Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-12T07:17:18.803Z Has data issue: false hasContentIssue false
  • English
  • Français

Local null controllability of a two-dimensional fluid-structure interaction problem

Published online by Cambridge University Press:  20 July 2007

Muriel Boulakia and
Axel Osses
Muriel Boulakia
Affiliation:
Laboratoire de Mathématiques Appliquées, Université de Versailles-St-Quentin, 45 avenue des États-Unis, 78035 Versailles Cedex, France; boulakia@math.uvsq.fr
Axel Osses
Affiliation:
Departamento de Ingenería Matemática and Centro de Modelamiento Matemático UMI 2807 CNRS, Facultad de Ciencias de Físicas y Matemáticas, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile; axosses@dim.uchile.cl
Get access

Abstract

In this paper, we prove a controllabilityresult for a fluid-structure interaction problem. In dimension two,a rigid structure moves into an incompressible fluid governed byNavier-Stokes equations. The control acts on a fixed subset of thefluid domain. We prove that, for small initial data, this system isnull controllable, that is, for a given T > 0, the system can bedriven at rest and the structure to its reference configuration attime T. To show this result, we first consider a linearizedsystem. Thanks to an observability inequality obtained from aCarleman inequality, we prove an optimal controllability result witha regular control. Next, with the help of Kakutani's fixed pointtheorem and a regularity result, we pass to the nonlinear problem.

Keywords

Controllabilityfluid-solid interactionNavier-Stokes equationsCarleman estimates

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 1 , January 2008, pp. 1 - 42
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Anita, S. and Barbu, V., Null controllability of nonlinear convective heat equations. ESAIM: COCV 5 (2000) 157–173. CrossRef
M. Boulakia and A. Osses, Local null controllability of a two-dimensional fluid-structure interaction problem. Prépublication 139, UVSQ (octobre 2005).
Conca, C., San Martin, J. and Tucsnak, M., 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. CrossRef
Coron, J.M. and Guerrero, S., Singular optimal control: A linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44 (2005) 237–257.
Desjardins, B. and Esteban, M.J., Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146 (1999) 59–71. CrossRef
Doubova, A. and Fernandez-Cara, E., Some control results for simplified one-dimensional models of fluid-solid interaction. Math. Models Methods Appl. Sci. 15 (2005) 783–824. CrossRef
Fabre, C. and Lebeau, G., Prolongement unique des solutions de l'équation de Stokes. Comm. Partial Diff. Equations 21 (1996) 573–596. CrossRef
Fabre, C., Puel, J.-P. and Zuazua, E., Approximate controllability of the semilinear heat equation. Proc. Royal Soc. Edinburgh 125A (1995) 31–61. CrossRef
Fernandez-Cara, E. and Zuazua, E., The cost of approximate controllability for heat equations: the linear case. Adv. Differential Equations 5 (2000) 465–514.
Fernandez-Cara, E., Guerrero, S., Imanuvilov, O. Yu. and Puel, J.-P., Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. CrossRef
A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996).
Imanuvilov, O.Yu., Remarks on exact controllability for the Navier-Stokes equations. ESAIM: COCV 6 (2001) 39–72. CrossRef
Imanuvilov, O.Yu. and Puel, J.-P., Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems. Internat. Math. Res. Notices 16 (2003) 883–913. CrossRef
O.Yu. Imanuvilov and T. Takahashi, Exact controllability of a fluid-rigid body system. Prépublication IECN (novembre 2005).
Nakoulima, O., Contrôlabilité à zéro avec contraintes sur le contrôle. C. R. Acad. Sci. Paris Ser. I 339 (2004) 405–410. CrossRef
Osses, A. and Puel, J.P., Approximate controllability for a linear model of fluid structure interaction. ESAIM: COCV 4 (1999) 497–513. CrossRef
Raymond, J.P. and Vanninathan, M., Exact controllability in fluid-solid structure: the Helmholtz model. ESAIM: COCV 11 (2005) 180–203. CrossRef
San Martin, J., Starovoitov, V. and Tucsnak, M., 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.
T. Takahashi, 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.
Temam, R., Behaviour at time $t=0$ of the solutions of semi-linear evolution equations. J. Diff. Equations 43 (1982) 73–92. CrossRef
Vázquez, J.L., Zuazua, E., Large time behavior for a simplified 1D model of fluid-solid interaction. Comm. Partial Differential Equations 28 (2003) 1705–1738. CrossRef
J.L. Vázquez and E. Zuazua, Lack of collision in a simplified 1-dimensional model for fluid-solid interaction. Math. Models Methods Apll. Sci., M3AS 16 (2006) 637–678.
Zhang, X. and Zuazua, E., Polynomial decay and control of a $1-d$ hyperbolic-parabolic coupled system. J. Differential Equations 204 (2004) 380–438. CrossRef