An optimal control problem for Maxwell's equations

TL;DR

This paper investigates the boundary control of Maxwell's equations, demonstrating how to recover optimal solutions via approximation methods despite the challenges posed by the energy space topology.

Contribution

It introduces a novel approach to obtain optimal control solutions for Maxwell's equations using approximation techniques, bypassing the need for Riccati equation solvability.

Findings

The Riccati operator can be approximated to recover optimal solutions.

Explicit solutions are derived for the zero conductivity case.

The approach applies to hyperbolic PDEs with energy space topology.

Abstract

This article is concerned with the optimal boundary control of the Maxwell system. We consider a Bolza problem, where the quadratic functional to be minimized penalizes the electromagnetic field at a given final time. Since the state is weighted in the energy space topology -- a physically realistic choice --, the property that the optimal cost operator does satisfy the Riccati equation (RE) corresponding to the optimization problem is missed, just like in the case of other significant hyperbolic partial differential equations; however, we prove that this Riccati operator as well as the optimal solution can be recovered by means of approximating problems for which the optimal synthesis holds via proper differential Riccati equations. In the case of zero conductivity, an explicit representation of the optimal pair is valid which does not demand the well-posedness of the RE, instead.