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We consider the feedback stabilization of a simplified 1d model for a fluid–structure interaction system. The fluid equation is the viscous Burgers equation whereas the motion of the particle is given by the Newton's laws. We stabilize this system around a stationary state by using feedbacks located at the exterior boundary of the fluid domain. With one input, we obtain a local stabilizability of the system with an exponential decay rate of order . An arbitrary order for the exponential decay rate can be proved if a unique continuation result holds true or if two inputs are used to stabilize the system. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domains of the stationary state and of the stabilized solution are different.
Keywords: Feedback stabilization, Fluid–structure interaction, Viscous Burgers equation
@article{AIHPC_2014__31_2_369_0,
author = {Badra, Mehdi and Takahashi, Tak\'eo},
title = {Feedback stabilization of a simplified 1d fluid{\textendash}particle system},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {369--389},
publisher = {Elsevier},
volume = {31},
number = {2},
year = {2014},
doi = {10.1016/j.anihpc.2013.03.009},
mrnumber = {3181675},
zbl = {1302.74057},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2013.03.009/}
}
TY - JOUR AU - Badra, Mehdi AU - Takahashi, Takéo TI - Feedback stabilization of a simplified 1d fluid–particle system JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 369 EP - 389 VL - 31 IS - 2 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2013.03.009/ DO - 10.1016/j.anihpc.2013.03.009 LA - en ID - AIHPC_2014__31_2_369_0 ER -
%0 Journal Article %A Badra, Mehdi %A Takahashi, Takéo %T Feedback stabilization of a simplified 1d fluid–particle system %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 369-389 %V 31 %N 2 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2013.03.009/ %R 10.1016/j.anihpc.2013.03.009 %G en %F AIHPC_2014__31_2_369_0
Badra, Mehdi; Takahashi, Takéo. Feedback stabilization of a simplified 1d fluid–particle system. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 369-389. doi: 10.1016/j.anihpc.2013.03.009
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