Optimal Control of the Thermistor Problem in Three Spatial Dimensions

TL;DR

This paper develops a mathematical framework for controlling the temperature in a 3D thermistor system, establishing existence, uniqueness, and optimality conditions, supported by numerical demonstrations.

Contribution

It introduces new theoretical results on the existence, uniqueness, and optimality conditions for the 3D thermistor control problem using maximal parabolic regularity.

Findings

Proved local and global existence of solutions.

Derived first-order necessary optimality conditions.

Validated results with numerical simulations.

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 are derived by employing maximal parabolic regularity in the fundamental theorem of Pr\"uss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results.