Optimal control of Maxwell’s equations with regularized state constraints

Abstract

This paper is devoted to an optimal control problem of Maxwell’s equations in the presence of pointwise state constraints. The control is given by a divergence-free three-dimensional vector function representing an applied current density. To cope with the divergence-free constraint on the control, we consider a vector potential ansatz. Due to the lack of regularity of the control-to-state mapping, existence of Lagrange multipliers cannot be guaranteed. We regularize the optimal control problem by penalizing the pointwise state constraints. Optimality conditions for the regularized problem can be derived straightforwardly. It also turns out that the solution of the regularized problem enjoys higher regularity which then allows us to establish its convergence towards the solution of the unregularized problem. The second part of the paper focuses on the numerical analysis of the regularized optimal control problem. Here the state and the control are discretized by Nédélec’s curl-conforming edge elements. Employing the higher regularity property of the optimal control, we establish an a priori error estimate for the discretization error in the $\boldsymbol{H}(\bold{curl})$-norm. The paper ends by numerical results including a numerical verification of our theoretical results.