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.
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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/cocv/2009034/
[1] and , Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences 87, Control Theory and Optimization II. Springer-Verlag, Berlin (2004). | Zbl
[2] , and , Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18 (2008) 277-302. | Zbl
[3] and , Bacteria swim by rotating their flagellar filaments. Nature 245 (1973) 380-382.
[4] , A finite model for ciliated micro-organisms. J. Biomech. 6 (1973) 133-140.
[5] , An oscil lating-boundary-layer theory for ciliary propulsion. J. Fluid Mech. 65 (1974) 799-824. | Zbl
[6] and , Contrôlabilité Bang Bang, contrôlabilité différentiable, et perturbation des systèmes non linéaires. Ann. Mat. Pura Appl. 105 (1975) 93-119. | Zbl
[7] , Mechanics of swimming and flying, Cambridge Studies in Mathematical Biology 2. Cambridge University Press, Cambridge (1981). | Zbl
[8] , and , On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete Contin. Dyn. Syst. 14 (2006) 643-672. | Zbl
[9] , 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] and , 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] and , Low Reynolds number hydrodynamics with special applications to particulate media. Prentice-Hall Inc., Englewood Cliffs, USA (1965). | Zbl
[12] , Geometric control theory, Cambridge Studies in Advanced Mathematics 52. Cambridge University Press, Cambridge (1997). | Zbl
[13] and , Control systems subordinated to a group action: accessibility. J. Differ. Equ. 39 (1980) 186-211. | Zbl
[14] and , Control systems on semi-simple Lie groups and their homogeneous sapces. Ann. Inst. Fourier 31 (1981) 151-179. | Numdam | Zbl
[15] and , Controllability properties of affine systems. SIAM J. Contr. Opt. 22 (1984) 501-508. | Zbl
[16] and , A porous prolate-spheroidal model for ciliated micro-organisms. J. Fluid Mech. 80 (1977) 259-278. | Zbl
[17] , 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] , Life at low Reynolds numbers. Am. J. Phys. 45 (1977) 3-11.
[19] , and , A control theoretic approach to the swimming of microscopic organisms. Quart. Appl. Math. 65 (2007) 405-424. | Zbl
[20] , Différentiation de problèmes aux limites par rapport au domaine. Lecture notes, University of Seville, Spain (1991).
[21] , Some properties of vector field systems that are not altered by small perturbations. J. Differ. Equ. 20 (1976) 292-315. | Zbl
[22] , Analysis of the swimming of microscopic organisms. Proc. Roy. Soc. London. Ser. A 209 (1951) 447-461. | Zbl
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