On the controllability of the relativistic Vlasov-Maxwell system

TL;DR

This paper investigates the controllability of the 2D relativistic Vlasov-Maxwell system on a torus, demonstrating local exact controllability under geometric conditions and controllability of the distribution function under specific assumptions.

Contribution

It provides new controllability results for the relativistic Vlasov-Maxwell system, including cases with and without geometric control conditions, using novel approximation and asymptotic methods.

Findings

Proves local exact controllability with geometric control condition.

Establishes controllability of the distribution function without geometric control, under certain conditions.

Utilizes the return method and asymptotic analysis for large speed of light.

Abstract

In this paper, we study the controllability of the two-dimensional relativistic Vlasov-Maxwell system in a torus, by means of an interior control. We give two types of results. With the geometric control condition on the control set, we prove the local exact controllability of the system in large time. Our proof in this case is based on the return method, on some results on the control of the Maxwell equations, and on a suitable approximation scheme to solve the non-linear Vlasov-Maxwell system on the torus with an absorption procedure. Without geometric control condition, but assuming that a strip of the torus is contained in the control set and under certain additional conditions on the initial data, we establish a controllability result on the distribution function only, also in large time. Here, we need some additional arguments based on the asymptotics of the Vlasov-Maxwell system with large speed of light and on our previous results concerning the controllability of the Vlasov-Poisson system with an external magnetic field [14].