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7 - Feedback stabilization for the Navier–Stokes equations: theory and calculations

Published online by Cambridge University Press:  05 November 2012

By
A.V. Fursikov  and
A.A. Kornev
Edited by
James C. Robinson ,
José L. Rodrigo  and
Witold Sadowski
A.V. Fursikov
Affiliation:
Moscow State University
A.A. Kornev
Affiliation:
Moscow State University
James C. Robinson
Affiliation:
University of Warwick
José L. Rodrigo
Affiliation:
University of Warwick
Witold Sadowski
Affiliation:
Uniwersytet Warszawski, Poland
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Summary

Abstract This paper considers the feedback stabilization problem for the Navier-Stokes equations defined in a bounded domain. Control via a forcing term (both distributed and impulsive) supported in a subdomain, via the initial condition, and via boundary conditions are studied, and relationships between these different kinds of control are presented. The precise meaning of ‘feedback control’ is discussed, and a feedback map providing control via the state variable is constructed for initial and distributed control (in the latter case for the linear Oseen equations only). Numerical algorithms for the calculation of stable invariant manifolds and projection operators on these sets are discussed. Finally the results of a numerical stabilization of a particular fluid flow are presented.

Introduction

The aim of this paper is to give a relatively short presentation of mathematical and numerical results concerning stabilization of the Navier-Stokes equations by feedback control. A description of the mathematical stabilization construction will be accompanied by a discussion of how mathematical notions of stabilization theory can be adapted to calculations.

Control theory for partial differential equations has been developed very intensively over the last few decades, and is now a very wide and farreaching field, even if one excludes extremal theory for partial differential equations. For some idea of this field more broadly, see the recent books by Coron (2007) and Tucsnak & Weiss (2009), as well as the earlier survey in Fursikov & Imanuilov (1999), together with the references in these publications.

The stabilization problem for the 2D Navier-Stokes system by a feedback distributed control supported in the whole fluid domain was studied by Barbu & Sritharan (1998).

Type
Chapter
Information
Mathematical Aspects of Fluid Mechanics , pp. 130 - 172
Publisher: Cambridge University Press
Print publication year: 2012

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