Control óptimo y resolución numérica de problemas parabólicos con controles finitos dispersos

Abstract

Optimal control of parabolic partial differential equations is an important problem in practice as it characterizes various phenomena such as heating control materials like steel, pest control evolve in a territory among others or therapies for the treatment of cancer such as hyperthermia. In fact, from a practical point of view finite controls are interesting because in many cases the technology that controls certain processes makes it through the selection of finite parameters that cause specific effects on the phenomenon to be controlled. The purpose of this project is to solve parabolic optimal control problems with finite controls, with and without dependence on time and a cuadratic functional. Then, we add sparcity to the problem for the finite case. We also make a characterization of the optimal control by a projection formula. We implement the methods. Primal Dual Active set and Coordinate descent that has parallelization properties. Finally, we present some numerical experiments obtained from the application of the developed algorithms.