Optimal control of a Vlasov-Poisson plasma by an external magnetic field

The aim of various technical applications (for example fusion research) is to control a plasma by magnetic fields in a desired fashion. In our model the plasma is described by the Vlasov-Poisson system that is equipped with an external magnetic field. We will prove that this model satisfies some basic properties that are necessary for calculus of variations. After that, we will analyze an optimal control problem with a tracking type cost functional with respect to the following topics: Necessary conditions of first order for local optimality, derivation of an optimality system, sufficient conditions of second order for local optimality, uniqueness of the optimal control under certain conditions.