Memoirs of the American Mathematical Society 2006; 128 pp; softcover Volume: 181 ISBN10: 0821838741 ISBN13: 9780821838747 List Price: US$66 Individual Members: US$39.60 Institutional Members: US$52.80 Order Code: MEMO/181/852
 The steadystate solutions to NavierStokes equations on a bounded domain \(\Omega \subset R^d\), \(d = 2,3\), are locally exponentially stabilizable by a boundary closedloop 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 nonlinearity 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 finitedimensional unstable subspace. In order to inject dissipation as to force local exponential stabilization of the steadystate solutions, an Optimal Control Problem (OCP) with a quadratic cost functional over an infinite timehorizon is introduced for the linearized NS equations. As a result, the same Riccatibased, optimal boundary feedback controller which is obtained in the linearized OCP is then selected and implemented also on the full NS 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 unboundednessbetween the unboundedness of the boundary control operator and the unboundedness of the penalization or observation operatoris strictly larger than \(\tfrac{3}{2}\), as expressed in terms of fractional powers of the freedynamics operator. In contrast, established (and rich) optimal control theory [LT.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 nonstandard OCPwith the aforementioned high level of unboundedness in control and observation operators and subject, moreover, to the additional constraint that the controllers be pointwise tangentialbe nonempty; that is, it satisfies the socalled Finite Cost Condition [LT.2]. Table of Contents  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): Wellposedness of the NavierStokes equations with Riccatibased boundary feedback control. Case \(d=3\)
 Theorem 2.3(ii): Local uniform stability of the NavierStokes equations with Riccatibased boundary feedback control
 A PDEinterpretation 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
