<p style='text-indent:20px;'>We consider an optimal control problem governed by parameterized stationary Maxwell's system with the Gauss's law. The parameters enter through dielectric, magnetic permeability, and charge density. Moreover, the parameter set is assumed to be compact. We discretize the electric field by a finite element method and use variational discretization concept for the control. We present a reduced basis method for the optimal control problem and establish the uniform convergence of the reduced order solutions to that of the original full-dimensional problem provided that the snapshot parameter sample is dense in the parameter set, with an appropriate parameter separability rule. Finally, we establish the absolute a posteriori error estimator for the reduced order solutions and the corresponding cost functions in terms of the state and adjoint residuals.</p>

In this paper we analyze the Maxwell system { ∇ × ( σ − 1 ( x ) ∇ × E ( x ) ) a m p ; =   ϵ ( x ) u ( x ) , x ∈ Ω ∇ ⋅ ( ϵ ( x ) E ( x ) ) a m p ; =   ρ ( x ) , x x x i x ∈ Ω E ( x ) × n → ( x ) a m p ; =   0 , x x x x x x x ∈ ∂ Ω ∇ ⋅ ( ϵ ( x ) u ( x ) ) a m p ; =   0 , x x x x x x x ∈ Ω \begin{equation*} \begin {cases} \nabla \times \left (\sigma ^{-1}(\mathbf {x} )\nabla \times \mathbf {E}(\mathbf {x} )\right )&=~ \epsilon (\mathbf {x} ) \mathbf {u}(\mathbf {x}), \quad \mathbf {x} \in \Omega \\ \nabla \cdot (\epsilon (\mathbf {x} ) \mathbf {E}(\mathbf {x} )) &=~ \rho (\mathbf {x} ), \quad \phantom {xxxi} \mathbf {x} \in \Omega \\ \mathbf {E}(\mathbf {x} )\times \vec {\mathbf {n}}(\mathbf {x}) &=~ \mathbf {0}, \quad \phantom {xxxxxx} \mathbf {x} \in \partial \Omega \\ \nabla \cdot \left ( \epsilon (\mathbf {x} ) \mathbf {u}(\mathbf {x})\right ) &=~0, \quad \phantom {xxxxxx}\mathbf {x} \in \Omega \end{cases} \end{equation*} with the Nédélec finite elements. We show the convergence of the finite element approximations as well as establish their error bounds.