Sobre a existência, unicidade e controlabilidade de soluções para algumas equações diferenciais parciais de evolução.

Abstract

In this work we study the existence, uniqueness and exact controllability at the boundary and internal to the linear wave equation with Dirichlet boundary condition. In addition, we did a study of the existence, uniqueness and controllability of the heat equation and also on the controllability of the semilinear wave equation. To this end, in the part of the existence of the linear wave equation, we use the Faedo-Galerkin method. Already for the heat equation we made the existence of solution through the semigroup theory of linear operators. For exact controllability, we essentially use the Hilbertian Uniqueness Method (HUM) and also by means of variational methods, we show that the exact controllability can be done through a minimization problem. For the controllability of the heat equation, we use a method that is based on obtaining a Carleman inequality through an inequality of observability and finally, in the controllability of the semilinear wave equation, we use a method based on the Fixed Point Theorem of Schauder.