Optimal control of a VlasovPoisson plasma by an external magnetic field  The basics for variational calculus
Abstract
We consider the three dimensional VlasovPoisson system that is equipped with an external magnetic field to describe a plasma. The aim of various concrete applications is to control a plasma in a desired fashion. This can be modeled by an optimal control problem. For that reason the basics for calculus of variations will be introduced in this paper. We have to find a suitable class of fields that are admissible for this procedure as they provide unique global solutions of the VlasovPoisson system. Then we can define a fieldstate operator that maps any admissible field onto its corresponding distribution function. We will show that this fieldstate operator is Lipschitz continuous and (weakly) compact. Last we will consider a model problem with a tracking type cost functional and we will show that this optimal control problem has at least one globally optimal solution.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 DOI:
 10.48550/arXiv.1708.02464
 arXiv:
 arXiv:1708.02464
 Bibcode:
 2017arXiv170802464K
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Mathematics  Optimization and Control;
 49J20;
 35Q83