Controlling a Vlasov–Poisson Plasma by a Particle-in-Cell Method Based on a Monte Carlo Framework

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 to 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 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.