Normal Feedback Boundary Stabilization to Trajectories for 3d Navier–stokes Equations

Given a nonstationary trajectory of the Navier–Stokes system, a finitedimensional feedback boundary controller stabilizing locally the system to the given solution is constructed. Moreover the controller is supported in a given open subset of the boundary of the domain containing the fluid and acts normal to the boundary. In a first step a controller is constructed that stabilizes the linear Oseen–Stokes system “around the given trajectory” to zero; for that a corollary of a suitable truncated boundary observability inequality, the regularizing property for the system, and some standard techniques of the optimal control theory are used. Then it is shown that the same controller also stabilizes, locally, the Navier–Stokes system to the given trajectory.