Optimal voltage control of non-stationary eddy current problems

Fredi Tröltzsch,Alberto Valli

doi:10.3934/mcrf.2018002

Abstract

A mathematical model is set up that can be useful for controlled voltage excitation in time-dependent electromagnetism.The well-posedness of the model is proved and an associated optimal control problem is investigated. Here, the controlfunction is a transient voltage and the aim of the control is the best approximation of desired electric and magnetic fields insuitable \begin{document} $L^2$ \end{document} -norms.Special emphasis is laid on an adjoint calculus for first-order necessary optimality conditions.Moreover, a peculiar attention is devoted to propose a formulation for which the computational complexity of the finite element solution method is substantially reduced.

Highlights

In the last two decades, the optimal control of electromagnetic fields received increasing attention
We find the eddy current model, in which wave propagation is not taken into account:
The voltage excitation problem reads: given VE : [0, T ] → R and VJ : [0, T ] → R, we look for a solution of the eddy current problem (2.2) satisfying for each t ∈ [0, T ] the boundary conditions μH · n = 0 on ∂Ω

Summary

Introduction

In the last two decades, the optimal control of electromagnetic fields received increasing attention. In the majority of these papers, distributed and/or time-dependent electrical currents were considered as controls. The control of electrical voltages was first investigated in the time-harmonic case, see [17, 18, 28, 24, 25]. It is more realistic to control the electrical voltage in a non-harmonic setting. This leads to specific issues of modeling and mathematical analysis. The mathematical analysis for the optimal control of voltages is the central aspect. The electric charge volume density ρ is assumed to vanish in non-conducting regions

Disregarding the displacement current term

We start by defining the space of harmonic fields

By the notation

We set ν

Hence we have found