Optimal control of inductive heating of ferromagnetic materials

Inductive heating is a technological process where a steel workpiece is surrounded by an electromagnetic coil to which currents at various frequencies and time-varying amplitudes are applied. The amplitudes are considered as the controls and the objective is to heat the workpiece up to a desired temperature profile at the final time of the heating process. The workpiece is then quenched, which due to a phase transition in the crystallographic structure of the steel leads to a hardening of the surface of the workpiece. For the inductive heating process, the state equations represent a coupled system of nonlinear partial differential equations consisting of the eddy current equations in the coil, the workpiece, and the ambient air, and a heat equation in the workpiece. The nonlinearity stems from the temperature-dependent nonlinear material laws for steel, both with regard to its electromagnetic and thermal behaviour. Following the principle 'Discretize first, then optimize', we consider a semi-discretization in time by the implicit Euler scheme, which leads to a discrete-time optimal control problem. We prove the existence of a minimizer for the discrete-time optimal control problem and derive the first-order necessary optimality conditions.