Document 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 Osses21 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