Problemas de control óptimo gobernados por ecuaciones semilineales con restricciones sobre el gradiente del estado

Résumé

The thesis deals with different optimal control problems. In the first part, the equations that appear in the control problems are studied. In Chapter 2, we introduce some regularity results for linear equations. These results will be applied later to stablish the regularity both of the state and the adjoint state. In Chapter 3, we study the state equations that govern the control problems. We show the continuity and differentiability relations between the control and the state. A sensitivity analysis of the state with respect to diffuse perturbations of the control is also carried out. The second part is the central kernel of the work. We study necessary as well as sufficient optimality conditions for the control problems. In Chapter 4 we show properties of the functionals that appear in our problems: the objective functional and the constraints. We study under what conditions they are differentiable and, in order to prove a Pontryagin principle, we give sensitivity results with respect to diffuse perturbations of the control. In Chapter 5, we state Pontryagin principle for our problems. In Chapter 6, we state first and second order optimality conditions. In Chapter 7, we introduce a new kind of second order optimality conditions that involve the Hamiltonian. Both the elliptic and parabolic cases are treated in each chapter. In the third part, we perform the numerical analysis of problems with pointwise constraints on the state. In Chapter 8, we provide results about about unform convergence of the discretization of elliptic semilinear equations. Finally, in Chpater 9, the control problem is discretized and we investigate how the solutions of the discrete problems converge towards the solution of the continuous problem.