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.
Classification : 35Q30, 93C20
Mots clés : controllability, fluid-solid interaction, Navier-Stokes equations, Carleman estimates
@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},
zbl = {1149.35068},
mrnumber = {2375750},
language = {en},
url = {http://www.numdam.org/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 DA - 2008/// SP - 1 EP - 42 VL - 14 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007031/ UR - https://zbmath.org/?q=an%3A1149.35068 UR - https://www.ams.org/mathscinet-getitem?mr=2375750 UR - https://doi.org/10.1051/cocv:2007031 DO - 10.1051/cocv:2007031 LA - en ID - COCV_2008__14_1_1_0 ER -
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. http://www.numdam.org/articles/10.1051/cocv:2007031/
[1] and , Null controllability of nonlinear convective heat equations. ESAIM: COCV 5 (2000) 157-173. | Numdam | MR 1744610 | Zbl 0938.93008
[2] and , Local null controllability of a two-dimensional fluid-structure interaction problem. Prépublication 139, UVSQ (octobre 2005).
[3] , and , Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Comm. Partial Differential Equations 25 (2000) 1019-1042. | MR 1759801 | Zbl 0954.35135
[4] and , Singular optimal control: A linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44 (2005) 237-257. | MR 2176274 | Zbl 1078.93009
[5] and , Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146 (1999) 59-71. | MR 1682663 | Zbl 0943.35063
[6] and , Some control results for simplified one-dimensional models of fluid-solid interaction. Math. Models Methods Appl. Sci. 15 (2005) 783-824. | MR 2139944 | Zbl 1122.93008
[7] and , Prolongement unique des solutions de l'équation de Stokes. Comm. Partial Diff. Equations 21 (1996) 573-596. | MR 1387461 | Zbl 0849.35098
[8] , and , Approximate controllability of the semilinear heat equation. Proc. Royal Soc. Edinburgh 125A (1995) 31-61. | MR 1318622 | Zbl 0818.93032
[9] and , The cost of approximate controllability for heat equations: the linear case. Adv. Differential Equations 5 (2000) 465-514. | MR 1750109 | Zbl 1007.93034
[10] , , and , Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501-1542. | MR 2103189
[11] and , Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR 1406566 | Zbl 0862.49004
[12] , Remarks on exact controllability for the Navier-Stokes equations. ESAIM: COCV 6 (2001) 39-72. | Numdam | MR 1804497 | Zbl 0961.35104
[13] and , Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems. Internat. Math. Res. Notices 16 (2003) 883-913. | MR 1959940 | Zbl 1146.35340
[14] and , Exact controllability of a fluid-rigid body system. Prépublication IECN (novembre 2005). | Zbl 1124.35056
[15] , Contrôlabilité à zéro avec contraintes sur le contrôle. C. R. Acad. Sci. Paris Ser. I 339 (2004) 405-410. | MR 2092753 | Zbl 1060.93015
[16] and , Approximate controllability for a linear model of fluid structure interaction. ESAIM: COCV 4 (1999) 497-513. | Numdam | MR 1713527 | Zbl 0931.35014
[17] and , Exact controllability in fluid-solid structure: the Helmholtz model. ESAIM: COCV 11 (2005) 180-203. | Numdam | MR 2141885 | Zbl 1125.93007
[18] , and , Global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Rational Mech. Anal. 161 (2002) 113-147. | MR 1870954 | Zbl 1018.76012
[19] , Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations 8 (2003) 1499-1532. | MR 2029294 | Zbl 1101.35356
[20] , Behaviour at time of the solutions of semi-linear evolution equations. J. Diff. Equations 43 (1982) 73-92. | MR 645638 | Zbl 0446.35057
[21] , , Large time behavior for a simplified 1D model of fluid-solid interaction. Comm. Partial Differential Equations 28 (2003) 1705-1738. | MR 2001181 | Zbl 1071.74017
[22] and , Lack of collision in a simplified 1-dimensional model for fluid-solid interaction. Math. Models Methods Apll. Sci., M3AS 16 (2006) 637-678. | MR 2226121
[23] and , Polynomial decay and control of a hyperbolic-parabolic coupled system. J. Differential Equations 204 (2004) 380-438. | MR 2085542 | Zbl 1064.93008
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