Optimal Control of the Inhomogeneous Relativistic Maxwell--Newton--Lorentz Equations

Abstract

This note is concerned with an optimal control problem governed by the relativistic Maxwell--Newton--Lorentz equations, which describe the motion of charged particles in electro-magnetic fields and consists of a hyperbolic PDE system coupled with a nonlinear ODE. An external magnetic field acts as control variable. Additional control constraints are incorporated by introducing a scalar magnetic potential which leads to an additional state equation in the form of a very weak elliptic PDE. Existence and uniqueness for the state equation is shown and the existence of a global optimal control is established. Moreover, first-order necessary optimality conditions in the form of Karush--Kuhn--Tucker conditions are derived. A numerical test illustrates the theoretical findings.