Practical course of the optimal control theory with examples

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«Практический курс теории оптимального управления с примерами» представляет собой учебное пособие, читаемого учащиеся различных специальностей на английском языке.

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Contents

Preface. 6

Introduction. 10

1. Problem formulation. 10

2. The maximum principle. 12

3. Example. 17

4. Approximate solution of the optimality conditions. 21

Summary. 24

Example 1. Insufficiency of the  optimality conditions. 26

1.1. Problem formulation. 27

1.2. The maximum principle. 27

1.3. Analysis of the optimality conditions. 29

1.4. Uniqueness of the optimal control33

1.5. Uniqueness of an optimal control in a specific example. 36

1.6. Further analysis of optimality conditions. 38

1.7. Sufficiency of the optimality conditions. 41

1.8. Sufficiency of the optimality conditions in  a specific example. 43

1.9. Conclusion of the analysis of the optimality conditions. 46

1.9. Final remarks. 50

Summary. 52

Example 2. The singular control54

2.1. Problem formulation. 54

2.2. The maximum principle. 55

2.3. Analysis of the optimality conditions. 56

2.4. Nonoptimality of singular controls. 61

2.5. Uniqueness of singular controls. 63

2.6. The Kelly Condition. 67

Summary. 70

Example 3. Nonexistence of optimal controls. 71

3.1. Problem formulation. 71

3.2. The maximum principle. 72

3.3. Analysis of the optimality conditions. 73

3.4. Unsolvability of the optimization problem… 77

3.5. Existence of optimal controls. 81

3.6. The proof of the solvability of an optimization problem… 83

3.7. Conclusion of the analysis. 86

Summary. 91

Example 4. Nonexistence of optimal controls  (Part 2)92

4.1. Problem formulation. 93

4.2. The maximum principle for systems with fixed final state. 94

4.3. Approximate solution of the optimality  conditions. 96

4.4. The optimality conditions for problem 4. 97

4.5. Direct investigation of problem 4. 99

4.6. Revising the problem analysis. 101

4.7. Problems with unbounded set of admissible controls. 102

4.8. The Cantor function. 106

4.9. Further analysis of the maximum condition. 108

4.10. Conclusion of the problem analysis. 110

Summary. 115

Example 5. Ill-posedness in the sense of  Tikhonov. 117

5.1. Problem formulation. 118

5.2. Solution of the problem… 118

5.3. Ill-posedness in the sense of Tikhonov. 119

5.4. Analysis of well-posedness in the sense of  Tikhonov. 124

5.5. The well-posed optimization problem… 125

5.6. Regularization of optimal control problems. 127

summary. 128

Example 7. Insufficiency of the optimality conditions (Part 2)129

7.1. Problem formulation. 129

7.2. The existence of an optimal control130

7.3. Necessary conditions for an extremum… 133

7.4. Transformation of the optimality conditions. 135

7.5. Analysis of the boundary value problem… 137

7.6. The nonlinear heat conduction equation with infinitely many equilibrium states  143

7.7. Conclusion of the analysis of the variational problem… 144

Summary. 146

Example 8. Extremal bifurcation. 147

8.1. Problem formulation. 147

8.2. The necessary condition for an extremum… 148

8.3. Solvability of the Chafee – Infante problem… 149

8.4. The set of solutions of the Chafee – Infante problem… 152

8.5. Bifurcation points. 154

Summary. 157

Conclusion. 159

Tasks. 160

Comments. 165

Bibliography. 167

Short Russian-English mathematical dictionary. 174

 

Additional Information

Переплет

Мягкий, Твердый

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