We consider a control-constrained optimal control problem subject to time-harmonic Maxwell's equations; the control variable belongs to a finite-dimensional set and enters the state equation as a coefficient. We derive existence of optimal solutions, and analyze first- and second-order optimality conditions. We devise an approximation scheme based on the lowest order N\'ed\'elec finite elements to approximate optimal solutions. We analyze convergence properties of the proposed scheme and prove a priori error estimates. We also design an a posteriori error estimator that can be decomposed as the sum two contributions related to the discretization of the state and adjoint equations, and prove that the devised error estimator is reliable and locally efficient. We perform numerical tests in order to assess the performance of the devised discretization strategy and the a posteriori error estimator.

翻译：本文研究受时间谐波麦克斯韦方程约束的控制受限最优控制问题；控制变量属于有限维集合，并以系数形式进入状态方程。我们证明了最优解的存在性，并分析了最优性的一阶和二阶条件。设计了基于最低阶Nédélec有限元的逼近方案来近似最优解。分析了所提出方案的收敛性，并证明了先验误差估计。此外，还设计了一个后验误差估计器，该估计器可分解为状态方程和伴随方程离散化相关的两项之和，并证明了所设计的误差估计器是可靠且局部有效的。通过数值实验评估了所设计离散策略和后验误差估计器的性能。