  The time evolution of a collisionless plasma is modeled by the relativistic
Vlasov-Maxwell system which couples the Vlasov equation (the transport
equation) with the Maxwell equations of electrodynamics. We consider the case
that the plasma is located in a bounded container $\Omega\subset\mathbb R^3$,
for example a fusion reactor. Furthermore, there are external currents,
typically in the exterior of the container, that may serve as a control of the
plasma if adjusted suitably. We model objects, that are placed in space, via
given matrix-valued functions $\varepsilon$ (the permittivity) and $\mu$ (the
permeability). A typical aim in fusion plasma physics is to keep the amount of
particles hitting $\partial\Omega$ as small as possible (since they damage the
reactor wall), while the control costs should not be too exhaustive (to ensure
efficiency). This leads to a minimizing problem with a PDE constraint. This
problem is analyzed in detail. In particular, we prove existence of minimizers
and establish an approach to derive first order optimality conditions.
