Optimal control of the full time-dependent maxwell equations

Universität Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Str. 9, 45127 Essen, Germany.
vera.bommer@uni-due.de; irwin.yousept@uni-due.de

Received: 5 January 2015
Revised: 24 April 2015

Abstract

This paper analyzes the optimal control of the full time-dependent Maxwell equations. Our goal is to find an optimal current density and its time-dependent amplitude which steer the electric and magnetic fields to the desired ones. The main difficulty of the optimal control problem arises from the complexity of the Maxwell equations, featuring a first-order hyperbolic structure. We present a rigorous mathematical analysis for the optimal control problem. Here, the semigroup theory and the Helmholtz decomposition theory are the key tools in the analysis. Our theoretical findings include existence, strong regularity, and KKT theory. The corresponding optimality system consists of forward-backward Maxwell equations for the optimal electromagnetic and adjoint fields, magnetostatic saddle point equations for the optimal current density, and a projection formula for the optimal time-dependent amplitude. A semismooth Newton algorithm in a function space is established for solving the nonlinear and nonsmooth optimality system. The paper is concluded by numerical results, where mixed finite elements and Crank–Nicholson schema are used.