of the differential system when the evolution system U(t,s) is equicontinuous (see Theorem4.2.1). Furthermore, we construct a new type of noncompact measure and also get the controllability of the above differential system only supposing the evolution system is strongly continuous, without any compactness and equicontinuity hypotheses to evolution system (see Theorem4.2.2). Here, we essentially improve the method in Chapter2. where the semigroup is supposed to be equicontinuous.
Chapter5is concerned with the approximate controllability of the following semilinear fractional differential equations where the state variable x(·) takes values in the Hilbert space X:Dq is the Caputo fractional derivative of order q with0 Due to system errors and technical errors in reality, approximate controllability can be applied more widely. By means of semigroup of bounded linear operators and fractional calculus, we discuss the approximate controllability of the above semilin-ear fractional differential equations. We give the definition of the mild solutions to the fractional differential system and get the approximate controllability results with the assumption that the associated linear control system is approximately con-trollable. In our results, the compactness condition and Lipschitz condition to the nonlocal function g are not needed.
本文主要研究Banach空间中的脉冲微分方程和分数阶微分方程,利用算子半群理论、非紧测度、逼近解等方法,先研究几类泛函微分方程解的存在性,然后将其应用到微分系统的控制问题中.本文具体内容由以下五个章节组成.
本文第一章简要地介绍与本论文研究问题有关的背景知识及我们的主要工作.
第二章主要研究如下具有非局部条件的脉冲微分方程解的存在性其中A是一个线性算子(不一定有界),生成强连续半群；9是非局部项；Ii是脉冲函数.
本章主要使用算子半群理论、非紧测度和不动点方法.在2.1节中,我们介绍一些非紧测度的概念和性质,并证明与脉冲函数密切相关的分段连续函数空间中非紧测度的一个重要性质(见引理2.1.5).2.2节给出本章的主要结论,即脉冲微分方程在不同条件下适度解的存在性条件.利用我们证明的非紧测度性质和Darbo不动点定理,我们在算子半群等度连续的条件下,分别得到了紧性条件,Lipschitz条件和混合型条件下,上述非局部脉冲微分方程的适度解.2.3节中给出这些结果在偏微分方程中的应用.
第三章主要讨论脉冲微分包含解的存在性其中A：D(A)(?)X→X是线性算子,生成Banach空间(X,||·||)中强连续半群T(·),F是上半连续的多值函数.
我们将第二章中单值脉冲微分方程的讨论扩展到多值的情形,但方法和主要着眼点与第二章是不同的.我们在算子半群为紧半群条件下,对非局部项g非紧非Lipschitz连续的情况进行研究.主要运用逼近解的技巧和多值分析的方法.
3.1节中回忆多值分析的一些概念及多值映射不动点定理.3.2节,构造脉冲微分包含的逼近问题,证明逼近解解集是相对紧的,进而得到原来脉冲微分包含问题解的存在性.
第四章致力于研究如下半线性脉冲微分系统的可控性其中A(t)是一族线性算子,生成一个发展算子U：△={(t,s)∈[0,b]×[0,b]：0≤s≤t≤b}→L(X),这里X是Banach空间,L(X)是空间X上有界线性算子的全体；B是从Banach空间V到X的有界线性算子,控制函数u(·)∈L2([0,b],V).
本章我们利用非紧测度的方法,在发展系统不具有紧性的条件下,研究脉冲微分系统的精确可控性.具体地,在定理4.2.1中,我们使用Monch不动点定理,讨论脉冲微分系统在发展系统U(t,s)等度连续条件下的可控性；在定理4.2.2中,通过构造一个新的非紧测度,我们只假设生成的发展系统是强连续的,既不需要紧性,甚至也不需要等度连续性,证得脉冲微分系统的可控性,这里我们对第二章中的方法做了实质性改进,那里需要算子半群是等度连续的.
第五章考虑如下半线性分数阶非局部微分方程的近似可控性其中x(·)取值于Hilbert空间X；Dq是q阶的Caputo分数阶导数,0
The theory of differential equations in Banach spaces is an important branch of nonlinear analysis, which is applied to many fields, such as partial differential equations, engineering, control theory. For example, by means of semigroup of lin-ear operators, evolution equations can be transformed into differential equations in abstract spaces, which implies that we can deal with partial differential equa-tions in a unified way. Modern control theory is closely related to many branches of mathematics and controllability is the fundamental concept of control theory. Therefore, it has vital theoretical and practical significance to study the existence and controllability of differential equations in Banach spaces.
The present dissertation focuses on the impulsive differential equations and fractional differential equations. By using semigroup of linear operators, measure of noncompactness and approximate solutions, we firstly study the existence of solutions to several types of functional differential equations. Then the results are extended to the control problems of differential systems. The present dissertation consists of five chapters.
Chapter1introduces briefly the background and our main work.
In Chapter2, we discuss the existence of mild solutions to the following nonlocal impulsive differential equations where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators{T(t):t≥0}, g is a nonlocal item,Ii are impulsive functions.
The tools used in this chapter are semigroup of linear operators, measure of non-compactness and fixed point theorems. In Section2.1, we introduce some concepts and properties of measure of noncompactness. and prove an important property of measure of noncompactness in PC([0, b]; X), which is closely related to impulsive functions (see Lemma2.1.5). In Section2.2, we give the main results of this chap-ter, i.e., the existence of impulsive differential equations are obtained under various conditions. By supposing that semigroup is equicontinuous, we get the existence results of impulsive differential equations when compactness conditions, Lipschitz conditions and mixed-type conditions are satisfied, respectively. In Section2.3, we give an example applied to partial differential equations.
Chapter3is concerned with the existence of the impulsive differential inclusions where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators{T(t):t≥0}, F is an upper semicontinuous multifunction.
We extend the discussions on impulsive differential equations to the scenario of multifunctions. However, the method and focus used in this chapter are different from Chapter2. By supposing the semigroup is compact, we discuss the existence results when nonlocal item g is not compact and not Lipschitz continuous.
In Section3.1, we recall some concepts and fixed point theorems on multi-valued analysis. In Section3.2, by constructing the approximate problem of the above impulsive differential inclusions, we get the existence of the impulsive differential inclusions by means of the compactness of approximate solutions set.
Chapter4is devoted to the controllability of the following semilinear impulsive differential system where A(t) is a family of linear operators which generates an evolution operator U:Δ={(t, s)∈[0,b] x [0,b]:0
Chapter5is concerned with the approximate controllability of the following semilinear fractional differential equations where the state variable x(·) takes values in the Hilbert space X:Dq is the Caputo fractional derivative of order q with0
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