Local null controllability for the thermistor problem

Abstract

In this paper, we deal with the local null controllability of an initial boundary value problem for a thermistor equation. The control is distributed, locally in space. The main ingredients of the proof are suitable Carleman estimates for an adjoint system and Liusternik’s Inverse Mapping Theorem in Hilbert spaces. As a complement to the theoretical results, we present formulations and numerical simulations for the controllability problem, the results were obtained through the finite element method and a quasi-Newton method, simulated in the software Freefem++. Finally, large time null controllability, stabilization and others possible extensions and open problems concerning other systems are presented.