By Samuel Eilenberg

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V. Fursikov, Stabilization for the 3D Navier-Stokes system of feedback boundary control, Disc. Cont. Dynam. Systems 10 (182) (2004)m 289–314. 1] D. Henry, Perturbation of the Boundary in Boundary Value Problems, London Mathematical Society Lecture Notes 318, Cambridge University Press, 2005. 1] T. Kato, Perturbation Theory of Linear Operators, Springer-Verlag, New YorkBerlin, 1966. 1] I. Lasiecka, Boundary control of parabolic systems: Regularity of optimal solutions, Appl. Math. & Optimiz. 4 (1978), 301–327.

4). 7b). ] Accordingly, by standard, ﬁnite-dimensional control theory, the linear dynamics in (Re zN ) can be stabilized by a K = 2N -dimensional real-valued, closed-loop feedback control of the same feedback law as its complexvalued counterpart. Thus, henceforth in this section, we set K = 2N (K = N if all unstable eigenvalues λj , j = 1, . . 16) are real). Let w ˜j ≡ Re wj , for j = 1, . . 4). for j = 1, . . 8b) 36 V. Barbu, I. Lasiecka and R. 6]. 2. Let d = 2 and assume the FDSA. Let Γ1 be any portion of the boundary Γ = ∂Ω, meas Γ1 > 0.

FDSA: We assume that for each of the distinct unstable eigenvalues λ1 , . . 16), algebraic and geometric multiplicity coincide. Denote by the same symbol H the complexiﬁcation of the original space H. 1) onto u : H −→ ZN onto u ∗ ) : H −→ (ZN Stabilization Strategies of the Navier-Stokes Equations 25 where C (respect. C) is a simple, closed curve surrounding {λi }M respect. i=1 M {λi }i=1 . 1, p. 2) u s where each of the subspaces ZN and ZN is invariant under A. 3) u s for the restrictions of A to ZN and ZN , respectively.