Local Exact Controllability for the Planar Compressible Magnetohydrodynamic Equations

Abstract

This paper is concerned with the boundary controllability for the planar compressible magnetohydrodynamic equations. For a constant target trajectory, we prove that the system is exactly controllable by the control functions acting on the whole boundary, given the small $H^2$ initial perturbation. The proof relies on the continuous dependence of the solution on the initial data and the Carleman inequality for the velocity and magnetic fields. A suitable space for the density is introduced in the fixed point argument to reduce the regularity of the initial data.