Controllability properties of a class of systems modeling swimming microscopic organisms
Institut Élie Cartan de Nancy, UMR 7502
INRIA/Nancy-Université/CNRS, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France. email@example.com
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.
Revised: 4 January 2009
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.
Mathematics Subject Classification: 37C20 / 70Q05 / 76Z10 / 93B05 / 93C10
Key words: Swimming micro-organisms / ciliata / high viscosity / nonlinear systems / controllability
© EDP Sciences, SMAI, 2009