2017/08/08 by Patrik Knopf, Knopf, Patrik · 3 citations
Mathematics · Physics and Astronomy · #35Q83 #49J20 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Optimization and Control (math.OC) #math-ph #math.AP #math.MP #math.OC #msc:35Q83 #msc:49J20
paper · pdf · doi:10.48550/arxiv.1708.02464
arxiv created 2017/08/08 · arxiv updated 2017/10/27
We consider the three dimensional Vlasov-Poisson 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 Vlasov-Poisson system. Then we can define a field-state operator that maps any admissible field onto its corresponding distribution function. We will show that this field-state 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.