Controllability properties of a class of systems modeling swimming microscopic organisms

¹ Institut Élie Cartan de Nancy, UMR 7502 INRIA/Nancy-Université/CNRS, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France. mario.sigalotti@inria.fr
² Équipe-projet CORIDA, INRIA Nancy – Grand Est, France.
³ Laboratoire et Département de Mathématiques, UMR 7122 Université de Metz/CNRS, Bât. A, Île du Saulcy, 57045 Metz Cedex 1, France.

Received: 15 November 2007
Revised: 4 January 2009

Abstract

We consider a finite-dimensional model for the motion of microscopic organisms whose propulsion exploits the action of a layer of cilia covering its surface. The model couples Newton's laws driving the organism, considered as a rigid body, with Stokes equations governing the surrounding fluid. The action of the cilia is described by a set of controlled velocity fields on the surface of the organism. The first contribution of the paper is the proof that such a system is generically controllable when the space of controlled velocity fields is at least three-dimensional. We also provide a complete characterization of controllable systems in the case in which the organism has a spherical shape. Finally, we offer a complete picture of controllable and non-controllable systems under the additional hypothesis that the organism and the fluid have densities of the same order of magnitude.