This note is concerned with an optimal control problem governed
by the relativistic Maxwell-Newton-Lorentz equations, which describes the mo-
tion of charges 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 intro-
ducing a scalar magnetic potential which leads to an additional state equation
in 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 form of Karush-Kuhn-
Tucker conditions are derived. A numerical test illustrates the theoretical
findings.