Feedback Stabilization and Observer Design for Linearized Mhd Channel Flow at Low Magnetic Reynolds Number

We study a 2D and a 3D incompressible magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow, which is electrically conducting and subject to an external transverse magnetic field. We consider periodic boundary condition along one axis (two axis, for the three-dimensional case). We consider the flow at low magnetic Reynolds number and obtain the so-called simplified magnetohydrodynamic equations (SMHD). We stabilize the linearized SMHD system by a boundary finite-dimensional feedback controller with vertical velocity observation which acts only on the normal component of the velocity field. Also, we design a feedback observer for the SMHD equations which assures the convergence of the estimates of the velocity field. The observer consists of a copy of the linearized SMHD equations combined with an injection of output estimation error of the wall-normal velocity, on the uncontrolled wall.