Optimization with Partial Differential Equations

In this article a boundary feedback stabilization approach for incompressible Navier-Stokes flow 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) we end up with a differential-algebraic system (DAE) 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] used to analyze and stabilize the continuous problem. Using the discrete projection, a linear quadratic regulator approach can be applied to stabilize the (discrete) linearized flow field with respect to small perturbations from a stationary trajectory. We show that the discrete Leray projector is nothing else but the numerical projection method proposed by Heinkenschloss et al. [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 NewtonADI 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 Kármán vortex shedding” within a finite element based flow solver. †Research Group Applied Mathematics III (AMIII), Friedrich-Alexander-Universität ErlangenNürnberg, Cauerstr. 11, 91058 Erlangen, Germany ‡Research Group Mathematics in Industry and Technology (MiIT), Technische Universität Chemnitz, Reichenhainer Str. 39/41, 09126 Chemnitz, Germany §Research Group Computational Methods in Systems and Control Theory (CSC), Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Sandtorstr. 1, 39106 Magdeburg, Germany