  In the paper "Optimal control of a Vlasov-Poisson plasma by an external
magnetic field - The basics for variational calculus" [arXiv:1708.02464] we
have already introduced a set of admissible magnetic fields and we have proved
that each of those fields induces a unique strong solution of the
Vlasov-Poisson system. We have also established that the field-state operator
that maps any admissible field onto its corresponding solution is continuous
and weakly compact. In this paper we will show that this operator is also
Fr\'echet differentiable and we will continue to analyze the optimal control
problem that was introduced in [arXiv:1708.02464]. More precisely, we will
establish necessary and sufficient conditions for local optimality and we will
show that an optimal solution is unique under certain conditions.
