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.
Keywords: Navier-Stokes equation, feedback stabilization, Dirichlet control, Riccati equation
@article{COCV_2009__15_4_934_0,
author = {Badra, Mehdi},
title = {Feedback stabilization of the {2-D} and {3-D} {Navier-Stokes} equations based on an extended system},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {934--968},
year = {2009},
publisher = {EDP-Sciences},
volume = {15},
number = {4},
doi = {10.1051/cocv:2008059},
mrnumber = {2567253},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2008059/}
}
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 SP - 934 EP - 968 VL - 15 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv:2008059/ DO - 10.1051/cocv:2008059 LA - en ID - COCV_2009__15_4_934_0 ER -
%0 Journal Article %A Badra, Mehdi %T Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 934-968 %V 15 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv:2008059/ %R 10.1051/cocv:2008059 %G en %F COCV_2009__15_4_934_0
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, Tome 15 (2009) no. 4, pp. 934-968. doi: 10.1051/cocv:2008059
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