WIAS Preprint No. 2238, (2016)

Optimal control of the thermistor problem in three spatial dimensions


  • Meinlschmidt, Hannes
    ORCID: 0000-0002-5874-8017
  • Meyer, Christian
  • Rehberg, Joachim

2010 Mathematics Subject Classification

  • 33K55 35M10 49J20 49K20


  • partial differential equations, optimal control problems, state constraints




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 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.

Appeared in

  • SIAM J. Control Optim., Part 1 (Existence of optimal solutions) 55:5 (2017) pp. 2876--2904, Part 2 (Optimality Conditions) 55:4 (2017) pp. 2368--2392.

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