Issue ESAIM: COCV Volume 14, Number 1, January-March 2008 Page(s) 1 - 42 DOI 10.1051/cocv:2007031

ESAIM: COCV, Vol. 14, N°1, pp. 1-42
DOI: 10.1051/cocv:2007031

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

Muriel Boulakia¹ and Axel Osses<sup>2

1</sup>  Laboratoire de Mathématiques Appliquées, Université de Versailles-St-Quentin, 45 avenue des États-Unis, 78035 Versailles Cedex, France; boulakia@math.uvsq.fr
²  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

(Received October 27, 2005. Revised April 5, 2006. Published online July 20, 2007.)

Abstract
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 T > 0, the system can be driven at rest and the structure to its reference configuration at time T. 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.

Mathematics Subject Classification. 35Q30, 93C20

Key words: Controllability, fluid-solid interaction, Navier-Stokes equations, Carleman estimates


© EDP Sciences, SMAI 2007

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