Chapter
4. Approximate solution of the optimality conditions
pp.：
19 – 22
Example 1. Insufficiency of the optimality conditions
pp.：
26 – 27
1.1. Problem formulation
pp.：
27 – 28
1.2. The maximum principle
pp.：
28 – 29
1.3. Analysis of the optimality conditions
pp.：
29 – 31
1.4. Uniqueness of the optimal control
pp.：
31 – 34
1.5. Uniqueness of an optimal control in a specific example
pp.：
34 – 36
1.6. Further analysis of optimality conditions
pp.：
36 – 39
1.7. Sufficiency of the optimality conditions
pp.：
39 – 41
1.8. Sufficiency of the optimality conditions in a specific example
pp.：
41 – 44
1.9. Conclusion of the analysis of the optimality conditions
pp.：
44 – 47
Example 2. The singular control
pp.：
50 – 55
2.2. The maximum principle
pp.：
55 – 56
2.1. Problem formulation
pp.：
55 – 55
2.3. Analysis of the optimality conditions
pp.：
56 – 57
2.4. Nonoptimality of singular controls
pp.：
57 – 61
2.5. Uniqueness of singular controls
pp.：
61 – 64
2.6. The Kelly condition
pp.：
64 – 67
Example 3. Nonexistence of optimal controls
pp.：
70 – 73
3.1. Problem formulation
pp.：
73 – 73
3.2. The maximum principle
pp.：
73 – 74
3.3. Analysis of the optimality conditions
pp.：
74 – 75
3.4. Unsolvability of the optimization problem
pp.：
75 – 79
3.5. Existence of optimal controls
pp.：
79 – 83
3.6. The proof of the solvability of an optimization problem
pp.：
83 – 85
3.7. Conclusion of the analysis
pp.：
85 – 88
Example 4. Nonexistence of optimal controls (Part 2)
pp.：
93 – 95
4.1. Problem formulation
pp.：
95 – 96
4.2. The maximum principle for systems with fixed final state
pp.：
96 – 97
4.3. Approximate solution of the optimality conditions
pp.：
97 – 99
4.4. The optimality conditions for Problem 4
pp.：
99 – 101
4.5. Direct investigation of Problem 4
pp.：
101 – 102
4.6. Revising the problem analysis
pp.：
102 – 104
4.7. Problems with unbounded set of admissible controls
pp.：
104 – 106
4.8. The Cantor function
pp.：
106 – 109
4.9. Further analysis of the maximum condition
pp.：
109 – 111
4.10. Conclusion of the problem analysis
pp.：
111 – 113
Example 5. Ill-posedness in the sense of Tikhonov
pp.：
118 – 121
5.1. Problem formulation
pp.：
121 – 122
5.2. Solution of the problem
pp.：
122 – 122
5.3. Ill-posedness in the sense of Tikhonov
pp.：
122 – 123
5.4. Analysis of well-posedness in the sense of Tikhonov
pp.：
123 – 128
5.5. The well-posed optimization problem
pp.：
128 – 129
5.6. Regularization of optimal control problems
pp.：
129 – 131
Example 6. Ill-posedness in the sense of Hadamard
pp.：
133 – 135
6.1. Problem formulation
pp.：
135 – 136
6.3. Well-posedness in the sense of Hadamard
pp.：
136 – 138
6.2. Ill-posedness in the sense of Hadamard
pp.：
136 – 136
6.4. A well-posed optimization problem
pp.：
138 – 139
Example 7. Insufficiency of the optimality conditions (Part 2)
pp.：
140 – 143
7.1. Problem formulation
pp.：
143 – 143
7.2. The existence of an optimal control
pp.：
143 – 144
7.3. Necessary condition for an extremum
pp.：
144 – 147
7.4. Transformation of the optimality conditions
pp.：
147 – 149
7.5. Analysis of the boundary value problem
pp.：
149 – 151
7.6. The nonlinear heat conduction equation with infinitely many equilibrium states
pp.：
151 – 157
7.7. Conclusion of the analysis of the variational problem
pp.：
157 – 158
Example 8. The Chafee–Infante problem
pp.：
160 – 161
8.2. The necessary condition for an extremum
pp.：
161 – 162
8.1. Problem formulation
pp.：
161 – 161
8.3. Solvability of the Chafee–Infante problem
pp.：
162 – 163
8.4. The set of solutions of the Chafee–Infante problem
pp.：
163 – 166
8.5. Bifurcation points
pp.：
166 – 169
Conclusion
pp.：
173 – 175
Bibliography
pp.：
175 – 177
Counterexamples in Optimal Control Theory
( Inverse and Ill-Posed problems Series )
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