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Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations

Author:

Jean-Pierre RaymondAuthors Info & Claims

SIAM Journal on Control and Optimization, Volume 45, Issue 3

Pages 790 - 828

https://doi.org/10.1137/050628726

Published: 01 March 2006 Publication History

Abstract

We study the exponential stabilization of the linearized Navier--Stokes equations around an unstable stationary solution, by means of a feedback boundary control, in dimension 2 or 3. The feedback law is determined by solving a linear-quadratic control problem. We do not assume that the normal component of the control is equal to zero. In the nonzero case the state equation, satisfied by the velocity field y, is decoupled into an evolution equation, satisfied by Py, where P is the so-called Helmholtz projection operator, and a quasi-stationary elliptic equation, satisfied by (I - P)y. Using this decomposition, we show that the feedback law can be expressed as a function only of Py. In the two-dimensional case we show that the linear feedback law provides a local exponential stabilization of the Navier--Stokes equations.

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Djebour IRamdani KValein J(2024)Observer-Based Feedback-Control for the Stabilization of a Class of Parabolic SystemsJournal of Optimization Theory and Applications10.1007/s10957-024-02496-1202:3(1217-1241)Online publication date: 1-Sep-2024
https://dl.acm.org/doi/10.1007/s10957-024-02496-1
Hamadi MJbilou KRatnani A(2022)An efficient extended block Arnoldi algorithm for feedback stabilization of incompressible Navier-Stokes flow problemsApplied Numerical Mathematics10.1016/j.apnum.2022.01.011174:C(142-162)Online publication date: 1-Apr-2022
https://dl.acm.org/doi/10.1016/j.apnum.2022.01.011
Benner PHeiland JWerner S(2022)Robust output-feedback stabilization for incompressible flows using low-dimensional -controllersComputational Optimization and Applications10.1007/s10589-022-00359-x82:1(225-249)Online publication date: 1-May-2022
https://dl.acm.org/doi/10.1007/s10589-022-00359-x
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Index Terms

Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations
1. Computing methodologies
  1. Artificial intelligence
    1. Control methods
      1. Computational control theory
2. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
      2. Partial differential equations
    2. Numerical analysis

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cover image SIAM Journal on Control and Optimization

SIAM Journal on Control and Optimization Volume 45, Issue 3

2006

374 pages

ISSN:0363-0129

Issue’s Table of Contents

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 March 2006

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Djebour IRamdani KValein J(2024)Observer-Based Feedback-Control for the Stabilization of a Class of Parabolic SystemsJournal of Optimization Theory and Applications10.1007/s10957-024-02496-1202:3(1217-1241)Online publication date: 1-Sep-2024
https://dl.acm.org/doi/10.1007/s10957-024-02496-1
Hamadi MJbilou KRatnani A(2022)An efficient extended block Arnoldi algorithm for feedback stabilization of incompressible Navier-Stokes flow problemsApplied Numerical Mathematics10.1016/j.apnum.2022.01.011174:C(142-162)Online publication date: 1-Apr-2022
https://dl.acm.org/doi/10.1016/j.apnum.2022.01.011
Benner PHeiland JWerner S(2022)Robust output-feedback stabilization for incompressible flows using low-dimensional -controllersComputational Optimization and Applications10.1007/s10589-022-00359-x82:1(225-249)Online publication date: 1-May-2022
https://dl.acm.org/doi/10.1007/s10589-022-00359-x
Borggaard JGugercin SZietsman L(2021)Feedback stabilization of fluids using reduced-order models for control and compensator design2016 IEEE 55th Conference on Decision and Control (CDC)10.1109/CDC.2016.7799440(7579-7585)Online publication date: 10-Mar-2021
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Rodrigues S(2021)Feedback Boundary Stabilization to Trajectories for 3D Navier–Stokes EquationsApplied Mathematics and Optimization10.1007/s00245-017-9474-584:Suppl 2(1149-1186)Online publication date: 1-Dec-2021
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Benner PHeinkenschloss MSaak JWeichelt H(2020)Efficient solution of large-scale algebraic Riccati equations associated with index-2 DAEs via the inexact low-rank Newton-ADI methodApplied Numerical Mathematics10.1016/j.apnum.2019.11.016152:C(338-354)Online publication date: 1-Jun-2020
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Kundu SPani A(2020)Global Stabilization of Two Dimensional Viscous Burgers’ Equation by Nonlinear Neumann Boundary Feedback Control and Its Finite Element AnalysisJournal of Scientific Computing10.1007/s10915-020-01294-x84:3Online publication date: 24-Aug-2020
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Sène ANgom TNgom E(2019)Global Stabilization of the Navier-Stokes Equations Around an Unstable Steady State with Mixed Boundary Kinetic Energy ControllerJournal of Dynamical and Control Systems10.1007/s10883-018-9406-y25:2(197-218)Online publication date: 1-Apr-2019
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Benner PHeiland J(2017)Nonlinear feedback stabilization of incompressible flows via updated Riccati-based gains2017 IEEE 56th Annual Conference on Decision and Control (CDC)10.1109/CDC.2017.8263813(1163-1168)Online publication date: 12-Dec-2017
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