Optimal control of an eddy current problem with a dipole source

TL;DR

This paper develops a mathematical framework for optimal control of eddy current systems with a dipole source, addressing well-posedness and deriving first order optimality conditions.

Contribution

It introduces a novel approach using the fundamental solution of a curl-curl operator to handle the dipole source in eddy current control problems.

Findings

Well-posedness of the state problem established

First order optimality conditions derived

Effective control strategy for electric and magnetic fields

Abstract

This paper is concerned with the analysis of a class of optimal control problems governed by a time-harmonic eddy current system with a dipole source, which is taken as the control variable. A mathematical model is set up for the state equation where the dipole source takes the form of a Dirac mass located in the interior of the conducting domain. A non-standard approach featuring the fundamental solution of a $\operatorname{curl} \operatorname{curl} - \operatorname{Id} $ operator is proposed to address the well-posedness of the state problem, leading to a split structure of the state field as the sum of a singular part and a regular part. The aim of the control is the best approximation of desired electric and magnetic fields via a suitable $L^2$-quadratic tracking cost functional. Here, special attention is devoted to establishing an adjoint calculus which is consistent with the form of the state variable and in this way first order optimality conditions are eventually derived.