A closed-form feedback controller for stabilization of magnetohydrodynamic channel flow

Abstract

We present a PDE boundary controller that stabilizes the velocity, pressure, and electromagnetic fields in a magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow, a benchmark model for applications such as cooling systems, hypersonic flight and propulsion. This flow is characterized by an electrically conducting fluid moving between parallel plates in the presence of an externally imposed transverse magnetic field. The system is described by the inductionless MHD equations, a combination of the Navier-Stokes equations and a Poisson equation for the electric potential under the so-called MHD approximation in a low magnetic Reynolds number regime, and is unstable for large Reynolds numbers. Our control design needs actuation of velocity and the electric potential at only one of the walls. The backstepping method for stabilization of parabolic PDEs is applied to the velocity field system written in some appropriate coordinates; this system is very similar to the Orr-Sommerfeld-Squire system of PDE's and presents the same difficulties. Thus we use actuation not only to guarantee stability but also to decouple the system in order to prevent transients. Control gains are computed solving linear hyperbolic PDEs - a much simpler task than, for instance, solving nonlinear Riccati equations. Stabilization of non-discretized 3-D MHD channel flow has so far been an open problem.