Controllability properties of a class of systems modeling swimming microscopic organisms
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, p. 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.

Classification:  37C20,  70Q05,  76Z10,  93B05,  93C10
Keywords: swimming micro-organisms, ciliata, high viscosity, nonlinear systems, controllability
     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},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {4},
     year = {2010},
     pages = {1053-1076},
     doi = {10.1051/cocv/2009034},
     zbl = {pre05821909},
     mrnumber = {2744162},
     language = {en},
     url = {}
Sigalotti, Mario; Vivalda, Jean-Claude. Controllability properties of a class of systems modeling swimming microscopic organisms. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 1053-1076. doi : 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 1062.93001

[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 1146.76062

[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 0291.76043

[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 0316.93007

[7] S. Childress, Mechanics of swimming and flying, Cambridge Studies in Mathematical Biology 2. Cambridge University Press, Cambridge (1981). | Zbl 0499.76118

[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 1134.93313

[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 0949.35004

[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 0703.93014

[11] J. Happel and H. Brenner, Low Reynolds number hydrodynamics with special applications to particulate media. Prentice-Hall Inc., Englewood Cliffs, USA (1965). | Zbl 0612.76032

[12] V. Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics 52. Cambridge University Press, Cambridge (1997). | Zbl 0940.93005

[13] V. Jurdjevic and I. Kupka, Control systems subordinated to a group action: accessibility. J. Differ. Equ. 39 (1980) 186-211. | Zbl 0531.93008

[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 0453.93011

[15] V. Jurdjevic and G. Sallet, Controllability properties of affine systems. SIAM J. Contr. Opt. 22 (1984) 501-508. | Zbl 0549.93010

[16] S. Keller and T. Wu, A porous prolate-spheroidal model for ciliated micro-organisms. J. Fluid Mech. 80 (1977) 259-278. | Zbl 0352.76080

[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 0312.76076

[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 1135.76058

[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 0346.49036

[22] G. Taylor, Analysis of the swimming of microscopic organisms. Proc. Roy. Soc. London. Ser. A 209 (1951) 447-461. | Zbl 0043.40302