# Numerical Solution of the Feedback Control Problem for Navier-Stokes Equations

Peter Benner^{*}, Jens Saak and Heiko Weichelt

We discuss the feedback stabilization of flow problems described by the incompressible Navier-Stokes equations. In the last decade, a series of papers by Raymond and co-workers showed that for small perturbations, the deviation from a nominal flow, defined by a possibly unstable solution of the steady Navier-Stokes equations, can be steered to zero at an exponential convergence rate using an LQR problem for the velocity field projected onto a suitable space of divergence-free functions. We show how to solve this LQR problem numerically using the associated algebraic (operator) Riccati equation. The key idea is to avoid the explicit Helmholtz projection onto the divergence-free vector fields by employing a saddle point formulation discussed already by Heinkenschloss, Sorensen, and Sun (SIAM J. Sci. Comp. 30:1038-1063, 2008) in the context of balanced truncation model reduction. Also, a number of other issues such as initializing Newton's method for the algebraic Riccati equations, need to be solved to derive a working algorithm for the numerical solution of the flow stabilization problem. We will show how the computed feedback control using this approach effectively stabilizes unstable flows using as test examples von Karman vortex shedding and the coupled systems of a reactive substance transported by an incompressible fluid.

Mathematics Subject Classification: 76D55

Keywords: flow control; feedback stabilization; algebraic Riccati equation; Navier-Stokes equations

Minisymposion: On Optimal Feedback Control for Partial Differential Equations: Theory and Numerical Methods