本文關(guān)鍵詞：發(fā)展方程的時(shí)間最優(yōu)控制問題的bang-bang性　出處：《武漢大學(xué)》2017年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 時(shí)間最優(yōu)控制 bang-bang性 范數(shù)最優(yōu)控制 等價(jià)性 正可測集上的零能控性 Pontryagin最大值原理

【摘要】：本文主要研究一些發(fā)展方程的時(shí)間最優(yōu)控制問題的bang-bang性.通過引入范數(shù)最優(yōu)控制問題,建立時(shí)間最優(yōu)控制問題和范數(shù)最優(yōu)控制問題之間的等價(jià)性,將時(shí)間最優(yōu)控制問題的bang-bang性轉(zhuǎn)化為范數(shù)最優(yōu)控制問題的bang-bang性,然后我們研究以下發(fā)展方程類型:(1)當(dāng)發(fā)展方程為時(shí)不變時(shí),研究能達(dá)子空間并建立相應(yīng)的兩個(gè)表示定理,得到了范數(shù)最優(yōu)控制問題的Pontryagin最大值原理,結(jié)合方程的某種弱的任意正可測集上的唯一延拓性得到了范數(shù)最優(yōu)控制問題的bang-bang性,由此和前面所說的等價(jià)性導(dǎo)出時(shí)間最優(yōu)控制問題的bang-bang性;(2)當(dāng)發(fā)展方程為帶時(shí)變位勢的熱方程時(shí),直接利用帶估計(jì)的任意正可測集上的能控性得到范數(shù)最優(yōu)控制問題的bang-bang性,然后結(jié)合前面所說的等價(jià)性導(dǎo)出時(shí)間最優(yōu)控制問題的bang-bang性.本文共包括四章.第一章為前言,主要闡述本文的研究背景和研究動(dòng)機(jī).在這章中,列出了本文中經(jīng)常使用到的數(shù)學(xué)記號(hào).然后以常微分方程為例,介紹了時(shí)間最優(yōu)控制問題.接著回顧了導(dǎo)出時(shí)間最優(yōu)控制問題的bang-bang性的方法的發(fā)展和研究現(xiàn)狀.第二章的主要內(nèi)容來自[WZ].在這章中,主要研究時(shí)不變抽象控制系統(tǒng)的時(shí)間最優(yōu)控制問題的bang-bang性.這里考慮的目標(biāo)集為狀態(tài)空間的原點(diǎn),以及控制系統(tǒng)可能沒有任意區(qū)間上的L∞-零能控性和倒向唯一性.更加確切地說,我們研究時(shí)間最優(yōu)控制問題的bang-bang性是如何依賴于參數(shù)(M,y0),其中M0是控制的球型約束集的半徑以及y0為初始狀態(tài).對(duì)于時(shí)間最優(yōu)控制問題的參數(shù)空間,我們將它分為了幾個(gè)部分,并且對(duì)每個(gè)部分回答了相應(yīng)的bang-bang性是否成立,除了一條臨界曲線.值得一提的是,這條臨界曲線在系統(tǒng)具有任意區(qū)間上的L∞-零能控時(shí)為空集.第三章的主要內(nèi)容來自[WXZ].在這章中,主要研究時(shí)變熱方程的時(shí)間最優(yōu)控制問題的bang-bang性.其核心思想是引入范數(shù)最優(yōu)控制問題,建立時(shí)間最優(yōu)控制問題和范數(shù)最優(yōu)控制問題的等價(jià)性,然后將時(shí)間最優(yōu)控制問題的bang-bang性轉(zhuǎn)化為范數(shù)最優(yōu)控制問題的bang-bang性,而范數(shù)最優(yōu)控制問題的bang-bang性可以利用任意時(shí)間正可測集上的帶估計(jì)的L∞-零能控導(dǎo)出.在建立上述等價(jià)性的過程中,最困難的是證明最優(yōu)范數(shù)關(guān)于時(shí)間的左連續(xù)性,我們需要某種假設(shè)條件保證其成立,從而得出此假設(shè)是時(shí)間最優(yōu)控制問題的bang-bang性成立的充分條件.最后證明了在某些特殊時(shí)變情形(包含時(shí)不變情形)下這個(gè)假設(shè)成立.第四章的主要內(nèi)容來自[Z1].從前面兩章中可以看出,為了得到時(shí)間最優(yōu)控制問題的bang-bang性,我們建立了它與范數(shù)最優(yōu)控制問題的等價(jià)性,然后研究范數(shù)最優(yōu)控制問題的bang-bang性.這個(gè)等價(jià)性是研究時(shí)間最優(yōu)控制問題的有力工具.它提供了另一個(gè)視角來看待時(shí)間最優(yōu)控制問題.在這章中,主要研究Schrodinger方程的時(shí)間最優(yōu)控制問題和范數(shù)最優(yōu)控制問題的等價(jià)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.

【學(xué)位授予單位】：武漢大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O232

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 ;Bang-Bang Principle of Time Optimal Controls and Null Controllability of Fractional Order Parabolic Equations[J];Acta Mathematica Sinica(English Series);2010年12期

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本文編號(hào)：1362566