Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system

Mehdi Badra

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • Volume: 15, Issue: 4, page 934-968
  • ISSN: 1292-8119

Abstract

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We study the local exponential stabilization of the 2D and 3D Navier-Stokes equations in a bounded domain, around a given steady-state flow, by means of a boundary control. We look for a control so that the solution to the Navier-Stokes equations be a strong solution. In the 3D case, such solutions may exist if the Dirichlet control satisfies a compatibility condition with the initial condition. In order to determine a feedback law satisfying such a compatibility condition, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the boundary of the domain. We determine a linear feedback law by solving a linear quadratic control problem for the linearized extended system. We show that this feedback law also stabilizes the nonlinear extended system.

How to cite

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Badra, Mehdi. "Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system." ESAIM: Control, Optimisation and Calculus of Variations 15.4 (2009): 934-968. <http://eudml.org/doc/245958>.

@article{Badra2009,
abstract = {We study the local exponential stabilization of the 2D and 3D Navier-Stokes equations in a bounded domain, around a given steady-state flow, by means of a boundary control. We look for a control so that the solution to the Navier-Stokes equations be a strong solution. In the 3D case, such solutions may exist if the Dirichlet control satisfies a compatibility condition with the initial condition. In order to determine a feedback law satisfying such a compatibility condition, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the boundary of the domain. We determine a linear feedback law by solving a linear quadratic control problem for the linearized extended system. We show that this feedback law also stabilizes the nonlinear extended system.},
author = {Badra, Mehdi},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Navier-Stokes equation; feedback stabilization; Dirichlet control; Riccati equation},
language = {eng},
number = {4},
pages = {934-968},
publisher = {EDP-Sciences},
title = {Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system},
url = {http://eudml.org/doc/245958},
volume = {15},
year = {2009},
}

TY - JOUR
AU - Badra, Mehdi
TI - Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2009
PB - EDP-Sciences
VL - 15
IS - 4
SP - 934
EP - 968
AB - We study the local exponential stabilization of the 2D and 3D Navier-Stokes equations in a bounded domain, around a given steady-state flow, by means of a boundary control. We look for a control so that the solution to the Navier-Stokes equations be a strong solution. In the 3D case, such solutions may exist if the Dirichlet control satisfies a compatibility condition with the initial condition. In order to determine a feedback law satisfying such a compatibility condition, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the boundary of the domain. We determine a linear feedback law by solving a linear quadratic control problem for the linearized extended system. We show that this feedback law also stabilizes the nonlinear extended system.
LA - eng
KW - Navier-Stokes equation; feedback stabilization; Dirichlet control; Riccati equation
UR - http://eudml.org/doc/245958
ER -

References

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