Counterexamples in Optimal Control Theory

Preface
Introduction
Problem formulation
The maximum principle
Example
Approximate solution of the optimality conditions
Example 1. Insufficiency of the optimality conditions
Problem formulation
The maximum principle
Analysis of the optimality conditions
Uniqueness of the optimal control
Uniqueness of an optimal control in a specific sample
Further analysis of optimality conditions
Sufficiency of the optimality conditions
Sufficiency of the optimality conditions in a specific example
Conclusion of the analysis of the optimality conditions
Example 2. The singular control
Problem formulation
The maximum principle
Analysis of the optimality conditions
Nonoptimality of singular controls
Uniqueness of singular controls
Example 3. Nonexistence of optimal controls
Problem formulation
The maximum principle
Analysis of the optimality conditions
Unsolvability of the optimisation problem
Existence of optimal controls
The proof of the solvability of an optimisation problem
Conclusion of the anaysis
Example 4. Nonexistence of optimal controls (Part 2)
Problem formulation
The maximum principle for systems with fixed final state
Approximate solution of the optimality conditions
The optimality conditions for Problem 4
Direct investigation of Problem 4
Revising the problem analysis
Problems with unbounded set of admissible controls
The Cantor function
Further analysis of the maximum condition
Conclusion of the problem analysis
Example 5. Ill-Posedness in the sense of Tikhonov
Problem formulation
Solution of the problem
Ill-posedness in the sense of Tikhonov
Analysis of well-posedness in the sense of Tikhonov
The well-posed optimisation problem
Regularization of optimal control problems
Example 6. Ill-posedness in the sense of Hadamard
Problem formulation
Ill-posedness in the sense of Hadamard
Well-posedness in the sense of Hadamard
A well-posed optimisation problem
Example 7. Insufficiency of the optimality conditions (Part 2)
Problem formulation
The existence of an optimal control
Necessary condition for an extremum
Transformation of the optimality conditions
Analysis of the boundary value problem
The nonlinear heat conduction equation with infinitely many equilibrium states
Conclusion of the analysis of the variational problem
Example 8. The Chafee - Infante problem
Problem formulation
The necessary condition for an extremum
Solvability of the Chafee - Infante problem
The set of solutions of the Chafee - Infante problem
Bifurcation points
Comments
Conclusion
Bibliography