Stabilization of Navier--Stokes Equations by Oblique Boundary Feedback Controllers

One designs a linear stabilizing boundary feedback controller u for the equilibrium solutions to Navier--Stokes system on a bounded and open domain Oᵈ, d=2, u=η1\!\!\!\;_Γ∑Nⱼ₌₁μⱼ<y,φ^*ⱼ> (φⱼ (x)+α(x) n(x)), where Γ is a connected component of the boundary and 1\!\!\!\;_Γ is the characteristic function of Γ. Here, φⱼ are related to the eigenfunction system \φ^*ⱼ\ for the adjoint Stokes--Oseen system, n is the normal to , <·,·> is the scalar product in(L²(O))ᵈ, and α is an arbitrary C²-function with circulation zero on . For $d=2,3$ this controller stabilizes the corresponding linearized (Oseen--Stokes) system.