Exponential decay of the linear Maxwell system due to conductivity near the boundary

We study the anisotropic linear Maxwell system on a bounded domain $\mathrm{\Omega }$ with perfectly conducting boundary conditions. It is damped via a conductivity $\sigma$ which is strictly positive on a collar at the boundary. We prove that solutions decay exponentially to 0, if the fields have no magnetic charges on $\mathrm{\Omega }$ and no electric charges off the support of $\sigma$. Our approach relies on a splitting of the solution via a Helmholtz decomposition and an observability-type estimate for a related second-order system without charges, shown using Morawetz multipliers. Corresponding exact observability and controllability results are also established.