We discuss the Maxwell system with nonlinear instantaneous material laws and a strictly positive conductivity in the domain. The coefficients are matrix-valued. For small initial data we can show that the solution exponentially decays to 0 in H^3. We use higher order energy bounds and observability-type estimates both with error terms arising from the quasilinearity. A detailed regularity analysis is needed to control these error terms. This is joint work with Irena Lasiecka (Memphis) and Michael Pokojovy (El Paso).