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

¹ 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: 27 October 2005
Revised: 5 April 2006

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.