Optimal Control of the Thermistor Problem in Three Spatial Dimensions, Part 2: Optimality Conditions

Abstract

This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness, and continuity for the state system as well as existence of optimal solutions, admitting global-in-time solutions, to the optimization problem were shown in the the companion paper of this work. In this part, we address further properties of the set of controls whose associated solutions exist globally, such as openness, which includes analysis of the linearized state system via maximal parabolic regularity. The adjoint system involving measures is investigated using a duality argument. These results allow us to derive first-order necessary conditions for the optimal control problem in the form of a qualified optimality system in which we do not need to refer to the set of controls admitting global solutions. The theoretical findings are illustrated by numerical results. This work is the second of two papers on the three-dimensional thermistor problem