Exact Boundary Controllability for the Ideal Magneto-hydrodynamic Equations

TL;DR

This paper investigates the exact boundary controllability of the ideal magneto-hydrodynamic equations in a rectangular domain, establishing conditions for null controllability and methods for state manipulation using boundary controls.

Contribution

It provides a necessary and sufficient condition for null controllability of the MHD system and demonstrates how to steer states using boundary controls or external magnetic forces.

Findings

Identifies a precise condition for null controllability.

Shows states can be manipulated to each other under this condition.

Provides a method to move between states with external magnetic forces.

Abstract

We address the problem of controllability of the MHD system in a rectangular domain with a control prescribed on the side boundary. We identify a necessary and sufficient condition on the data to be null controllable, i.e., can be driven to the zero state. We also show that the validity of this condition allows the states to be stirred to each other. If the condition is not satisfied, one can move from one state to another with the help of a simple shear external magnetic force.