The steady-state solutions to Navier-Stokes equations on a bounded domain $/Omega /subset R^d$, $d = 2,3$, are locally exponentially stabilizable by a boundary closed-loop feedback controller, acting tangentially on the boundary $/partial /Omega$, in the Dirichlet boundary conditions. The greatest challenge arises from a combination between the control as acting on the boundary and the dimensionality $d=3$. If $d=3$, the non-linearity imposes and dictates the requirement that stabilization must occur in the space $(H^{/tfrac{3}{2}+/epsilon}(/Omega))^3$, $/epsilon > 0$, a high topological level. A first implication thereof is that, due to compatibility conditions that now come into play, for $d=3$, the boundary feedback stabilizing controller must be infinite dimensional.Moreover, it generally acts on the entire boundary $/partial /Omega$. Instead, for $d=2$, where the topological level for stabilization is $(H^{/tfrac{3}{2}-/epsilon}(/Omega))^2$, the boundary feedback stabilizing controller can be chosen to act on an arbitrarily small portion of the boundary. Moreover, still for $d=2$, it may even be finite dimensional, and this occurs if the linearized operator is diagonalizable over its finite-dimensional unstable subspace. In order to inject dissipation as to force local exponential stabilization of the steady-state solutions, an Optimal Control Problem (OCP) with a quadratic cost functional over an infinite time-horizon is introduced for the linearized N-S equations.As a result, the same Riccati-based, optimal boundary feedback controller which is obtained in the linearized OCP is then selected and implemented also on the full N-S system. For $d=3$, the OCP falls definitely outside the boundaries of established optimal control theory for parabolic systems with boundary controls, in that the combined index of unboundedness - between the unboundedness of the boundary control operator and the unboundedness of the penalization or observation operator - is strictly larger than $/tfrac{3}{2}$, as expressed in terms of fractional powers of the free-dynamics operator.In contrast, established (and rich) optimal control theory [L-T.2 ] of boundary control parabolic problems and corresponding algebraic Riccati theory requires a combined index of unboundedness strictly less than 1. An additional preliminary serious difficulty to overcome lies at the outset of the program, in establishing that the present highly non-standard OCP - with the aforementioned high level of unboundedness in control and observation operators and subject, moreover, to the additional constraint that the controllers be pointwise tangential - be non-empty; that is, it satisfies the so-called Finite Cost Condition [L-T.2].
Introduction Main results Proof of Theorems 2.1 and 2.2 on the linearized system (2.4): $d=3$ Boundary feedback uniform stabilization of the linearized system (3.1.4) via an optimal control problem and corresponding Riccati theory. Case $d=3$ Theorem 2.3(i): Well-posedness of the Navier-Stokes equations with Riccati-based boundary feedback control. Case $d=3$ Theorem 2.3(ii): Local uniform stability of the Navier-Stokes equations with Riccati-based boundary feedback control A PDE-interpretation of the abstract results in Sections 5 and 6 Appendix A. Technical material complementing Section 3.1 Appendix B. Boundary feedback stabilization with arbitrarily small support of the linearized system (3.1.4a) at the $(H^{/tfrac{3}{2}-/epsilon}(/Omega))^d /cap H$-level, with I.C. $y^0/in (H^{/frac{1}{2}-/epsilon}(/Omega))^d /cap H$. Cases $d=2,3$. Theorem 2.5 for $d=2$ Appendix C. Equivalence between unstable and stable versions of the optimal control problem of Section 4 Appendix D. Proof that $FS(/cdot) /in /mathcal {L}(W;L^2(0,/infty;(L^2(/Gamma))^d)$ Bibliography.
