Tangential Boundary Stabilization of Navier-Stokes Equations. (=Memoirs of the American Mathematical Society; Vol. 852).

Barbu, Viorel, Irena Lasiecka and Roberto Triggiani:

ISBN 10: 0821838741 / ISBN 13: 9780821838747

Published by American Mathematical Society, 2006

About this Item

128 S.; Ill. Like new. Shrink wrapped. Sprache: Englisch Gewicht in Gramm: 370.

Seller Inventory # 829209

Bibliographic Details

• Title: Tangential Boundary Stabilization of Navier-Stokes Equations. (=Memoirs of the American Mathematical Society; Vol. 852).
• Publisher: American Mathematical Society
• Publication Date: 2006
• Language: English
• ISBN 10: 0821838741
• ISBN 13: 9780821838747
• Binding: Paperback
• Condition: Wie neu
• Weight: 370 grams
• Seller catalogs:
  • Medizin

About this title

Synopsis

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].