Tangential boundary stabilization of Navier-Stokes equations

by Barbu, Viorel

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Title Statements

Full title :Tangential boundary stabilization of Navier-Stokes equations / Viorel Barbu, Irena Lasiecka, Roberto Triggiani.

Summary

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].

Edition Statements

Publication Date:©2006.

Publication Statements

Publisher(s):American mathematical Society
Place of Publication:Providence, R.I
Language:eng
Publication Date:©2006.

Notes

"Volume 181, number 852 (first of 5 numbers)."

The Physical Object

Description:ix, 128 pages ; : 26 cm.

Edition Identifiers

ISBN:0821838741, 9780821838747
oclc:ocm63108468
LC Classification:QA3
Dewey Decimal:510 s

1. Introduction -- 2. Main results -- 3. Proof of Theorems 2.1 and 2.2 on the linearized system (2.4): d=3 -- 4. Boundary feedback uniform stabilization of the linearized system (3.1.4) via an optimal control problem and corresponding Riccati theory. Case d=3 -- 5. Theorem 2.3(i): Well-posedness of the Navier-Stokes equations with Riccati-based boundary feedback control. Case d=3 -- 6. Theorem 2.3(ii): Local uniform stability of the Navier-Stokes equations with Riccati-based boundary feedback control -- 7. A PDE-interpretation of the abstract results in Sections 5 and 6

Subjects

31.45 partial differential equations Analise matematica Boundary layer Couche limite Equações de navier-stokes Equações diferenciais parciais Équation de Riccati Équations de Navier-Stokes Grenslagen Grenzschicht Mathematical optimization Navier-Stokes equations Navier-Stokes-Gleichung Navier-Stokes-vergelijkingen Optimaliseren Optimierung Optimisation mathématique RICCATI EQUATION Riccati-Differentialgleichung Vergelijking van Riccati