Structure-Preserving Optimal Control of Maxwell's Equations with Applications to Source Cloaking

TL;DR

This paper presents a structure-preserving numerical framework for the optimal control of Maxwell's equations, ensuring physical properties are maintained and enabling effective source cloaking applications.

Contribution

It introduces a novel structure-preserving finite element and time-stepping scheme for Maxwell's control problems, with proven convergence and applicability to source cloaking.

Findings

The solver preserves de Rham structure and enforces Gauss law discretely.

The control-to-state map is well-posed and continuous.

Numerical experiments demonstrate effective source cloaking.

Abstract

We develop a structure-preserving solution framework for the optimal control of the time-dependent Maxwell's equations. Building on a well-posedness theory for a weak form of the forward problem, we first analyze a forward solver that couples N\'ed\'elec and Raviart--Thomas finite elements with Crank--Nicolson time stepping. The solver preserves the de~Rham structure, enforces a discrete Gauss law, exactly satisfies a per-time-step energy balance, and converges to the weak solution under low regularity assumptions on the problem data, which are dictated by the optimal control setting. To control the Maxwell system, we add the curl of a space-time current density as a source to Amp\'ere's law. The curl form yields charge conservation without auxiliary constraints. We prove the well-posedness and continuity of the control-to-state map, derive the adjoint system and a gradient representation for a tracking-type objective functional, and formulate a discrete optimization scheme that inherits structure preservation from the forward solver. Our discrete stationarity conditions are consistent with their continuous counterparts, and the discrete optimal controls converge, with mesh and time refinements, to the continuous optima. We demonstrate the merits of our optimal control formulation and the theoretical developments by numerically solving a series of source-cloaking model problems.