The present paper provides a solution in the affirmative to a recognized open problem in the theory of uniform stabilization of 3-dimensional Navier-Stokes equations in the vicinity of an unstable equilibrium solution, by means of a 'minimal' and 'least' invasive feedback strategy which consists of a control pair {v, u} [L-T.3]. Here v is a tangential boundary feedback control, acting on an arbitrary small part Γ of the boundary Γ; while u is a localized, interior feedback control, acting tangentially on an arbitrarily small subset ω of the interior supported by Γ. The ideal strategy of taking u = 0 on ω is not sufficient. A question left open in the literature was: Can such feedback control v of the pair {v, u} be asserted to be finite dimensional also in the dimension d = 3? We here give an affirmative answer to this question, thus establishing an optimal result. To achieve the desired finite dimensionality of the feedback tangential boundary control v, it is here then necessary to abandon the Hilbert-Sobolev functional setting of past literature and replace it with a critical Besov space setting. These spaces are 'close' to L 3 (Ω) for d = 3. This functional setting is significant. It is in line with recent critical well-posedness in the full space of the non-controlled N-S equations. A key feature of such Besov spaces with tight indices is that they do not recognize compatibility conditions. The proof is constructive and is "optimal" also regarding the "minimal" number of tangential boundary controllers needed. The new setting requires the solution of novel technical and conceptual issues. These include establishing maximal regularity in the required critical Besov setting for the overall closed-loop linearized problem with tangential feedback control applied on the boundary. This result is also a new contribution to the area of maximal regularity. It escapes perturbation theory. Finally, the minimal amount of tangential boundary action is linked to the issue of unique continuation of over-determined Oseen eigenproblems.
An abstract framework guaranteeing the local continuous differentiability of the value function associated with optimal stabilization problems subject to abstract semilinear parabolic equations subject to a norm constraint on the controls is established. It guarantees that the value function satisfies the associated Hamilton–Jacobi–Bellman equation in the classical sense. The applicability of the developed framework is demonstrated for specific semilinear parabolic equations.
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