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Table of Contents
Intro; Preface; Acknowledgments; Contents; Acronyms; 1 Mathematical Preliminaries; 1.1 Elements in Functional Analysis; 1.1.1 Spaces and Operators; 1.1.2 Some Geometric Aspects of Banach Spaces; 1.1.3 Semigroups and Evolution Equations; 1.1.4 Minimization of Functionals; 1.2 Set-Valued Map; 1.3 Controllability and Observability Estimate; Miscellaneous Notes; References; 2 Time Optimal Control Problems; 2.1 Overview on Time Optimal Control Problems; 2.1.1 Introduction; 2.1.2 Different Ingredients in Time Optimal Control Problems; 2.2 Connections of Minimal and Maximal Time Controls
2.3 Several Examples2.4 Main Subjects on Time Optimal Control Problems; Miscellaneous Notes; References; 3 Existence of Admissible Controls and Optimal Controls; 3.1 Admissible Controls and Controllability; 3.1.1 General Results in Abstract Setting; 3.1.2 Linear ODEs with Ball-Type Control Constraints; 3.1.3 Heat Equations with Ball-Type Control Constraints; 3.2 Admissible Controls and Minimal Norm Problems; 3.2.1 General Results in Abstract Setting; 3.2.2 Heat Equations with Ball-Type Control Constraints; 3.3 Admissible Controls and Reachable Sets; 3.3.1 General Results in Abstract Setting
3.3.2 Reachable Sets of Linear ODEs3.4 Existence of Optimal Controls; 3.4.1 Existence of Optimal Controls for Regular Cases; 3.4.2 Existence of Optimal Controls for Blowup Case; Miscellaneous Notes; References; 4 Maximum Principle of Optimal Controls; 4.1 Classical Maximum Principle of Minimal Time Controls; 4.1.1 Geometric Intuition; 4.1.2 Classical Maximum Principle; 4.1.3 Conditions on Separation in Classical Maximum Principle; 4.2 Local Maximum Principle and Minimal Time Controls; 4.2.1 Some Properties on Controllable Sets and Reachable Sets; 4.2.2 Separability and Local Maximum Principle
4.2.3 Conditions on Separation in Local Maximum Principle4.3 Weak Maximum Principle and Minimal Time Controls; 4.3.1 Separability and Weak Maximum Principle; 4.3.2 Two Representation Theorems Related to the Separability; 4.3.3 Conditions on Separation in Weak Maximum Principle; 4.4 Maximum Principle for Maximal Time Controls; 4.4.1 Classical Maximum Principle for Maximal Time Controls; 4.4.2 Local Maximum Principle for Maximal Time Controls; Miscellaneous Notes; References; 5 Equivalence of Several Kinds of Optimal Controls
5.1 Minimal Time Controls, Minimal Norm Controls, and Time Invariant Systems5.2 Minimal Time Controls, Minimal Norm Controls, and Time-Varying Systems: Part I; 5.2.1 Properties of Minimal Time and Minimal Norm Functions; 5.2.2 Proof of the Main Result; 5.3 Minimal Time Controls, Minimal Norm Controls, and Time-Varying Systems: Part II; 5.4 Maximal Time, Minimal Norm, and Optimal Target Controls; 5.4.1 Existence of Optimal Controls for These Problems; 5.4.2 Connections Between Problems of Optimal Target Control and Minimal Norm Control
2.3 Several Examples2.4 Main Subjects on Time Optimal Control Problems; Miscellaneous Notes; References; 3 Existence of Admissible Controls and Optimal Controls; 3.1 Admissible Controls and Controllability; 3.1.1 General Results in Abstract Setting; 3.1.2 Linear ODEs with Ball-Type Control Constraints; 3.1.3 Heat Equations with Ball-Type Control Constraints; 3.2 Admissible Controls and Minimal Norm Problems; 3.2.1 General Results in Abstract Setting; 3.2.2 Heat Equations with Ball-Type Control Constraints; 3.3 Admissible Controls and Reachable Sets; 3.3.1 General Results in Abstract Setting
3.3.2 Reachable Sets of Linear ODEs3.4 Existence of Optimal Controls; 3.4.1 Existence of Optimal Controls for Regular Cases; 3.4.2 Existence of Optimal Controls for Blowup Case; Miscellaneous Notes; References; 4 Maximum Principle of Optimal Controls; 4.1 Classical Maximum Principle of Minimal Time Controls; 4.1.1 Geometric Intuition; 4.1.2 Classical Maximum Principle; 4.1.3 Conditions on Separation in Classical Maximum Principle; 4.2 Local Maximum Principle and Minimal Time Controls; 4.2.1 Some Properties on Controllable Sets and Reachable Sets; 4.2.2 Separability and Local Maximum Principle
4.2.3 Conditions on Separation in Local Maximum Principle4.3 Weak Maximum Principle and Minimal Time Controls; 4.3.1 Separability and Weak Maximum Principle; 4.3.2 Two Representation Theorems Related to the Separability; 4.3.3 Conditions on Separation in Weak Maximum Principle; 4.4 Maximum Principle for Maximal Time Controls; 4.4.1 Classical Maximum Principle for Maximal Time Controls; 4.4.2 Local Maximum Principle for Maximal Time Controls; Miscellaneous Notes; References; 5 Equivalence of Several Kinds of Optimal Controls
5.1 Minimal Time Controls, Minimal Norm Controls, and Time Invariant Systems5.2 Minimal Time Controls, Minimal Norm Controls, and Time-Varying Systems: Part I; 5.2.1 Properties of Minimal Time and Minimal Norm Functions; 5.2.2 Proof of the Main Result; 5.3 Minimal Time Controls, Minimal Norm Controls, and Time-Varying Systems: Part II; 5.4 Maximal Time, Minimal Norm, and Optimal Target Controls; 5.4.1 Existence of Optimal Controls for These Problems; 5.4.2 Connections Between Problems of Optimal Target Control and Minimal Norm Control