Optimal Control of the Thermistor Problem in Three Spatial Dimensions, Part 1: Existence of Optimal Solutions

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 are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. 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.for all ξ ∈ X. Moreover, in the proof of Theorem 4.5, ∂ θ B(θ, u) was found to be a topological isomorphism between the spaces X and L s (J; W −1,q ∅ ) × X s and consequently ∂ θ B(θ, u) * is also a topological isomorphism between L s (J; W 1,q ) × X s and X . In particular, for every f ∈ X there exists a unique p = p f ∈ L s (J; W 1,q ) × X s such that ∂ θ B(θ, u) * p = f. Hence, settinḡ

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

“…General assumptions and known results. We collect known results from the companion paper of this work [30] and recall some basic definitions.…”

“…The particular assumption of being volume-preserving is needed in [30] as a technical assumption to obtain that the set of global controls is nonempty and for a certain embedding of interpolation spaces, cf. [30,Lem. 3.19/Cor.…”

“…The property is, for instance, always satisfied if Ω is in fact a strong Lipschitz domain. We refer to [30,Rem. 2.4] for this, and more information about the geometric setting.…”

“…First of all, the discussion of the quasilinear state system alone requires sophisticated up-to-date tools from maximal elliptic and parabolic regularity theory. This concerns already local-in-time existence for solutions of (1.5)-(1.6) as carried out in the companion paper [30], but also the characterization of global-in-time solutions and continuous differentiability for the control-to-state operator. The corresponding maximal regularity results were established only recently, see e.g.…”

“…In particular, the fact the set of "global" controls is open allows to obtain in essence the same optimality system as one would obtain if one always had global-in-time solutions for every admissible control. In the derivation of first order necessary optimality conditions, we also have to consider the nonstandard second and third term in the objective functional which allowed to show existence of optimal controls in the companion paper [30]. Lastly, we also give a numerical example which underlines the necessity of both control-and state bounds in (P).…”