Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T01:18:46.679Z Has data issue: false hasContentIssue false
  • English
  • Français

Controllability properties of a class of systems modelingswimming microscopic organisms

Published online by Cambridge University Press:  11 August 2009

Mario Sigalotti  and
Jean-Claude Vivalda
Mario Sigalotti
Affiliation:
Institut Élie Cartan de Nancy, UMR 7502 INRIA/Nancy-Université/CNRS, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France. [email protected] Équipe-projet CORIDA, INRIA Nancy – Grand Est, France.
Jean-Claude Vivalda
Affiliation:
Équipe-projet CORIDA, INRIA Nancy – Grand Est, France. Laboratoire et Département de Mathématiques, UMR 7122 Université de Metz/CNRS, Bât. A, Île du Saulcy, 57045 Metz Cedex 1, France.
Get access

Abstract

We consider a finite-dimensional model for the motion of microscopic organisms whose propulsion exploitsthe action of a layer of cilia covering its surface. The model couples Newton's laws driving the organism, considered as a rigid body, withStokes equations governing the surrounding fluid.The action of thecilia 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 systemis 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 whichthe organism has a spherical shape. Finally, we offer a complete picture of controllable and non-controllable systems under the additional hypothesis thatthe organism and the fluid have densities of the same order of magnitude.

Keywords

Swimming micro-organismsciliatahigh viscositynonlinear systemscontrollability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 4 , October 2010 , pp. 1053 - 1076
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A.A. Agrachev and Y.L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences 87, Control Theory and Optimization II. Springer-Verlag, Berlin (2004).
Alouges, F., DeSimone, A. and Lefebvre, A., Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18 (2008) 277–302. CrossRef
Berg, H.C. and Anderson, R., Bacteria swim by rotating their flagellar filaments. Nature 245 (1973) 380–382. CrossRef
Blake, J., A finite model for ciliated micro-organisms. J. Biomech. 6 (1973) 133–140. CrossRef
Brennen, C., An oscil lating-boundary-layer theory for ciliary propulsion. J. Fluid Mech. 65 (1974) 799–824. CrossRef
Brunovský, P. and Lobry, C., Contrôlabilité Bang Bang, contrôlabilité différentiable, et perturbation des systèmes non linéaires. Ann. Mat. Pura Appl. 105 (1975) 93–119. CrossRef
S. Childress, Mechanics of swimming and flying, Cambridge Studies in Mathematical Biology 2. Cambridge University Press, Cambridge (1981).
Chitour, Y., Coron, J.-M. and Garavello, M., On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete Contin. Dyn. Syst. 14 (2006) 643–672.
G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations I: Linearized steady problems, Springer Tracts in Natural Philosophy 38. Springer-Verlag, New York (1994)
K.A. Grasse and H.J. Sussmann, Global controllability by nice controls, in Nonlinear controllability and optimal control, Monogr. Textbooks Pure Appl. Math. 133, Dekker, New York (1990) 33–79.
J. Happel and H. Brenner, Low Reynolds number hydrodynamics with special applications to particulate media. Prentice-Hall Inc., Englewood Cliffs, USA (1965).
V. Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics 52. Cambridge University Press, Cambridge (1997).
Jurdjevic, V. and Kupka, I., Control systems subordinated to a group action: accessibility. J. Differ. Equ. 39 (1980) 186–211. CrossRef
Jurdjevic, V. and Kupka, I., Control systems on semi-simple Lie groups and their homogeneous sapces. Ann. Inst. Fourier 31 (1981) 151–179. CrossRef
Jurdjevic, V. and Sallet, G., Controllability properties of affine systems. SIAM J. Contr. Opt. 22 (1984) 501–508. CrossRef
Keller, S. and Wu, T., A porous prolate-spheroidal model for ciliated micro-organisms. J. Fluid Mech. 80 (1977) 259–278. CrossRef
J. Lighthill, Mathematical Biofluiddynamics, Regional Conference Series in Applied Mathematics 17. Society for Industrial and Applied Mathematics, Philadelphia, USA (1975). (Based on the lecture course delivered to the Mathematical Biofluiddynamics Research Conference of the National Science Foundation held from July 16–20 1973, at Rensselaer Polytechnic Institute, Troy, New York, USA.)
Purcell, E.M., Life at low Reynolds numbers. Am. J. Phys. 45 (1977) 3–11. CrossRef
San Martín, J., Takahashi, T. and Tucsnak, M., A control theoretic approach to the swimming of microscopic organisms. Quart. Appl. Math. 65 (2007) 405–424. CrossRef
J. Simon, Différentiation de problèmes aux limites par rapport au domaine. Lecture notes, University of Seville, Spain (1991).
Sussmann, H.J., Some properties of vector field systems that are not altered by small perturbations. J. Differ. Equ. 20 (1976) 292–315. CrossRef
Taylor, G., Analysis of the swimming of microscopic organisms. Proc. Roy. Soc. London. Ser. A 209 (1951) 447–461. CrossRef