RICCATI-BASED BOUNDARY FEEDBACK STABILIZATION OF INCOMPRESSIBLE NAVIER-STOKES FLOWS

In this article a boundary feedback stabilization approach for incompressible Navier-Stokes flows is studied. One of the main difficulties encountered is the fact that after space discretization by a mixed finite element method (because of the solenoidal condition) one ends up with a differential algebraic system of index 2. The remedy here is to use a discrete realization of the Leray projection used by Raymond [J.-P. Raymond, SIAM J. Control Optim., 45 (2006), pp. 790-828] to analyze and stabilize the continuous problem. Using the discrete projection, a linear quadratic regulator (LQR) approach can be applied to stabilize the (discrete) linearized flow field with respect to small perturbations from a stationary trajectory. We provide a novel argument that the discrete Leray projector is nothing else but the numerical projection method proposed by Heinkenschloss and colleagues in [M. Heinkenschloss, D. C. Sorensen, and K. Sun, SIAM J. Sci. Comput., 30 (2008), pp. 1038-1063]. The nested iteration resulting from applying this approach within the Newton-ADI method to solve the LQR algebraic Riccati equation is the key to compute a feedback matrix that in turn can be applied within a closed-loop simulation. Numerical examples for various parameters influencing the different levels of the nested iteration are given. Finally, the stabilizing property of the computed feedback matrix is demonstrated using the von Karman vortex street within a finite element based flow solver.