  We consider the three dimensional Vlasov-Poisson system that is equipped with
an external magnetic field to describe a plasma. The aim of various concrete
applications is to control a plasma in a desired fashion. This can be modeled
by an optimal control problem. For that reason the basics for calculus of
variations will be introduced in this paper. We have to find a suitable class
of fields that are admissible for this procedure as they provide unique global
solutions of the Vlasov-Poisson system. Then we can define a field-state
operator that maps any admissible field onto its corresponding distribution
function. We will show that this field-state operator is Lipschitz continuous
and (weakly) compact. Last we will consider a model problem with a tracking
type cost functional and we will show that this optimal control problem has at
least one globally optimal solution.
