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Tangential Boundary Stabilization of Navier-Stokes Equations
Viorel Barbu and Irena Lasiecka, University of Virginia, Charlottesville, VA, and Roberto Triggiani
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Memoirs of the American Mathematical Society
2006; 128 pp; softcover
Volume: 181
ISBN-10: 0-8218-3874-1
ISBN-13: 978-0-8218-3874-7
List Price: US$62 Individual Members: US$37.20
Institutional Members: US\$49.60
Order Code: MEMO/181/852

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