WIAS Preprint No. 1422, (2009)

Optimal control of 3D state-constrained induction heating problems with nonlocal radiation effects


  • Druet, Pierre-Étienne
    ORCID: 0000-0001-5303-0500
  • Klein, Olaf
    ORCID: 0000-0002-4142-3603
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604
  • Tröltzsch, Fredi
  • Yousept, Irwin

2010 Mathematics Subject Classification

  • 49K20 35D10 35J60 78M50 80M50


  • State-constrained optimization, Maxwell equations, nonlocal radiation boundary conditions, induction heating, crystal growth




The paper is concerned with a class of optimal heating problems in semiconductor single crystal growth processes. To model the heating process, time-harmonic Maxwell equations are considered in the system of the state. Due to the high temperatures characterizing crystal growth, it is necessary to include nonlocal radiation boundary conditions and a temperature-dependent heat conductivity in the description of the heat transfer process. The first goal of this paper is to prove the existence and uniqueness of the solution to the state equation. The regularity analysis associated with the time harmonic Maxwell equations is also studied. In the second part of the paper, the existence and uniqueness of the solution to the corresponding linearized equation is shown. With this result at hand, the differentiability of the control-to-state mapping operator associated with the state equation is derived. Finally, based on the theoretical results, first oder necessary optimality conditions for an associated optimal control problem are established.

Appeared in

  • SIAM J. Control Optim., 49 (2011) pp. 1707--1736.

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