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

Buffe, Rémi; Takahashi, Takéo

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

Buffe, Rémi ¹ ; Takahashi, Takéo ¹

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

Comptes Rendus. Mathématique, Tome 361 (2023) no. G9, 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
Keywords: Null controllability, Navier–Stokes systems, Carleman estimates, fluid-structure interaction systems

Affiliations des auteurs :

Buffe, Rémi ¹ ; Takahashi, Takéo ¹

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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},
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Buffe, Rémi; Takahashi, Takéo. Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation. Comptes Rendus. Mathématique, Tome 361 (2023) no. G9, pp. 1541-1576. doi: 10.5802/crmath.509

Bibliographie
Cité par

[1] Badra, Mehdi; Takahashi, Takéo 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] Badra, Mehdi; Takahashi, Takéo 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] Badra, Mehdi; Takahashi, Takéo 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 | MR | DOI | Zbl

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

[8] Boulakia, Muriel; Guerrero, Sergio 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] Boulakia, Muriel; Osses, Axel 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 | MR | Numdam | Zbl

[10] Buffe, Rémi 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] Buffe, Rémi; Takahashi, Takéo 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] Čanić, Sunčica; Muha, Boris; Bukač, Martina 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 | MR | DOI | Zbl

[13] Chambolle, Antonin; Desjardins, Benoît; Esteban, Maria J.; Grandmont, Céline 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] Chen, Shu Ping; Triggiani, Roberto 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] Coron, Jean-Michel; Guerrero, Sergio 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] Doubova, Anna; Fernández-Cara, Enrique 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] Fernández-Cara, Enrique; González-Burgos, Manuel; Guerrero, Sergio; Puel, Jean-Pierre 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 | Zbl | MR | Numdam

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

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

[20] Grandmont, Céline 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] Grandmont, Céline; Hillairet, Matthieu 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] Grandmont, Céline; Hillairet, Matthieu; Lequeurre, Julien 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] Hörmander, Lars The analysis of linear partial differential operators. III: Pseudo-differential operators, Classics in Mathematics, Springer, 2007 | DOI | Zbl

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

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

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

[33] Lengeler, Daniel 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] Lengeler, Daniel; Růžička, Michael 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] Lequeurre, Julien 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] Lions, Jacques-Louis; Magenes, Enrico 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] Liu, Yuning; Takahashi, Takéo; Tucsnak, Marius Single input controllability of a simplified fluid-structure interaction model, ESAIM, Control Optim. Calc. Var., Volume 19 (2013) no. 1, pp. 20-42 | DOI | Zbl | MR | Numdam

[38] Maity, Debayan; Roy, Arnab; Takahashi, Takéo 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] Maity, Debayan; Takahashi, Takéo 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] Maity, Debayan; Takahashi, Takéo $L^{p}$ 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 | Zbl | MR

[41] Mitra, Sourav 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 | Zbl | MR

[42] Mitra, Sourav 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] Muha, Boris; Canić, Suncica 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] Muha, Boris; Canić, Suncica 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] Muha, Boris; Čanić, Sunčica 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] Muha, Boris; Čanić, Sunčica 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] Quarteroni, Alfio; Tuveri, Massimiliano; Veneziani, Alessandro Computational vascular fluid dynamics: problems, models and methods, Comput. Vis. Sci., Volume 2 (2000) no. 4, pp. 163-197 | DOI | Zbl

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

[49] Roy, Arnab; Takahashi, Takéo 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] Temam, Roger 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] Trifunović, Srđan; Wang, Ya-Guang 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] Trifunović, Srđan; Wang, Ya-Guang 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] Tucsnak, Marius; Weiss, George Observation and control for operator semigroups, Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser, 2009, xii+483 pages | DOI | MR

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