A Closed-Form Full-State Feedback Controller for Stabilization of 3D Magnetohydrodynamic Channel Flow

We present a boundary feedback law that stabilizes the velocity, pressure, and electromagnetic fields in a magnetohydrodynamic (MHD) channel flow. The MHD channel flow, also known as Hartmann flow, is a benchmark for applications such as cooling, hypersonic flight, and propulsion. It involves 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 MHD approximation in a low magnetic Reynolds number regime. This model is unstable for large Reynolds numbers and is stabilized by actuation of velocity and the electric potential at only one of the walls. The backstepping method for stabilization of parabolic partial differential equations (PDEs) is applied to the velocity field system written in appropriate coordinates. Control gains are computed by solving a set of linear hyperbolic PDEs. Stabilization of nondiscretized 3D MHD channel flow has so far been an open problem.