奇异摄动最优控制问题的空间对照结构研究
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摘要
通过对奇异摄动最优控制问题状态解极限性质的深入研究,本文探讨了奇异摄动最优控制问题中空间对照结构的存在性.近年来,对空间对照结构的研究已取得了非常深入的成果,从而为奇异摄动最优控制问题中空间对照结构的研究提供了理论依据.空间对照结构主要分为阶梯状空间对照结构和脉冲状空间对照结构两大类.本文主要讨论阶梯状空间对照结构,它的基本特点是在所讨论区间内存在一点t*(当然也可以存在多点t*),t*称为转移点,因为在每个转移点的讨论完全一样,所以我们只讨论存在一个转移点的情况.事先t*的位置是未知的,需要在渐近解的构造过程中确定.在t*的某个小邻域内,问题的解会发生剧烈的结构变化,当小参数趋于零时,解会趋向于不同的退化解.
本文由两部分组成,第一部分研究奇异摄动最优控制问题中的阶梯状空间对照结构,第二部分研究奇异摄动混合动态系统的最优控制.
第一章回顾了奇异摄动最优控制的发展过程,引入了与本文研究内容相关的一些基本定义和引理,介绍了本文的工作和创新之处.
第二章研究了数量情形的线性奇异摄动最优控制问题和含有积分边界条件的奇异摄动最优控制问题.利用指数二分法的一些性质和Fredholm交换引理以及k+σ交换引理,证明了阶梯状空间对照结构解的存在性.同时,根据解的结构,运用边界层函数法和直接展开法构造了其一致有效的形式渐近解.
第三章研究了数量情形非线性奇异摄动最优控制问题的阶梯状空间对照结构,利用必要最优性条件的等价性证明了阶梯状空间对照结构解的存在性.同时,利用直接展开法构造了该问题一致有效的形式渐近解.
第四章研究了高维奇异摄动最优控制问题的阶梯状空间对照结构,利用k+σ交换引理证明了阶梯状空间对照结构解的存在性,结合直接展开法构造了该问题一致有效的形式渐近解.
第五章研究了奇异摄动混合动态系统最优控制的渐近解.借助变分学的方法得到了混合动态系统的最优性条件,并利用边界层函数法构造了形式渐近解.运用缝接法对轨道进行了缝接,在整个区间上得到了解的存在性和渐近解的一致有效性.
In this dissertation, by means of further investigation of state solution limiting behavior of singularly perturbed optimal control problems, we dis-cuss the existence of contrast structure of singularly perturbed optimal con-trol problems. In recent years, the research of contrast structure is well de-veloped and thus it will provide theoretical basis for the research of contrast structure of singularly perturbed optimal control problems. The contrast structure in singularly perturbed problems is mainly classified as a step-like contrast structure and a spike-like contrast structure, a step-like contrast structure problem is only concerned in this dissertation. Its fundamental characteristic is that there exists t*(or multiple t*) within the domain of in-terest, t*is called as an internal transition point. The discussion at each t*is exactly the same, so we only consider the case that there exists one internal transition point. The position of t*is unknown in advance and it needs to be determined thereafter. In the neighborhood of t*, the solution will have an abrupt structure change and the solution will approach different reduced solutions when the small parameter μ→0.
The thesis consists of two parts. The first part considers the step-like contrast structure of singularly perturbed optimal control problems. The second part considers the optimal control of singularly perturbed hybrid dy-namical systems.
Chapter1introduces the development of singularly perturbed optimal control, gives some basic concepts and lemmas which are relevant to our study, and introduces the main research work and innovative points in this thesis.
Chapter2studies the linear singularly perturbed optimal control prob-lem and singularly perturbed optimal control problem with integral boundary conditions. By means of the properties of exponential dichotomies, Fredholm alternatives and k+σ exchange lemma, we prove the existence of step-like contrast structure solution. Meanwhile, based on the structure of the so-lution, we construct the uniformly valid formal asymptotic solution by the boundary layer function method and direct scheme method.
Chapter3is devoted to investigate the step-like contrast structure of nonlinear singularly perturbed optimal control problem. The existence of step-like contrast structure solution is proved by equivalence, which is based on the necessary optimality conditions. Meanwhile, the uniformly valid for-mal asymptotic solution is constructed by the direct scheme method.
Chapter4considers the step-like contrast structure of higher dimen-sional singularly perturbed optimal control problem. The existence of step-like contrast structure solution is proved by the k+σ exchange lemma. Using the direct scheme method, we construct the uniformly valid formal asymp-totic solution for the singularly perturbed optimal control problem.
