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

Abstract

Abstract. This paper is concerned with the state-constrained optimal control of the threedimensional 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 to derive first-order necessary conditions for the optimal control problem in 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.

Interpolation Theory for Sobolev Functions with Partially Vanishing Trace on Irregular Open Sets

“…We leave the care of writing them down to the interested readers. Here, we only present one such result that turned out useful in the W −1,p -theory of divergence form operators [11,32,33] and previously was available only in the restrictive setup of [19,Lemma 3.4]. The proof of this result will be given in Section 5.…”

“…Let s ∈ R \ Z. If s ≥ 1 − 1/p, then E : W s,p (R d−1 ) → X s+1/p,p (R d + ) follows by (33). Otherwise, we choose m ∈ N such that 2m + s ≥ 1 − 1/p.…”

“…Otherwise, we choose m ∈ N such that 2m + s ≥ 1 − 1/p. We use the decomposition (30) (32), (33) and (31).…”

Optimal control of a quasilinear parabolic equation and its time discretization

“…( 4) is still in its infancy. Most literature known to us treat quasilinearities with coefficients depending on x, t and on the function y but not on its gradient [7,26,11,29,30]. For quasilinearities involving spatial derivatives of y see for example [33,10].…”

A-posteriori reduced basis error-estimates for a semi-discrete in space quasilinear parabolic PDE

“…For additional state constraints we refer to [36]. A quasilinear version of the so-called thermistor problem has been addressed in [48,49], and convergence of the SQP method applied to the model problem from [9] has been proven in [35]. For earlier literature on quasilinear parabolic optimal control problems, and optimal control of quasilinear elliptic PDEs with refer to the introductions of [9,10].…”