Boundary control of time-harmonic eddy current equations

TL;DR

This paper develops a boundary control framework for time-harmonic eddy current equations, extending complex derivatives, deriving optimality conditions, and validating through finite element discretization and numerical experiments.

Contribution

It introduces a novel boundary control approach for Maxwell's equations in the frequency domain, including complex derivative extension and rigorous discretization analysis.

Findings

Well-posed boundary control formulation using surface curl

Extension of Wirtinger derivatives to complex Hilbert spaces

Convergence of the finite element scheme demonstrated

Abstract

Motivated by various applications, this article develops the notion of boundary control for Maxwell's equations in the frequency domain. Surface curl is shown to be the appropriate regularization in order for the optimal control problem to be well-posed. Since, all underlying variables are assumed to be complex valued, the standard results on differentiability do not directly apply. Instead, we extend the notion of Wirtinger derivatives to complexified Hilbert spaces. Optimality conditions are rigorously derived and higher order boundary regularity of the adjoint variable is established. The state and adjoint variables are discretized using higher order N\'ed\'elec finite elements. The finite element space for controls is identified, as a space, which preserves the structure of the control regularization. Convergence of the fully discrete scheme is established. The theory is validated by numerical experiments, in some cases, motivated by realistic applications.