Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation

Rémi Buffe; Takéo Takahashi

doi:10.5802/crmath.509

Théorie du contrôle

Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation

Rémi Buffe ¹ ; Takéo Takahashi ¹

¹ Université de Lorraine, CNRS, Inria, IECL, 54000 Nancy, France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1541-1576.

Résumé

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.

Reçu le : 2022-11-29
Révisé le : 2023-04-20
Accepté le : 2023-04-21
Publié le : 2023-11-10

DOI : 10.5802/crmath.509

Classification : 76D05, 35Q30, 74F10, 76D55, 76D27, 93B05, 93B07, 93C10
Mots clés : Null controllability, Navier–Stokes systems, Carleman estimates, fluid-structure interaction systems

Affiliations des auteurs :

Rémi Buffe ¹ ; Takéo Takahashi ¹

¹ Université de Lorraine, CNRS, Inria, IECL, 54000 Nancy, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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/

Bibliographie
Cité par

[1] Mehdi Badra; Takéo Takahashi Feedback boundary stabilization of 2D fluid-structure interaction systems, Discrete Contin. Dyn. Syst., Volume 37 (2017) no. 5, pp. 2315-2373 | DOI | MR | Zbl

[2] Mehdi Badra; Takéo Takahashi Gevrey regularity for a system coupling the Navier–Stokes system with a beam equation, SIAM J. Math. Anal., Volume 51 (2019) no. 6, pp. 4776-4814 | DOI | MR | Zbl

[3] Mehdi Badra; Takéo Takahashi Gevrey regularity for a system coupling the Navier–Stokes system with a beam: the non-flat case, Funkc. Ekvacioj, Volume 65 (2022) no. 1, pp. 63-109 | DOI | MR | Zbl

[4] Mehdi Badra; Takéo Takahashi Maximal regularity for the Stokes system coupled with a wave equation: application to the system of interaction between a viscous incompressible fluid and an elastic wall, J. Evol. Equ., Volume 22 (2022) no. 3, 71, 55 pages | DOI | MR | Zbl

[5] Hugo Beirão da Veiga On the existence of strong solutions to a coupled fluid-structure evolution problem, J. Math. Fluid Mech., Volume 6 (2004) no. 1, pp. 21-52 | DOI | MR

[6] Mourad Bellassoued; Jérôme Le Rousseau Carleman estimates for elliptic operators with complex coefficients. Part I: Boundary value problems, J. Math. Pures Appl., Volume 104 (2015) no. 4, pp. 657-728 | DOI | MR | Zbl

[7] Mourad Bellassoued; Jérôme Le Rousseau Carleman estimates for elliptic operators with complex coefficients. Part II: Transmission problems, J. Math. Pures Appl., Volume 115 (2018), pp. 127-186 | DOI | MR | Zbl

[8] Muriel Boulakia; Sergio Guerrero Local null controllability of a fluid-solid interaction problem in dimension 3, J. Eur. Math. Soc., Volume 15 (2013) no. 3, pp. 825-856 | DOI | MR | Zbl

[9] Muriel Boulakia; Axel Osses Local null controllability of a two-dimensional fluid-structure interaction problem, ESAIM, Control Optim. Calc. Var., Volume 14 (2008) no. 1, pp. 1-42 | DOI | Numdam | MR | Zbl

[10] Rémi Buffe Stabilization of the wave equation with Ventcel boundary condition, J. Math. Pures Appl., Volume 108 (2017) no. 2, pp. 207-259 | DOI | MR | Zbl

[11] Rémi Buffe; Takéo Takahashi Controllability of a Stokes system with a diffusive boundary condition, ESAIM, Control Optim. Calc. Var., Volume 28 (2022), 63, 29 pages | DOI | MR | Zbl

[12] Sunčica Čanić; Boris Muha; Martina Bukač Fluid-structure interaction in hemodynamics: modeling, analysis, and numerical simulation, Fluid-structure interaction and biomedical applications (Advances in Mathematical Fluid Mechanics), Springer, 2014, pp. 79-195 | DOI | MR | Zbl

[13] Antonin Chambolle; Benoît Desjardins; Maria J. Esteban; Céline Grandmont Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., Volume 7 (2005) no. 3, pp. 368-404 | DOI | MR | Zbl

[14] Shu Ping Chen; Roberto Triggiani Proof of extensions of two conjectures on structural damping for elastic systems, Pac. J. Math., Volume 136 (1989) no. 1, pp. 15-55 | DOI | MR | Zbl

[15] Jean-Michel Coron; Sergio Guerrero Null controllability of the $N$ -dimensional Stokes system with $N - 1$ scalar controls, J. Differ. Equations, Volume 246 (2009) no. 7, pp. 2908-2921 | DOI | MR | Zbl

[16] Anna Doubova; Enrique Fernández-Cara Some control results for simplified one-dimensional models of fluid-solid interaction, Math. Models Methods Appl. Sci., Volume 15 (2005) no. 5, pp. 783-824 | DOI | MR | Zbl

[17] Enrique Fernández-Cara; Manuel González-Burgos; Sergio Guerrero; Jean-Pierre Puel Null controllability of the heat equation with boundary Fourier conditions: the linear case, ESAIM, Control Optim. Calc. Var., Volume 12 (2006), pp. 442-465 | DOI | Numdam | MR | Zbl

[18] Enrique Fernández-Cara; Sergio Guerrero; Oleg Yu Imanuvilov; Jean-Pierre Puel Local exact controllability of the Navier–Stokes system, J. Math. Pures Appl., Volume 83 (2004) no. 12, pp. 1501-1542 | DOI | MR | Zbl

[19] Andrei Fursikov; Oleg Imanuvilov Controllability of evolution equations, Lecture Notes Series, Seoul, 34, Seoul National University, 1996, iv+163 pages | MR

[20] Céline Grandmont Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., Volume 40 (2008) no. 2, pp. 716-737 | DOI | MR | Zbl

[21] Céline Grandmont; Matthieu Hillairet Existence of global strong solutions to a beam-fluid interaction system, Arch. Ration. Mech. Anal., Volume 220 (2016) no. 3, pp. 1283-1333 | DOI | MR | Zbl

[22] Céline Grandmont; Matthieu Hillairet; Julien Lequeurre Existence of local strong solutions to fluid-beam and fluid-rod interaction systems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 36 (2019) no. 4, pp. 1105-1149 | DOI | MR | Zbl

[23] Lars Hörmander The analysis of linear partial differential operators. III: Pseudo-differential operators, Classics in Mathematics, Springer, 2007 | DOI | Zbl

[24] Oleg Imanuvilov; Takéo Takahashi Exact controllability of a fluid-rigid body system, J. Math. Pures Appl., Volume 87 (2007) no. 4, pp. 408-437 | DOI | MR | Zbl

[25] Jérôme Le Rousseau; Matthieu Léautaud; Luc Robbiano Controllability of a parabolic system with a diffuse interface, J. Eur. Math. Soc., Volume 15 (2013) no. 4, pp. 1485-1574 | DOI | MR | Zbl

[26] Jérôme Le Rousseau; Gilles Lebeau; Luc Robbiano Elliptic Carleman estimates and applications to stabilization and controllability. Volume I. Dirichlet boundary conditions on Euclidean space, Progress in Nonlinear Differential Equations and their Applications, 97, Birkhäuser, 2021 | DOI

[27] Jérôme Le Rousseau; Gilles Lebeau; Luc Robbiano Elliptic Carleman estimates and applications to stabilization and controllability. Volume II. General Boundary Conditions on Riemannian Manifolds, Progress in Nonlinear Differential Equations and their Applications, 97, Birkhäuser, 2022 | DOI

[28] Jérôme Le Rousseau; Nicolas Lerner Carleman estimates for anisotropic elliptic operators with jumps at an interface, Anal. PDE, Volume 6 (2013) no. 7, pp. 1601-1648 | DOI | MR | Zbl

[29] Jérôme Le Rousseau; Luc Robbiano Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Ration. Mech. Anal., Volume 195 (2010) no. 3, pp. 953-990 | DOI | MR | Zbl

[30] Jérôme Le Rousseau; Luc Robbiano Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces, Invent. Math., Volume 183 (2011) no. 2, pp. 245-336 | DOI | MR | Zbl

[31] Matthieu Léautaud Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems, J. Funct. Anal., Volume 258 (2010) no. 8, pp. 2739-2778 | DOI | MR | Zbl

[32] Gilles Lebeau; Luc Robbiano Contrôle exact de l’équation de la chaleur, Commun. Partial Differ. Equations, Volume 20 (1995) no. 1-2, pp. 335-356 | DOI | MR

[33] Daniel Lengeler Weak solutions for an incompressible, generalized Newtonian fluid interacting with a linearly elastic Koiter type shell, SIAM J. Math. Anal., Volume 46 (2014) no. 4, pp. 2614-2649 | DOI | MR | Zbl

