We show the local null-controllability of a fluid-structure interaction system coupling a viscous incompressible fluid with a damped beam located on a part of its boundary. The controls act on arbitrary small parts of the fluid domain and of the beam domain. In order to show the result, we first use a change of variables and a linearization to reduce the problem to the null-controllability of a Stokes-beam system in a cylindrical domain. We obtain this property by combining Carleman inequalities for the heat equation, for the damped beam equation and for the Laplace equation with high-frequency estimates. Then, the result on the nonlinear system is obtained by a fixed-point argument.
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Keywords: Null controllability, Navier–Stokes systems, Carleman estimates, fluid-structure interaction systems
Rémi Buffe 1; Takéo Takahashi 1
@article{CRMATH_2023__361_G9_1541_0, author = {R\'emi Buffe and Tak\'eo Takahashi}, title = {Controllability of a fluid-structure interaction system coupling the {Navier{\textendash}Stokes} system and a damped beam equation}, journal = {Comptes Rendus. Math\'ematique}, pages = {1541--1576}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.509}, language = {en}, }
TY - JOUR AU - Rémi Buffe AU - Takéo Takahashi TI - Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation JO - Comptes Rendus. Mathématique PY - 2023 SP - 1541 EP - 1576 VL - 361 PB - Académie des sciences, Paris DO - 10.5802/crmath.509 LA - en ID - CRMATH_2023__361_G9_1541_0 ER -
%0 Journal Article %A Rémi Buffe %A Takéo Takahashi %T Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation %J Comptes Rendus. Mathématique %D 2023 %P 1541-1576 %V 361 %I Académie des sciences, Paris %R 10.5802/crmath.509 %G en %F CRMATH_2023__361_G9_1541_0
Rémi Buffe; Takéo Takahashi. Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1541-1576. doi : 10.5802/crmath.509. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.509/
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