Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. CONTROL OPTIM. c  2010 Society for Industrial and Applied Mathematics Vol. 48, No. 5, pp. 3449–3481 OPTIMAL CONTROL FOR THE THERMISTOR PROBLEM ∗ D. H ¨ OMBERG † , C. MEYER † , J. REHBERG † , AND W. RING ‡ Abstract. This paper is concerned with the state-constrained optimal control of the two- dimensional thermistor problem, a quasi-linear 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. Existence, uniqueness, and continuity for the state system are derived by employing maximal elliptic and parabolic regularity. By similar arguments the linearized state system is dis- cussed, while 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. Key words. partial differential equations, optimal control problems, state constraints AMS subject classifications. 35K55, 35M10, 49J20, 49K20 DOI. 10.1137/080736259 1. Introduction. In this paper we consider state-constrained optimal control of the two-dimensional thermistor problem. In detail the optimal control problem under consideration looks as follows: (P) minimize J (θ, ϕ, u) := 1 2  D |θ(T ) - θ d | 2 dx + β 2  ΣN u 2 ds dt subject to (1.1)–(1.7) and θ(x, t) ≤ θ max (x, t) a.e. in Q, 0 ≤ u(x, t) ≤ u max (x, t) a.e. on Σ N , ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ where (1.1)–(1.7) refer to the following coupled PDE system consisting of the insta- tionary heat equation and the quasi-static potential equation, which is also known as the thermistor problem: ∂ t θ - div(κ∇θ)=(σ(θ)∇ϕ) ·∇ϕ in Q := Ω× ]0,T [, (1.1) ν · κ∇θ + αθ = αθ l on Σ := ∂ Ω× ]0,T [, (1.2) θ(0) = θ 0 in Ω, (1.3) - div(σ(θ)∇ϕ)=0 in Q, (1.4) ν · σ(θ)∇ϕ = u on Σ N := Γ N × ]0,T [, (1.5) ϕ =0 on Σ D := Γ D × ]0,T [, (1.6) ν · σ(θ)∇ϕ =0 on (∂ Ω\ Γ N ∪ Γ D )× ]0,T [. (1.7) Here θ is the temperature in a conducting material covered by the two-dimensional domain Ω, while ϕ refers to the electric potential. The boundary of Ω is denoted by ∂ Ω with unit normal ν facing outward from Ω. In addition, Γ D is a closed part of ∗ Received by the editors September 24, 2008; accepted for publication (in revised form) Octo- ber 30, 2009; published electronically February 17, 2010. http://www.siam.org/journals/sicon/48-5/73625.html † Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Ger- many (hoemberg@wias-berlin.de, cmeyer@gsc.tu-darmstadt.de, rehberg@wias-berlin.de). ‡ Institute of Mathematics, University of Graz, Heinrichstr. 36, A-8010 Graz, Austria (wolfgang. ring@uni-graz.at). 3449 Downloaded 07/10/19 to 143.50.47.147. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php