Optimal Control of the Two-Dimensional Vlasov-Maxwell System

TL;DR

This paper develops an optimal control framework for the two-dimensional Vlasov-Maxwell system, establishing existence, differentiability, and optimality conditions for controlling plasma shapes with external currents.

Contribution

It introduces a novel control approach for the 2D Vlasov-Maxwell system, including existence proofs, differentiability of the control-to-state map, and derivation of optimality conditions.

Findings

Proved global existence of solutions with control inputs.

Established differentiability of the control-to-state operator.

Derived first-order optimality conditions and the adjoint equation.

Abstract

The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We only consider a 'two-dimensional' version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma in a proper way. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applications, on the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, a cost term penalizing the external currents shall be as small as possible. These two aims lead to minimizing some objective function. We restrict ourselves to only such control currents that are realizable in applications. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation.