几类偏微分方程最优控制相关问题研究

发布时间：2018-05-19 06:50

本文选题：偏微分方程 + 最优控制　；参考：《华中科技大学》2016年博士论文

【摘要】：本文主要研究几类偏微分方程最优控制相关问题,其内容包括状态方程的适定性、最优控制的存在性以及最优控制满足的极大值原理。全文共分六章：第一章主要阐述偏微分方程最优控制问题的来源和研究对象,并简要介绍国内外的研究现状以及本文的主要结果。第二章研究了一类非线性色散浅水波方程的分布最优控制问题。首先利用Garlekin逼近方法和能量估计,在合适的状态空间中证明了控制系统解的存在唯一性。其次引入所要讨论的最优控制问题,并且证明每一个给定的初始状态都对应着控制系统唯一的最优控制。最后借助由A.Ya. Dubovitskii和A.A. Milyutin提出的泛函分析方法成功地推导出最优控制成立的一阶必要性条件,并指出最优控制具有Bang-Bang性质。本章得到的结果推广和改进了部分已知结果。第三章考虑了具有一个捕食者和两个被捕食者的生态模型的最优控制理论,其目的在于通过人工干预使得此生态系统中三个物种的种群密度最大化。给定初始种群密度(均为正),借助强连续算子半群理论和抛物方程的相关理论,首先证明了控制系统存在唯一的正强解,并且指出对于给定的初始种群密度,人工干预具有最优的策略(即控制系统存在唯一的最优控制)。然后借助于对偶原理推导出最优控制的一阶必要性条件和二阶充要性条件。本章对相应的最优控制问题给出了全面的回答,其结果是新的。第四章处理了一类带有记忆项的非线性耦合波动方程的最优控制问题。首先利用经典的变分原理和紧性原理给出了最优控制变量的存在性证明。其次通过对近似最优控制变量(惩罚掉了状态限制)的极值条件在合理的意义下取极限,推导出了最优控制变量所满足的庞氏极大值原理。作为主要结果的直接应用,同时得到了由非线性Klein-Gordon方程组描述的最优控制问题的必要性条件,其结果推广和改进了文献[83]中关于Klein-Gordon方程最优控制问题的主要结果。第五章讨论了带有逐点状态限制的Boussinesq方程组的最优控制问题。首先证明了状态函数关于控制函数的连续依耐性,然后利用Ekeland变分原理和改进的针状变分技术,推导出了最优控制满足的逐点形式的庞氏极大值原理。另外,在某些强稳定性条件假设下,进一步得到了一种形式更强的庞氏极大值原理,其特点在于拉格朗日乘子可以取固定的常数。本章结果推广和改进了文献[86]中带有积分型状态限制最优控制问题的结论。第六章研究了一类广义Camassa-Holm方程的局部适定性、强解的爆破性以及解析解的存在性。首先利用Littlewood-Paley分解原理和传输方程理论证明方程在Besov空间中的局部适定性。其次证明方程在临界Besov空间中的局部适定性。借助粒子轨道方法讨论了解的爆破性,并在特殊情况下得到强解的整体存在性。最后讨论了方程解的Gevrey正则性和解析性,同时给出解存在时间的上界估计。本章结果推广和改进了部分已知的结论,其中关于解的爆破性和解析性的讨论是新的。
[Abstract]:This paper mainly deals with the optimal control related problems of partial differential equations. The content includes the fitness of the state equation, the existence of optimal control and the maximum principle of optimal control. The full text is divided into six chapters. The first chapter mainly expounds the source and research object of the optimal control problem of partial differential equations, and briefly introduces the domestic and foreign countries. The second chapter studies the optimal control problem of the distribution of a class of nonlinear dispersion shallow water wave equations. First, the existence and uniqueness of the solution of the control system are proved in the appropriate state space by using the Garlekin approximation method and the energy estimation. Secondly, the optimal control problem is introduced, and the proof of the optimal control problem is introduced. Each given initial state corresponds to the only optimal control of the control system. Finally, the first order necessary conditions for the establishment of optimal control are successfully derived with the help of the functional analysis method proposed by A.Ya. Dubovitskii and A.A. Milyutin, and the optimal control has Bang-Bang quality. The results obtained in this chapter are extended and improved. The third chapter considers the optimal control theory of an ecological model with a predator and two predators. The aim is to maximize the population density of three species in the ecosystem by artificial intervention. Given the initial population density (all positive), the theory of the strong continuous operator semigroup and the phase of the parabolic equation are used. In the theory, we first prove the existence of the only positive strong solution of the control system, and point out that the artificial intervention has the best strategy for the given initial population density (i.e. the existence of the only optimal control in the control system). Then the first order necessary condition and the two order sufficient and necessary condition of the optimal control are derived with the aid of the dual principle. The optimal control problem is given a comprehensive answer. The results are new. The fourth chapter deals with the optimal control problem for a class of nonlinear coupled wave equations with memory terms. First, the existence of the optimal control variable is given by using the classical variational principle and the compactness principle. The extreme conditions of the state limit are taken in a reasonable sense, and the Ponzi maximum principle of the optimal control variable is derived. As a direct application of the main result, the necessary condition for the optimal control problem described by the nonlinear Klein-Gordon equations is obtained. The result is extended and improved in the literature [83]. The main result of the optimal control problem of the Klein-Gordon equation. The fifth chapter discusses the optimal control problem of the Boussinesq equations with the point by point state restriction. First, the state function is proved to be continuous dependent on the control function. Then the optimal control satisfaction is derived by using the Ekeland variational principle and the improved needle variational technique. In addition, under some strong stability conditions, a more form of Ponzi maximum principle is obtained, which is characterized by the Lagrange multiplier can take a fixed constant. The result of this chapter generalizes and improves the knot with the optimal control problem with integral state restriction in [86]. The sixth chapter studies the local well - being of a class of generalized Camassa-Holm equations, the explosive property of strong solutions and the existence of analytic solutions. First, we use the Littlewood-Paley decomposition principle and the transfer equation theory to prove the local well posed of the equation in the Besov space. Secondly, it proves the local well posed of the equation in the critical Besov space. The orbit method discusses the blasting property of the understanding and obtains the overall existence of the strong solution under special circumstances. Finally, the Gevrey regularity and the analytic property of the solution are discussed. At the same time, the upper bounds of the existence time of the solution are given. The results of this chapter extend and improve some known conclusions, in which the discussion of the explosive and analytic solution is new.
【学位授予单位】：华中科技大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.2;O232

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