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

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