半個多世紀(jì)以來,最優(yōu)控制理論已經(jīng)成為研究各類最優(yōu)控制問題、控制目標(biāo)和求解方法的一個廣闊領(lǐng)域.控制目標(biāo)的不斷復(fù)雜化必然導(dǎo)致其數(shù)學(xué)模型和求精確解及漸近解的復(fù)雜性.由于研究對象的復(fù)雜性以及數(shù)學(xué)公式和計算變得更加復(fù)雜,尋找解析解變的更加困難甚至不可能做到.因此,諸如數(shù)值方法的漸近方法可用來尋找問題的漸近解.通常情況下,迭代法被用作漸近方法.然而只有當(dāng)方程具有合適的初始近似時,才能得到理想的解.由于尚未建立普遍的計算方法和對初始近似的恰當(dāng)選擇的一般化結(jié)果,因此在這方面的研究對于提高迭代過程的計算效率是非常重要的.本文中,對最優(yōu)控制問題的漸近分析應(yīng)用擴(kuò)張原理、局部化原理、Krotov最優(yōu)解的充分條件、退化問題理論和Krotov極小化極大原理.特別地,建立了全新有效的迭代控制改進(jìn)方法和最優(yōu)控制問題漸近分析的一般方法.全文共分為六章.第一章主要介紹本文的研究背景、研究動機(jī)和主要結(jié)果.第二章提出了一種基于擴(kuò)張原理和局部化原理的迭代最優(yōu)化抽象方案.將該方法應(yīng)用于連續(xù)最優(yōu)控制問題,并給出了一般的最優(yōu)性充分條件.此外,考慮到一些在實(shí)踐中有用的矯正,得到了迭代控制的改進(jìn)算法.本文的結(jié)果推廣和補(bǔ)充了其他涉及最優(yōu)性... 

【文章頁數(shù)】：119 頁

【學(xué)位級別】：博士

【文章目錄】：
中文摘要
Abstract
1 Introduction
    1.1 Background
    1.2 Motivation
    1.3 Dissertation outline
2 Local control improvement for continuous systems
    2.1 General approach to approximate optimization
    2.2 Optimal control problem
    2.3 Control improvement algorithm
    2.4 Modifications
    2.5 Problems
        2.5.1 Problem 1
        2.5.2 Problem 2
3 Local control improvement for discrete systems
    3.1 Problem formulation and preliminary results
    3.2 Control improvement
        3.2.1 Derivation of the improvement algorithm
        3.2.2 Iterative improvement algorithm
    3.3 Transformations to discrete-continuous systems
    3.4 Problems
        3.4.1 Problem 1
        3.4.2 Problem 2. Parallel reaction problem
        3.4.3 Problem 3
4 Multistage approximate control optimization procedure
    4.1 Abstract optimization problem and principles of solving it
    4.2 General optimal control problem for a continuous system
        4.2.1 Constructing estimates for the boundaries of the admissible reachability set
    4.3 Transforming a controllable differential system in the general form to a system with linear controls
        4.3.1 Algorithm for convex hull construction
    4.4 General scheme for an approximate study of the original problem
    4.5 Problems
        4.5.1 Problem 1
        4.5.2 Problem 2
        4.5.3 Problem 3. Optimal control for a medical process
        4.5.4 Problem 4. Excitation transfer along a spin chain
5 Global control improvement
    5.1 Abstract iterative optimization scheme
    5.2 Global control improvement for continuous systems
    5.3 Global control improvement for discrete systems
    5.4 Different implementations of global improvement methods
    5.5 Problems
        5.5.1 Problem 1. Controlling a linear oscillator
        5.5.2 Problem 2. Nonlinear nonconvex problem with special modes
        5.5.3 Problem 3. Classical nonconvex quadratic problem
        5.5.4 Problem 4. Case of a degenerate second variation of the functional
        5.5.5 Problem 5. Optimizing the control over treating a viral disease based on a simple model of the immune process
        5.5.6 Problem 6
6 Conclusions and expectations
Appendix
    A1. The extension principle
    A2. Derivative and limit systems
    A3. Theorem 4.1
References
Acknowledgements
Resume



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