This monograph deals with cases where optimal control either does not exist or is not unique, cases where optimality conditions are insufficient of degenerate, or where extremum problems in the sense of Tikhonov and Hadamard are ill-posed, and other situations. A formal application of classical optimisation methods in such cases either leads to wrong results or has no effect. The detailed analysis of these examples should provide a better understanding of the modern theory of optimal control and the practical difficulties of solving extremum problems.

Preface

1

(4)

Introduction

5

(14)

1. Problem formulation

5

(2)

2. The maximum principle

7

(4)

3. Example

11

(3)

4. Approximate solution of the optimality conditions

14

(4)

Summary

18

(1)

Example 1. Insufficiency of the optimality conditions

19

(28)

1.1. Problem formulation

20

(1)

1.2. The maximum principle

21

(2)

1.3. Analysis of the optimality conditions

23

(3)

1.4. Uniqueness of the optimal control

26

(2)

1.5. Uniqueness of an optimal control in a specific example

28

(3)

1.6. Further analysis of optimality conditions

31

(2)

1.7. Sufficiency of the optimality conditions

33

(3)

1.8. Sufficiency of the optimality conditions in a specific example

36

(3)

1.9. Conclusion of the analysis of the optimality conditions

39

(3)

Summary

42

(5)

Example 2. The singular control

47

(18)

2.1. Problem formulation

47

(1)

2.2. The maximum principle

48

(1)

2.3. Analysis of the optimality conditions

49

(4)

2.4. Nonoptimality of singular controls

53

(3)

2.5. Uniqueness of singular controls

56

(3)

2.6. The Kelly condition

59

(3)

Summary

62

(3)

Example 3. Nonexistence of optimal controls

65

(22)

3.1. Problem formulation

65

(1)

3.2. The maximum principle

66

(1)

3.3. Analysis of the optimality conditions

67

(4)

3.4. Unsolvability of the optimization problem

71

(4)

3.5. Existence of optimal controls

75

(2)

3.6. The proof of the solvability of an optimization problem

77

(3)

3.7. Conclusion of the analysis

80

(5)

Summary

85

(2)

Example 4. Nonexistence of optimal controls (Part 2)

87

(26)

4.1. Problem formulation

88

(1)

4.2. The maximum principle for systems with fixed final state

89

(2)

4.3. Approximate solution of the optimality conditions

91

(2)

4.4. The optimality conditions for Problem 4

93

(1)

4.5. Direct investigation of Problem 4

94

(2)

4.6. Revising the problem analysis

96

(2)

4.7. Problems with unbounded set of admissible controls

98

(3)

4.8. The Cantor function

101

(2)

4.9. Further analysis of the maximum condition

103

(2)

4.10. Conclusion of the problem analysis

105

(5)

Summary

110

(3)

Example 5. Ill-posedness in the sense of Tikhonov

113

(14)

5.1. Problem formulation

114

(1)

5.2. Solution of the problem

114

(1)

5.3. Ill-posedness in the sense of Tikhonov

115

(5)

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

120

(1)

5.5. The well-posed optimization problem

121

(2)

5.6. Regularization of optimal control problems

123

(2)

Summary

125

(2)

Example 6. Ill-posedness in the sense of Hadamard

127

(8)

6.1. Problem formulation

128

(1)

6.2. Ill-posedness in the sense of Hadamard

128

(2)

6.3. Well-posedness In the sense of Hadamard

130

(1)

6.4. A well-posed optimization problem

131

(1)

Summary

132

(3)

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

135

(18)

7.1. Problem formulation

135

(1)

7.2. The existence of an optimal control

136

(3)

7.3. Necessary condition for an extremum

139

(2)

7.4. Transformation of the optimality conditions

141

(2)

7.5. Analysis of the boundary value problem

143

(6)

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

149

(1)

7.7. Conclusion of the analysis of the variational problem

150

(2)

Summary

152

(1)

Example 8. The Chafee-Infante problem

153

(12)

8.1. Problem formulation

153

(1)

8.2. The necessary condition for an extremum

154

(1)

8.3. Solvability of the Chafee-Infante problem

155

(3)

8.4. The set of solutions of the Chafee-Infante problem

158

(3)

8.5. Bifurcation points

161

(2)

Summary

163

(2)

Comments

165

(2)

Conclusion

167

(2)

Bibliography

169

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Counterexamples in Optimal Control Theory

9789067644006

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