OPTIMAL CONTROL OF THE INHOMOGENEOUS RELATIVISTIC MAXWELL-NEWTON-LORENTZ EQUATIONS

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 electromagnetic 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.