Optimal control and Riccati equation for a degenerate parabolic system

In this paper, we consider a stabilization problem for a fluid flow. For a perturbation of the velocity of an incoming flow on a flat plate, the laminar-to-turbulent transition location varies. We want to stabilize it by a suction velocity trough the plate. The linearization of the nonlinear model around a steady state solution leads to a linear degenerate parabolic equation. We look for a suction velocity in a feedback form, determined by solving a LQR problem with an infinite time horizon. We derive the associated optimality system and the optimal control. The study of the Riccati equation is difficult because the state equation is a degenerate parabolic equation for which the results in the literature cannot be directly applied. The existence of solution is established by studying the asymptotic behaviour of the minimal solution to a Differential Riccati Equation. Numerical tests show that the feedback law stabilizes the laminar-to-turbulent transition location of the flow.