Optimal control of the two-dimensional Vlasov-Maxwell system

The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples Vlasov equation (the transport equation) with Maxwell equations electrodynamics. We only consider 'two-dimensional' version problem since existence global, classical solutions full three-dimensional not known. add external currents to system, in applications generated coils, control proper way. After considering global this differentiability control-to-state operator proved. In applications, on one hand, we want shape be close some desired shape. On other cost term penalizing shall as small possible. These two aims lead minimizing objective function. restrict ourselves such that are realizable applications. that, prove minimizer and deduce first order optimality conditions adjoint equation.