[34] Daniel Lengeler; Michael Růžička Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell, Arch. Ration. Mech. Anal., Volume 211 (2014) no. 1, pp. 205-255 | DOI | MR | Zbl

[35] Julien Lequeurre Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal., Volume 43 (2011) no. 1, pp. 389-410 | DOI | MR | Zbl

[36] Jacques-Louis Lions; Enrico Magenes Non-homogeneous boundary value problems and applications. Vol. I, Grundlehren der Mathematischen Wissenschaften, 181, Springer, 1972, xvi+357 pages (Translated from the French by P. Kenneth) | MR

[37] Yuning Liu; Takéo Takahashi; Marius Tucsnak Single input controllability of a simplified fluid-structure interaction model, ESAIM, Control Optim. Calc. Var., Volume 19 (2013) no. 1, pp. 20-42 | DOI | Numdam | MR | Zbl

[38] Debayan Maity; Arnab Roy; Takéo Takahashi Existence of strong solutions for a system of interaction between a compressible viscous fluid and a wave equation, Nonlinearity, Volume 34 (2021) no. 4, pp. 2659-2687 | DOI | MR | Zbl

[39] Debayan Maity; Takéo Takahashi Existence and uniqueness of strong solutions for the system of interaction between a compressible Navier–Stokes–Fourier fluid and a damped plate equation, Nonlinear Anal., Real World Appl., Volume 59 (2021), 103267, 34 pages | DOI | MR | Zbl

[40] Debayan Maity; Takéo Takahashi $L^{p}$ theory for the interaction between the incompressible Navier–Stokes system and a damped plate, J. Math. Fluid Mech., Volume 23 (2021) no. 4, 103, 23 pages | DOI | MR | Zbl

[41] Sourav Mitra Carleman estimate for an adjoint of a damped beam equation and an application to null controllability, J. Math. Anal. Appl., Volume 484 (2020) no. 1, 123718, 29 pages | DOI | MR | Zbl

[42] Sourav Mitra Observability and unique continuation of the adjoint of a linearized simplified compressible fluid-structure model in a 2D channel, ESAIM, Control Optim. Calc. Var., Volume 27 (2021), S18, 51 pages | DOI | MR | Zbl

[43] Boris Muha; Suncica Canić Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch. Ration. Mech. Anal., Volume 207 (2013) no. 3, pp. 919-968 | DOI | MR | Zbl

[44] Boris Muha; Suncica Canić Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch. Ration. Mech. Anal., Volume 207 (2013) no. 3, pp. 919-968 | DOI | MR | Zbl

[45] Boris Muha; Sunčica Čanić A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof, Commun. Inf. Syst., Volume 13 (2013) no. 3, pp. 357-397 | DOI | MR | Zbl

[46] Boris Muha; Sunčica Čanić Fluid-structure interaction between an incompressible, viscous 3D fluid and an elastic shell with nonlinear Koiter membrane energy, Interfaces Free Bound., Volume 17 (2015) no. 4, pp. 465-495 | DOI | MR | Zbl

[47] Alfio Quarteroni; Massimiliano Tuveri; Alessandro Veneziani Computational vascular fluid dynamics: problems, models and methods, Comput. Vis. Sci., Volume 2 (2000) no. 4, pp. 163-197 | DOI | Zbl

[48] Jean-Pierre Raymond Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., Volume 48 (2010) no. 8, pp. 5398-5443 | DOI | MR | Zbl

[49] Arnab Roy; Takéo Takahashi Local null controllability of a rigid body moving into a Boussinesq flow, Math. Control Relat. Fields, Volume 9 (2019) no. 4, pp. 793-836 | DOI | MR | Zbl

[50] Roger Temam Navier–Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications, 2, North-Holland, 1979, x+519 pages (with an appendix by F. Thomasset) | MR

[51] Srđan Trifunović; Ya-Guang Wang Existence of a weak solution to the fluid-structure interaction problem in 3D, J. Differ. Equations, Volume 268 (2020) no. 4, pp. 1495-1531 | DOI | MR | Zbl

[52] Srđan Trifunović; Ya-Guang Wang Weak solution to the incompressible viscous fluid and a thermoelastic plate interaction problem in 3D, Acta Math. Sci., Ser. B, Engl. Ed., Volume 41 (2021) no. 1, pp. 19-38 | DOI | MR | Zbl

[53] Marius Tucsnak; George Weiss Observation and control for operator semigroups, Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser, 2009, xii+483 pages | DOI | MR

Cité par Sources :

Commentaires - Politique