  The Vlasov-Poisson system describes the time evolution of a plasma in the
so-called collisionless regime. The investigation of a high-temperature plasma
that is influenced by an exterior magnetic field is one of the most significant
aspects of thermonuclear fusion research. In this paper, we formulate and
analyze a kinetic optimal control problem for the Vlasov-Poisson system where
the control is represented by an external magnetic field. The main goal of such
optimal control problems is to confine the plasma in a certain region in phase
space. We first investigate the optimal control problem in terms of
mathematical analysis, i.e., we show the existence of at least one global
minimizer and we rigorously derive a first-order necessary optimality condition
for local minimizers by the adjoint approach. Then, we build a Monte Carlo
framework to solve the state equations as well as the adjoint equations by
means of a Particle-In-Cell method, and we apply a nonlinear conjugate gradient
method to solve the optimization problem. Eventually, we present numerical
experiments that successfully validate our optimization framework.
