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.
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},
mrnumber = {2375750},
zbl = {1149.35068},
language = {en},
url = {http://archive.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 SP - 1 EP - 42 VL - 14 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/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://archive.numdam.org/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. http://archive.numdam.org/articles/10.1051/cocv:2007031/
[1] and , Null controllability of nonlinear convective heat equations. ESAIM: COCV 5 (2000) 157-173. | Numdam | MR | Zbl
[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 | Zbl
[4] and , Singular optimal control: A linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44 (2005) 237-257. | MR | Zbl
[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 | Zbl
[6] and , Some control results for simplified one-dimensional models of fluid-solid interaction. Math. Models Methods Appl. Sci. 15 (2005) 783-824. | MR | Zbl
[7] and , Prolongement unique des solutions de l'équation de Stokes. Comm. Partial Diff. Equations 21 (1996) 573-596. | MR | Zbl
[8] , and , Approximate controllability of the semilinear heat equation. Proc. Royal Soc. Edinburgh 125A (1995) 31-61. | MR | Zbl
[9] and , The cost of approximate controllability for heat equations: the linear case. Adv. Differential Equations 5 (2000) 465-514. | MR | Zbl
[10] , , and , Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501-1542. | MR
[11] and , Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR | Zbl
[12] , Remarks on exact controllability for the Navier-Stokes equations. ESAIM: COCV 6 (2001) 39-72. | Numdam | MR | Zbl
[13] and , Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems. Internat. Math. Res. Notices 16 (2003) 883-913. | MR | Zbl
[14] and , Exact controllability of a fluid-rigid body system. Prépublication IECN (novembre 2005). | Zbl
[15] , Contrôlabilité à zéro avec contraintes sur le contrôle. C. R. Acad. Sci. Paris Ser. I 339 (2004) 405-410. | MR | Zbl
[16] and , Approximate controllability for a linear model of fluid structure interaction. ESAIM: COCV 4 (1999) 497-513. | Numdam | MR | Zbl
[17] and , Exact controllability in fluid-solid structure: the Helmholtz model. ESAIM: COCV 11 (2005) 180-203. | Numdam | MR | Zbl
[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 | Zbl
[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 | Zbl
[20] , Behaviour at time of the solutions of semi-linear evolution equations. J. Diff. Equations 43 (1982) 73-92. | MR | Zbl
[21] , , Large time behavior for a simplified 1D model of fluid-solid interaction. Comm. Partial Differential Equations 28 (2003) 1705-1738. | MR | Zbl
[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
[23] and , Polynomial decay and control of a hyperbolic-parabolic coupled system. J. Differential Equations 204 (2004) 380-438. | MR | Zbl
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