In chapter5, we focus on the asymptotic solution for optimal control of singularly perturbed hybrid dynamical systems. The variational method is used to obtain optimality conditions of hybrid dynamical systems and the for-mal asymptotic solution is constructed by boundary layer function method. By sewing orbit, the existence of solution is shown and the asymptotic solu-tion is proved to be uniformly valid in the whole interval.
本文由两部分组成,第一部分研究奇异摄动最优控制问题中的阶梯状空间对照结构,第二部分研究奇异摄动混合动态系统的最优控制.
第一章回顾了奇异摄动最优控制的发展过程,引入了与本文研究内容相关的一些基本定义和引理,介绍了本文的工作和创新之处.
第二章研究了数量情形的线性奇异摄动最优控制问题和含有积分边界条件的奇异摄动最优控制问题.利用指数二分法的一些性质和Fredholm交换引理以及k+σ交换引理,证明了阶梯状空间对照结构解的存在性.同时,根据解的结构,运用边界层函数法和直接展开法构造了其一致有效的形式渐近解.
第三章研究了数量情形非线性奇异摄动最优控制问题的阶梯状空间对照结构,利用必要最优性条件的等价性证明了阶梯状空间对照结构解的存在性.同时,利用直接展开法构造了该问题一致有效的形式渐近解.
第四章研究了高维奇异摄动最优控制问题的阶梯状空间对照结构,利用k+σ交换引理证明了阶梯状空间对照结构解的存在性,结合直接展开法构造了该问题一致有效的形式渐近解.
第五章研究了奇异摄动混合动态系统最优控制的渐近解.借助变分学的方法得到了混合动态系统的最优性条件,并利用边界层函数法构造了形式渐近解.运用缝接法对轨道进行了缝接,在整个区间上得到了解的存在性和渐近解的一致有效性.
In this dissertation, by means of further investigation of state solution limiting behavior of singularly perturbed optimal control problems, we dis-cuss the existence of contrast structure of singularly perturbed optimal con-trol problems. In recent years, the research of contrast structure is well de-veloped and thus it will provide theoretical basis for the research of contrast structure of singularly perturbed optimal control problems. The contrast structure in singularly perturbed problems is mainly classified as a step-like contrast structure and a spike-like contrast structure, a step-like contrast structure problem is only concerned in this dissertation. Its fundamental characteristic is that there exists t*(or multiple t*) within the domain of in-terest, t*is called as an internal transition point. The discussion at each t*is exactly the same, so we only consider the case that there exists one internal transition point. The position of t*is unknown in advance and it needs to be determined thereafter. In the neighborhood of t*, the solution will have an abrupt structure change and the solution will approach different reduced solutions when the small parameter μ→0.
The thesis consists of two parts. The first part considers the step-like contrast structure of singularly perturbed optimal control problems. The second part considers the optimal control of singularly perturbed hybrid dy-namical systems.
Chapter1introduces the development of singularly perturbed optimal control, gives some basic concepts and lemmas which are relevant to our study, and introduces the main research work and innovative points in this thesis.
Chapter2studies the linear singularly perturbed optimal control prob-lem and singularly perturbed optimal control problem with integral boundary conditions. By means of the properties of exponential dichotomies, Fredholm alternatives and k+σ exchange lemma, we prove the existence of step-like contrast structure solution. Meanwhile, based on the structure of the so-lution, we construct the uniformly valid formal asymptotic solution by the boundary layer function method and direct scheme method.
Chapter3is devoted to investigate the step-like contrast structure of nonlinear singularly perturbed optimal control problem. The existence of step-like contrast structure solution is proved by equivalence, which is based on the necessary optimality conditions. Meanwhile, the uniformly valid for-mal asymptotic solution is constructed by the direct scheme method.
Chapter4considers the step-like contrast structure of higher dimen-sional singularly perturbed optimal control problem. The existence of step-like contrast structure solution is proved by the k+σ exchange lemma. Using the direct scheme method, we construct the uniformly valid formal asymp-totic solution for the singularly perturbed optimal control problem.
In chapter5, we focus on the asymptotic solution for optimal control of singularly perturbed hybrid dynamical systems. The variational method is used to obtain optimality conditions of hybrid dynamical systems and the for-mal asymptotic solution is constructed by boundary layer function method. By sewing orbit, the existence of solution is shown and the asymptotic solu-tion is proved to be uniformly valid in the whole interval.
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