Burgersʹ equation: Galerkin least-squares approximations and feedback control

The standard Galerkin finite-element approximation of Burgersʹ equation is known to be numerically unstable for small values of viscosity. One way to overcome this difficulty is to use a stabilized finite-element method, such as Galerkin least squares, and this is a standard approach taken in the simulation literature. In this paper, we investigate the effect of a stabilized finite-element approximation of Burgersʹ equation in the context of feedback control design. The issue at hand is how the additional stabilizing terms affect the resulting controller. We show that the controller designed for the stabilized system varies little from the unstabilized controller.