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

Abstract

Abstract. This note is concerned with an optimal control problem governed by the relativistic Maxwell-Newton-Lorentz equations, which describes the motion 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 introducing 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-KuhnTucker conditions are derived. A numerical test illustrates the theoretical findings.

A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system

“…Let us have a look at related work: There are quite a few contributions available regarding the optimal control of coupled PDE-ODE systems, cf. [17,1,21,16,12,4] and the references therein. The authors mainly focus on the analysis of their specific model and the derivation of first order necessary optimality conditions and provide tailored algorithms for the numerical solution of the optimal control problems.…”