Optimal control of a Vlasov-Poisson plasma by an external magnetic field - Analysis of a tracking type optimal control problem

TL;DR

This paper analyzes the mathematical properties of an optimal control problem for a Vlasov-Poisson plasma using an external magnetic field, establishing differentiability and optimality conditions for the control-to-state operator.

Contribution

It proves the Fréchet differentiability of the field-state operator and derives necessary and sufficient conditions for local optimality in the control problem.

Findings

The field-state operator is Fréchet differentiable.

Necessary and sufficient optimality conditions are established.

Under certain conditions, the optimal solution is unique.

Abstract

In the paper "Optimal control of a Vlasov-Poisson plasma by an external magnetic field - The basics for variational calculus" [arXiv:1708.02464] we have already introduced a set of admissible magnetic fields and we have proved that each of those fields induces a unique strong solution of the Vlasov-Poisson system. We have also established that the field-state operator that maps any admissible field onto its corresponding solution is continuous and weakly compact. In this paper we will show that this operator is also Fr\'echet differentiable and we will continue to analyze the optimal control problem that was introduced in [arXiv:1708.02464]. More precisely, we will establish necessary and sufficient conditions for local optimality and we will show that an optimal solution is unique under certain conditions.