Convergence analysis and a posteriori error estimates of reduced order solutions for optimal control problem of parameterized Maxwell system

In this paper we investigate the reduced order solution of the optimal control problem governed by a parameterized stationary Maxwell system with the Gauss law. In this context the dielectric, the magnetic permeability and the charge density are assumed to be known, where the control is constrained of general type and the parameter set is compact. We approximate the electric field of the Maxwell system in finite element spaces. Adopting the variational discretization concept, we consider a weighted parameterized optimal control problem. Utilizing techniques from the primal reduced basis approach, we construct a reduced basis surrogate model for the aforementioned optimal control problem. We prove the uniform convergence of reduced order solutions to that of the original high dimensional problem provided the snapshot parameter sample being dense in the parameter set and with an appropriate parameter separability rule. Furthermore, we establish the absolute a posteriori error estimator for the reduced order solutions and the corresponding cost functional which deals with the norm of the residuals of the state and adjoint equations.