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

¹ School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R. China.
² School of Mathematical Sciences, Center for Applied Mathematics, MOE-LSC, CMA-Shanghai and SHL-MAC, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China.

^* Corresponding author: manuel.rissel@sjtu.edu.cn

Received: 13 May 2021
Accepted: 2 November 2021

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 Elsässer type system by applying the return method, introduced in Coron [Math. Control Signals Syst. 5 (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.