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In this paper, we prove a controllability result for a fluid-structure interaction problem. In dimension two, a rigid structure moves into an incompressible fluid governed by Navier-Stokes equations. The control acts on a fixed subset of the fluid domain. We prove that, for small initial data, this system is null controllable, that is, for a given , the system can be driven at rest and the structure to its reference configuration at time . To show this result, we first consider a linearized system. Thanks to an observability inequality obtained from a Carleman inequality, we prove an optimal controllability result with a regular control. Next, with the help of Kakutani’s fixed point theorem and a regularity result, we pass to the nonlinear problem.
Keywords: controllability, fluid-solid interaction, Navier-Stokes equations, Carleman estimates
Boulakia, Muriel ; Osses, Axel 1
@article{COCV_2008__14_1_1_0,
author = {Boulakia, Muriel and Osses, Axel},
title = {Local null controllability of a two-dimensional fluid-structure interaction problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1--42},
publisher = {EDP-Sciences},
volume = {14},
number = {1},
year = {2008},
doi = {10.1051/cocv:2007031},
mrnumber = {2375750},
zbl = {1149.35068},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007031/}
}
TY - JOUR AU - Boulakia, Muriel AU - Osses, Axel TI - Local null controllability of a two-dimensional fluid-structure interaction problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 1 EP - 42 VL - 14 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007031/ DO - 10.1051/cocv:2007031 LA - en ID - COCV_2008__14_1_1_0 ER -
%0 Journal Article %A Boulakia, Muriel %A Osses, Axel %T Local null controllability of a two-dimensional fluid-structure interaction problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 1-42 %V 14 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007031/ %R 10.1051/cocv:2007031 %G en %F COCV_2008__14_1_1_0
Boulakia, Muriel; Osses, Axel. Local null controllability of a two-dimensional fluid-structure interaction problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 1-42. doi: 10.1051/cocv:2007031
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