Normal feedback stabilization for linearized periodic MHD channel flow, at low magnetic Reynolds number

We study 2D (and 3D) incompressible magnetohydrodynamic (MHD) channel flow, also known as Hartmann’s flow, which is electrically conducting and subject to an external transverse magnetic field. We consider periodic boundary conditions along one axis (two axes, for the three-dimensional case). We consider the flow at low magnetic Reynolds number and obtain the so-called simplified magnetohydrodynamic (SMHD) equations. We stabilize the linearized SMHD system by using a boundary finite-dimensional feedback controller with vertical velocity observation which acts only on the normal component of the velocity field. The stability is achieved without any a priori condition on the viscosity coefficient, that is, on the Reynolds number.