Controllability properties of a class of systems modeling swimming microscopic organisms

ESAIM - Control, Optimisation and Calculus of Variations

Volumn 16, Issue 4, 2010, Pages 1053-1076

  Sigalotti, Mario ^a,b   Vivalda, Jean Claude ^b,c  

^a UNIVERSITÉ HENRI POINCARÉ NANCY 1   (France)

^b INRIA   (France)

^c UNIVERSITÉ DE LORRAINE   (France)

Author keywords

Ciliata; Controllability; High viscosity; Nonlinear systems; Swimming micro organisms

Indexed keywords

CILIATA; CONTROLLABLE SYSTEMS; FINITE-DIMENSIONAL MODELS; HIGH VISCOSITY; MICROSCOPIC ORGANISMS; NEWTON'S LAWS; ORDER OF MAGNITUDE; RIGID BODY; SPHERICAL SHAPE; STOKES EQUATIONS; SYSTEMS MODELING; VELOCITY FIELD;

CONTROLLABILITY; NONLINEAR SYSTEMS; PROTOZOA; VISCOSITY;

BIOLOGY;

EID: 77958597644     PISSN: 12928119     EISSN: 12623377     Source Type: Journal    
DOI: 10.1051/cocv/2009034     Document Type: Article

References (22)

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).
2. F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18 (2008) 277-302.
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.
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.
7. S. Childress, Mechanics of swimming and flying, Cambridge Studies in Mathematical Biology 2. Cambridge University Press, Cambridge (1981).
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.
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)
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.
11. J. Happel and H. Brenner, Low Reynolds number hydrodynamics with special applications to particulate media. Prentice-Hall Inc., Englewood Cliffs, USA (1965).
12. V. Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics 52. Cambridge University Press, Cambridge (1997).
13. V. Jurdjevic and I. Kupka, Control systems subordinated to a group action: accessibility. J. Differ. Equ. 39 (1980) 186-211.
14. V. Jurdjevic and I. Kupka, Control systems on semi-simple Lie groups and their homogeneous sapces. Ann. Inst. Fourier 31 (1981) 151-179.
15. V. Jurdjevic and G. Sallet, Controllability properties of affine systems. SIAM J. Contr. Opt. 22 (1984) 501-508.
16. S. Keller and T. Wu, A porous prolate-spheroidal model for ciliated micro-organisms. J. Fluid Mech. 80 (1977) 259-278.
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.)
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.
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.
22. G. Taylor, Analysis of the swimming of microscopic organisms. Proc. Roy. Soc. London. Ser. A 209 (1951) 447-461.