Optimal Control of the Inhomogeneous Relativistic Maxwell Newton Lorentz Equations

TL;DR

This paper develops an optimal control framework for the relativistic Maxwell-Newton-Lorentz equations, incorporating a magnetic potential as control, proving existence, uniqueness, and deriving optimality conditions with numerical validation.

Contribution

It introduces a novel control approach for the coupled PDE-ODE system with additional state constraints via a scalar potential, and establishes theoretical properties and optimality conditions.

Findings

Existence and uniqueness of the state equation are proven.

Global optimal control existence is established.

Numerical tests confirm theoretical results.

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-Kuhn-Tucker conditions are derived. A numerical test illustrates the theoretical findings.