Reduced order feedback synthesis for viscous incompressible flows

We discuss the application of the reduced basis method for the simulation and control of unsteady viscous flows governed by the incompressible Navier-Stokes equations. We describe how to use this method in terms of the construction of a lower-order compensator design. Our approach includes a construction method of the optimal state feedback law for finite-dimensional nonlinear regulator problems. The method is applied to construct a feedback law for the reduced order control model of the Navier-Stokes equations, and then we apply our feedback law to the original control system. Our method is demonstrated on a control problem formulated in a channel flow using a boundary velocity control. We also show how these ideas can be extended to control problems governed by partial differential equations. Numerical results are reported for the open and closed loop controls, and a compensator design is proposed to complete the closed loop dynamics.