Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations

We study the exponential stabilization of the linearized Navier-Stokes equations around an unstable stationary solution, by means of a feedback boundary control, in dimension 2 or 3. The feedback law is determined by solving a Linear-Quadratic control problem. We do not assume that the normal component of the control is equal to zero. In that case the state equation, satisfied by the velocity field y, is decoupled into an evolution equation satisfied by Py, where P is the so-called Helmholtz projection operator, and a quasi-stationary elliptic equation satisfied by (I − P )y. Using this decomposition we show that the feedback law can be expressed only in function of Py. In the two dimensional case we show that the linear feedback law provides a local exponential stabilization of the Navier-Stokes equations.