WIAS Preprint No. 1363, (2008)

Optimal control for the thermistor problem


  • Hömberg, Dietmar
    ORCID: 0000-0001-9460-5729
  • Meyer, Christian
  • Rehberg, Joachim
  • Ring, Wolfgang

2010 Mathematics Subject Classification

  • 35K55 35M10 49J20 49K20


  • Partial differential equations, optimal control problems, state constraints




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 discussed, while 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.

Appeared in

  • SIAM J. Control Optim., 48 (2010) pp. 3449--3481.

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