Global exact controllability of ideal incompressible magnetohydrodynamic flows through a planar duct

TL;DR

This paper proves that ideal incompressible magnetohydrodynamic flows in a rectangular domain can be exactly controlled from the boundary using the return method, overcoming nonlinear coupling challenges with innovative constructions.

Contribution

It establishes the first global exact boundary controllability results for ideal MHD flows in a planar duct, employing the return method and novel analytical techniques.

Findings

Boundary controllability for Els"asser system proved

Results extended to original MHD system with pressure correction

Overcomes nonlinear coupling challenges with new constructions

Abstract

This article is concerned with the global exact controllability for ideal incompressible magnetohydrodynamics in a rectangular domain where the controls are situated in both vertical walls. First, global exact controllability via boundary controls is established for a related Els\"asser type system by applying the return method, introduced in [Coron J.M., Math. Control Signals Systems, 5(3) (1992) 295--312]. Similar results are then inferred for the original magnetohydrodynamics system with the help of a special pressure-like corrector in the induction equation. Overall, the main difficulties stem from the nonlinear coupling between the fluid velocity and the magnetic field in combination with the aim of exactly controlling the system. In order to overcome some of the obstacles, we introduce ad-hoc constructions, such as suitable initial data extensions outside of the physical part of the domain and a certain weighted space.