Optimal Control of a Vlasov-Poisson Plasma by Fixed Magnetic Field Coils

We consider the Vlasov-Poisson system that is equipped with an external magnetic field to describe the time evolution of the distribution function of a plasma. An optimal control problem where the external magnetic field is the control itself has already been investigated by Knopf (Calc Var 57:134, 2018). However, in real technical applications it will not be possible to choose the control field in such a general fashion as it will be induced by fixed field coils. In this paper we will use the fundamentals that were established by Knopf (Calc Var 57:134, 2018) to analyze an optimal control problem where the magnetic field is a superposition of the fields that are generated by N fixed magnetic field coils. Thereby, the aim is to control the plasma in such a way that its distribution function matches a desired distribution function at some certain final time T as closely as possible. This problem will be analyzed with respect to the following topics: existence of a globally optimal solution, necessary conditions of first order for local optimality, derivation of an optimality system, sufficient conditions of second order for local optimality and uniqueness of the optimal control under certain conditions.