Optimal Control of Three-Dimensional State-Constrained Induction Heating Problems with Nonlocal Radiation Effects

Abstract

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 existence and uniqueness of the state. The regularity analysis associated with the time-harmonic Maxwell equations is also studied. In the second part of the paper, existence and uniqueness of the solution of the corresponding linearized equation are shown. With this result at hand, the differentiability of the control-to-state operator is derived. Finally, based on the theoretical results, first order necessary optimality conditions for an associated optimal control problem are established.