●Chapter 1 Introduction 1
1.1 Backgrounds and Motivations of the Book 1
1.2 Optimal Control Theory 3
1.3 Problem Formulation12
1.3.1 Some Examples of Optimal Control Problems 12
1.3.2 Mathematical Formulation 19
1.4 Organization 24
References 25
Chapter 2 Extrema of Functional via Variational Method 31
2.1 Fundamental Notions 32
2.1.1 Linearity of Function and Functional 32
2.1.2 Norm in Euclidean Space and Functional 34
2.1.3 Increment of Function and Functional 35
2.1.4 Di erential of Function and Variation of Functional 37
2.2 Extrema of Functional 39
2.2.1 Extrema with Fixed Final Time and Fixed Final State 43
2.2.2 Speci c Forms of Euler Equation in Di erent Cases 47
2.2.3 Su cient Condition for Extrema 55
2.2.4 Extrema with Fixed Final Time and Free Final State 58
2.2.5 Extrema with Free Final Time and Fixed Final State 61
2.2.6 Extrema with Free Final Time and Free Final State 66
2.3 Extrema of Functional with Multiple Independent Functions 72
2.4 Extrema of Function with Constraints 80
2.4.1 Elimination/Direct Method 81
2.4.2 Lagrange Multiplier Method 82
2.5 Extrema of Functional with Constraints 84
2.5.1 Extrema of Functional with Di erential Constraints 84
2.5.2 Extrema of Functional with Isoperimetric Constraints 89
2.6 Summary 91
2.7 Exercises92
Chapter 3 Optimal Control via Variational Method 96
3.1 Necessary and Su cient Condition for Optimal Control 96
3.2 Optimal Control Problems with Di erent Boundary Conditions 104
3.2.1 Optimal Control with Fixed Final Time and State 104
3.2.2 Optimal Control with Fixed Final Time and Free Final State 106
3.2.3 Optimal Control with Free Final Time and Fixed Final State 111
3.2.4 Optimal Control with Free Final Time and State 112
3.3 Linear Quadratic Regulator Problems 122
3.3.1 In nite-interval Time-invariant LQR Problems 130
3.4 Linear Quadratic Tracking Problems 132
3.5 Summary 140
3.6 Exercises 140
Chapter 4 Pontryagin's Minimum Principle 145
4.1 Pontryagin's Minimum Principle with Constrained Control 145
4.2 Pontryagin's Minimum Principle with Constrained State Variable 155
4.3 Minimum Time Problems159
4.3.1 Optimal Control Solution for Minimum Time Problems 159
4.3.2 Minimum Time Problems for Linear Time-invariant Systems 161
4.4 Minimum Fuel Problems 171
4.5 Performance Cost Composed of Elapsed Time and Consumed Fuel 192
4.6 Minimum Energy Problems 204
4.7 Performance Cost Composed of Elapsed Time and Consumed Energy 212
4.8 Summary 221
4.9 Exercises 221
Chapter 5 Conclusions 226
內容簡介
本書主要討論如何通過變分法來實現很優控制問題。更具體地說 研究了如何應用變分法實現泛函極值。它涵蓋了具有不同邊界條件、涉及多個函數、具有一定約束條件等的泛函極值問題。 1.利用變分法給出了（連續時間）很優控制解的充要條件，求解了不同邊界條件下的很優控制問題，並分別對線性二次型調節器和跟蹤問題進行了詳細的分析。 2.通過應用基於變分法的Pontryagin很小原理，給出了具有狀態約束的很優控制問題的解。並將所得結果應用於實現幾種常見的很優控制問題，如很小時間、很小燃料和很小能量問題等。 作為很優控制方法的另一個重要分支，本文還介紹了如何通過動態規劃求解很優控制問題，並討論了變分法與動態規劃的關繫，以供比較。 3.關於涉及單個代理的繫統，還值得研究如何在微分模型框架內實現底層很優控制問題的分散解。應用龐特裡亞金很小原理和動態規劃方法實現了平衡。 由於離散時間很優控制問題在許多領域都很流行，所以......
