Resultados de controlabilidade pontual para a equação do calor unidimensional

Abstract

In this work we’ll study the pointwise null controllability problem for the onedimensional heat equation. First, will be shown that the pointwise controllability is equivalent to the convergence of an appropriate series depending only on the point, the controllability time and the eigenvalues and eigenfunctions of the one-dimensional Laplace operator. As a consequence, there exists a minimum time of controllability, which may be strictly positive. This phenomenon of positive minimum time of controllability doesn’t occur in the case of controls applied on open sets. Furthermore, using arguments from diophantine approximation theory, we show that the set of points for which the equation is pointwise controllable for all time is a dense subset of the domain. Finally, with the aim of understanding the phenomenon of minimum time, it was analyzed what happens when the control is applied to an open set that degenerates to a point.