Local exact controllability to the trajectories for the two-dimensional magnetohydrodynamic system with controls acting only on the velocity field

Abstract

In this paper, we study the local exact controllability to the trajectories for the two-dimensional incompressible magnetohydrodynamic system on a bounded domain with no-slip boundary condition on the velocity field and the perfect insulating condition on the magnetic field. The controls are distributed in an arbitrarily small nonempty open subset and act only on the velocity field. In this situation, the divergence free condition for the magnetic field can be inherited from the initial value. With this condition, we transform the magnetohydrodynamic system into a coupled system between the Navier-Stokes equations and a scalar equation. Our proof relies on a new Carleman inequality for two kinds of boundary conditions.