  The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell
system which couples the Vlasov equation (the transport equation) with the
Maxwell equations of electrodynamics. We only consider a 'two-dimensional'
version of the problem since existence of global, classical solutions of the
full three-dimensional problem is not known. We add external currents to the
system, in applications generated by coils, to control the plasma in a proper
way. After considering global existence of solutions to this system,
differentiability of the control-to-state operator is proved. In applications,
on the one hand, we want the shape of the plasma to be close to some desired
shape. On the other hand, a cost term penalizing the external currents shall be
as small as possible. These two aims lead to minimizing some objective
function. We restrict ourselves to only such control currents that are
realizable in applications. After that, we prove existence of a minimizer and
deduce first order optimality conditions and the adjoint equation.
