Optimal control-based feedback stabilization of multi-field flow problems

Abstract

We discuss the numerical solution of the feedback stabilization problem for multi-field flow problems. Our approach is based on an analytical Riccati feedback concept derived by Raymond which allows to steer a perturbed flow back to its desired state, assumed to be a stationary, possibly unstable, flow profile. This concept, originally derived for incompressible flow fields described by the Navier-Stokes equations, uses a linear-quadratic regulator (LQR) approach for the linearized Navier-Stokes equations formulated on the space of divergence-free velocity fields. We extend this approach to a setting where the Navier-Stokes equations are coupled to a diffusion-convection equation describing the transport of a reactive species in a fluid. The stabilizing feedback control resulting from the LQR problem is obtained via solving the associated operator Riccati equation. We describe a numerical procedure to solve this Riccati equation numerically. This involves several technical difficulties on the algebraic level that we address in this report. We illustrate the performance of our method by a numerical example.