(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Hömberg, Dietmar; Meyer, Christian; Rehberg, Joachim; Ring, Wolfgang
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.
