Controllability properties of a class of systems modeling swimming microscopic organisms
ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 1053-1076.

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.

DOI : 10.1051/cocv/2009034
Classification : 37C20, 70Q05, 76Z10, 93B05, 93C10
Mots clés : swimming micro-organisms, ciliata, high viscosity, nonlinear systems, controllability
@article{COCV_2010__16_4_1053_0,
     author = {Sigalotti, Mario and Vivalda, Jean-Claude},
     title = {Controllability properties of a class of systems modeling swimming microscopic organisms},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1053--1076},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {4},
     year = {2010},
     doi = {10.1051/cocv/2009034},
     mrnumber = {2744162},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2009034/}
}
TY  - JOUR
AU  - Sigalotti, Mario
AU  - Vivalda, Jean-Claude
TI  - Controllability properties of a class of systems modeling swimming microscopic organisms
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2010
SP  - 1053
EP  - 1076
VL  - 16
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2009034/
DO  - 10.1051/cocv/2009034
LA  - en
ID  - COCV_2010__16_4_1053_0
ER  - 
%0 Journal Article
%A Sigalotti, Mario
%A Vivalda, Jean-Claude
%T Controllability properties of a class of systems modeling swimming microscopic organisms
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2010
%P 1053-1076
%V 16
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2009034/
%R 10.1051/cocv/2009034
%G en
%F COCV_2010__16_4_1053_0
Sigalotti, Mario; Vivalda, Jean-Claude. Controllability properties of a class of systems modeling swimming microscopic organisms. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 1053-1076. doi : 10.1051/cocv/2009034. http://archive.numdam.org/articles/10.1051/cocv/2009034/

[1] 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). | Zbl

[2] F. Alouges, A. Desimone and A. Lefebvre, Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18 (2008) 277-302. | Zbl

[3] H.C. Berg and R. Anderson, Bacteria swim by rotating their flagellar filaments. Nature 245 (1973) 380-382.

[4] J. Blake, A finite model for ciliated micro-organisms. J. Biomech. 6 (1973) 133-140.

[5] C. Brennen, An oscil lating-boundary-layer theory for ciliary propulsion. J. Fluid Mech. 65 (1974) 799-824. | Zbl

[6] P. Brunovský and C. Lobry, Contrôlabilité Bang Bang, contrôlabilité différentiable, et perturbation des systèmes non linéaires. Ann. Mat. Pura Appl. 105 (1975) 93-119. | Zbl

[7] S. Childress, Mechanics of swimming and flying, Cambridge Studies in Mathematical Biology 2. Cambridge University Press, Cambridge (1981). | Zbl

[8] Y. Chitour, J.-M. Coron and M. Garavello, On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete Contin. Dyn. Syst. 14 (2006) 643-672. | Zbl

[9] 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) | Zbl

[10] 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. | Zbl

[11] J. Happel and H. Brenner, Low Reynolds number hydrodynamics with special applications to particulate media. Prentice-Hall Inc., Englewood Cliffs, USA (1965). | Zbl

[12] V. Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics 52. Cambridge University Press, Cambridge (1997). | Zbl

[13] V. Jurdjevic and I. Kupka, Control systems subordinated to a group action: accessibility. J. Differ. Equ. 39 (1980) 186-211. | Zbl

[14] V. Jurdjevic and I. Kupka, Control systems on semi-simple Lie groups and their homogeneous sapces. Ann. Inst. Fourier 31 (1981) 151-179. | Numdam | Zbl

[15] V. Jurdjevic and G. Sallet, Controllability properties of affine systems. SIAM J. Contr. Opt. 22 (1984) 501-508. | Zbl

[16] S. Keller and T. Wu, A porous prolate-spheroidal model for ciliated micro-organisms. J. Fluid Mech. 80 (1977) 259-278. | Zbl

[17] 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.) | Zbl

[18] E.M. Purcell, Life at low Reynolds numbers. Am. J. Phys. 45 (1977) 3-11.

[19] J. San Martín, T. Takahashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms. Quart. Appl. Math. 65 (2007) 405-424. | Zbl

[20] J. Simon, Différentiation de problèmes aux limites par rapport au domaine. Lecture notes, University of Seville, Spain (1991).

[21] H.J. Sussmann, Some properties of vector field systems that are not altered by small perturbations. J. Differ. Equ. 20 (1976) 292-315. | Zbl

[22] G. Taylor, Analysis of the swimming of microscopic organisms. Proc. Roy. Soc. London. Ser. A 209 (1951) 447-461. | Zbl

Cité par Sources :