Optimal control of a Vlasov-Poisson plasma by an external magnetic field

TL;DR

This paper investigates the optimal control of a plasma modeled by the Vlasov-Poisson system using an external magnetic field, establishing mathematical properties and deriving conditions for optimality.

Contribution

It provides a rigorous mathematical framework for controlling plasmas via magnetic fields, including necessary and sufficient optimality conditions and uniqueness results.

Findings

Proved the Vlasov-Poisson system with magnetic control satisfies key variational properties.

Derived first-order necessary optimality conditions for the control problem.

Established second-order sufficient conditions and conditions for control uniqueness.

Abstract

The aim of various technical applications (for example fusion research) is to control a plasma by magnetic fields in a desired fashion. In our model the plasma is described by the Vlasov-Poisson system that is equipped with an external magnetic field. We will prove that this model satisfies some basic properties that are necessary for calculus of variations. After that, we will analyze an optimal control problem with a tracking type cost functional with respect to the following topics: Necessary conditions of first order for local optimality, derivation of an optimality system, sufficient conditions of second order for local optimality, uniqueness of the optimal control under certain conditions.