发展方程的时间最优控制问题的bang-bang性

发布时间：2018-01-01 02:31

本文关键词：发展方程的时间最优控制问题的bang-bang性　出处：《武汉大学》2017年博士论文　论文类型：学位论文

【摘要】：本文主要研究一些发展方程的时间最优控制问题的bang-bang性.通过引入范数最优控制问题,建立时间最优控制问题和范数最优控制问题之间的等价性,将时间最优控制问题的bang-bang性转化为范数最优控制问题的bang-bang性,然后我们研究以下发展方程类型:(1)当发展方程为时不变时,研究能达子空间并建立相应的两个表示定理,得到了范数最优控制问题的Pontryagin最大值原理,结合方程的某种弱的任意正可测集上的唯一延拓性得到了范数最优控制问题的bang-bang性,由此和前面所说的等价性导出时间最优控制问题的bang-bang性;(2)当发展方程为带时变位势的热方程时,直接利用带估计的任意正可测集上的能控性得到范数最优控制问题的bang-bang性,然后结合前面所说的等价性导出时间最优控制问题的bang-bang性.本文共包括四章.第一章为前言,主要阐述本文的研究背景和研究动机.在这章中,列出了本文中经常使用到的数学记号.然后以常微分方程为例,介绍了时间最优控制问题.接着回顾了导出时间最优控制问题的bang-bang性的方法的发展和研究现状.第二章的主要内容来自[WZ].在这章中,主要研究时不变抽象控制系统的时间最优控制问题的bang-bang性.这里考虑的目标集为状态空间的原点,以及控制系统可能没有任意区间上的L∞-零能控性和倒向唯一性.更加确切地说,我们研究时间最优控制问题的bang-bang性是如何依赖于参数(M,y0),其中M0是控制的球型约束集的半径以及y0为初始状态.对于时间最优控制问题的参数空间,我们将它分为了几个部分,并且对每个部分回答了相应的bang-bang性是否成立,除了一条临界曲线.值得一提的是,这条临界曲线在系统具有任意区间上的L∞-零能控时为空集.第三章的主要内容来自[WXZ].在这章中,主要研究时变热方程的时间最优控制问题的bang-bang性.其核心思想是引入范数最优控制问题,建立时间最优控制问题和范数最优控制问题的等价性,然后将时间最优控制问题的bang-bang性转化为范数最优控制问题的bang-bang性,而范数最优控制问题的bang-bang性可以利用任意时间正可测集上的带估计的L∞-零能控导出.在建立上述等价性的过程中,最困难的是证明最优范数关于时间的左连续性,我们需要某种假设条件保证其成立,从而得出此假设是时间最优控制问题的bang-bang性成立的充分条件.最后证明了在某些特殊时变情形(包含时不变情形)下这个假设成立.第四章的主要内容来自[Z1].从前面两章中可以看出,为了得到时间最优控制问题的bang-bang性,我们建立了它与范数最优控制问题的等价性,然后研究范数最优控制问题的bang-bang性.这个等价性是研究时间最优控制问题的有力工具.它提供了另一个视角来看待时间最优控制问题.在这章中,主要研究Schrodinger方程的时间最优控制问题和范数最优控制问题的等价性.
[Abstract]:This paper mainly studies the bang-bang of time optimal control problem for some evolution equations. By introducing the norm optimal control problem, establish the equivalence between the time optimal control problem and norm optimal control problem, the time optimal control problem of bang-bang is transformed into bang-bang norm optimal control problem, and then we study the following equation (type: 1) when the development equation is time invariant, the research of subspace and establish two corresponding representation theorem, the norm optimal control problem of the Pontryagin maximum principle, combined with some arbitrary equations with weak positive measurable sets the unique continuation obtained bang-bang norm optimal control problems, bang-bang optimal equivalence derived time from this and said in front of the control problem; (2) when the heat equation development equation with time-varying potential, with direct estimation Is any measurable set on the controllability of bang-bang norm optimal control problem, bang-bang optimal equivalence derived time and then said in front of the control problem. The thesis consists of four chapters. The first chapter is the preface, mainly expounds the research background and motivation. In this chapter. This paper lists the mathematical mark that is often used. Then the ordinary differential equation as an example, introduces the time optimal control problems. Then it reviews the research status and development method of bang-bang optimal control problems are time. The main content of the second chapter from [WZ]. in this chapter, the same time optimal control Abstract bang-bang the system control problem. The main research here to consider the target set for the origin of the state space, and the control system may not L for arbitrary interval on the null controllability and backward uniqueness is more. All said, bang-bang we study the time optimal control problem is how to depend on the parameters (M, Y0), where M0 is the ball type constraint control set radius and Y0 is the initial state parameter space. For the time optimal control problem, we divide it to several parts, and each part of the answer whether the corresponding bang-bang was established, in addition to a critical curve. It is worth mentioning that this critical curve with L for arbitrary interval on the null controllability for the empty set in the system. The main content of the third chapter from [WXZ]. bang-bang in this chapter, the problem of time optimal control of heat equation the. Its core idea is to introduce norm optimal control problem, control the equivalence problem and norm optimal control problems of the establishing time optimal, then time optimal control problems of bang-bang into the B norm optimal control problem Ang-bang, bang-bang and norm optimal control problems can be used at any time are measurable sets with L estimation for null controllability is derived. In the process of establishing the equivalence, the most difficult is that the left continuity of the optimal norm of time, we need some assumptions to ensure its establishment thus, this assumption is bang-bang established sufficient conditions for the time optimal control problem. Finally it is proved that in some special circumstances (including time-varying time invariant case) under this hypothesis. The main content of the fourth chapter from the [Z1]. from the previous two chapters can be seen, in order to get the optimal time control problem of bang-bang, we to establish its equivalence with the norm of the optimal control problem, and then study the bang-bang norm optimal control problem. The equivalence is a powerful tool to study the time optimal control problem. It provides From another point of view, we consider the problem of time optimal control. In this chapter, we mainly study the equivalence between the time optimal control problem of Schrodinger equation and the norm optimal control problem.

【学位授予单位】：武汉大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O232

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