Optimal strokes for driftless swimmers: A general geometric approach
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 6.

Swimming consists by definition in propelling through a fluid by means of bodily movements. Thus, from a mathematical point of view, swimming turns into a control problem for which the controls are the deformations of the swimmer. The aim of this paper is to present a unified geometric approach for the optimization of the body deformations of so-called driftless swimmers. The class of driftless swimmers includes, among other, swimmers in a 3D Stokes flow (case of micro-swimmers in viscous fluids) or swimmers in a 2D or 3D potential flow. A general framework is introduced, allowing the complete analysis of five usual nonlinear optimization problems to be carried out. The results are illustrated with examples coming from the literature and with an in-depth study of a swimmer in a 2D potential flow. Numerical tests are also provided.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2017012
Classification : 74F10, 70S05, 76B03, 93B27
Mots-clés : Locomotion, swimmer, geometric control theory, optimal control
Chambrion, Thomas 1 ; Giraldi, Laetitia 1 ; Munnier, Alexandre 1

1
@article{COCV_2019__25__A6_0,
     author = {Chambrion, Thomas and Giraldi, Laetitia and Munnier, Alexandre},
     title = {Optimal strokes for driftless swimmers: {A} general geometric approach},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {25},
     year = {2019},
     doi = {10.1051/cocv/2017012},
     zbl = {1434.74047},
     mrnumber = {3943360},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2017012/}
}
TY  - JOUR
AU  - Chambrion, Thomas
AU  - Giraldi, Laetitia
AU  - Munnier, Alexandre
TI  - Optimal strokes for driftless swimmers: A general geometric approach
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2019
VL  - 25
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2017012/
DO  - 10.1051/cocv/2017012
LA  - en
ID  - COCV_2019__25__A6_0
ER  - 
%0 Journal Article
%A Chambrion, Thomas
%A Giraldi, Laetitia
%A Munnier, Alexandre
%T Optimal strokes for driftless swimmers: A general geometric approach
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2019
%V 25
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2017012/
%R 10.1051/cocv/2017012
%G en
%F COCV_2019__25__A6_0
Chambrion, Thomas; Giraldi, Laetitia; Munnier, Alexandre. Optimal strokes for driftless swimmers: A general geometric approach. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 6. doi : 10.1051/cocv/2017012. http://www.numdam.org/articles/10.1051/cocv/2017012/

[1] A.A. Agrachev, Non Linear and Optimal Control Theory. Springer Verlag (2008). | DOI | MR

[2] A.A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannan and Sub-Riemannian Geometry. Preprint SISSA 09/2012/M (2012).

[3] F. Alouges, A. Desimone, L. Heltai, A. Lefebvre and B. Merlet, Optimally swimming Stokesian robots. Discrete Contin. Dyn. Syst. Ser. B 18 (2013) 1189–1215. | MR | Zbl

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

[5] F. Alouges and L. Giraldi, Enhanced controllability of low Reynolds number swimmers in the presence of a wall. Acta Appl. Math. 128 (2013) 153–179. | DOI | MR | Zbl

[6] T. Chambrion and A. Munnier, Locomotion and control of a self-propelled shape-changing body in a fluid. J. Nonlinear Sci. 21 (2011) 325–385. | DOI | MR | Zbl

[7] T. Chambrion and A. Munnier, Generic controllability of 3D swimmers in a perfect fluid. SIAM J. Control Optim. 50 (2012) 2814–2835. | DOI | MR | Zbl

[8] G. Dal Maso, A. Desimone and M. Morandotti, An existence and uniqueness result for the motion of self-propelled micro-swimmers. SIAM J. Math. Anal. 43 (2011) 1345–1368. | DOI | MR | Zbl

[9] D. Gérard-Varet and L. Giraldi, Rough Wall Effect on Microswimmers. Preprint (2013). | arXiv | Numdam | MR

[10] J. Lighthill, Mathematical biofluiddynamic. Society for Industrial and Applied Mathematics. Philadelphia, PA (1975). | DOI | MR | Zbl

[11] M.J. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Math. 5 (1952) 109–118. | DOI | MR | Zbl

[12] J. Lohéac and A. Munnier, Controllability of 3D low Reynolds swimmers. ESAIM: COCV 20 (2014) 236–268. | Numdam | MR | Zbl

[13] J. Lohéac, J.F. Scheid and M. Tucsnak, Controllability and time optimal control for low Reynolds numbers swimmers. Acta Appl. Math. 123 (2013) 175–200. | DOI | MR | Zbl

[14] A. Munnier, On the self-displacement of deformable bodies in a potential fluid flow. Math. Model. Methods Appl. Sci. 18 (2008) 1945–1981. | DOI | MR | Zbl

[15] A. Munnier, Passive and self-propelled locomotion of an elastic swimmer in a perfect fluid. SIAM J. Appl. Dyn. Syst. 10 (2011) 1363–1403. | DOI | MR | Zbl

[16] E.M. Purcell, Life at low Reynolds number. Am. J. Phys. 45 (1977) 3–11. | DOI

[17] G. Taylor, Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209 (1951) 447–461. | DOI | MR | Zbl

[18] E.F. Whittlesey, Analytic functions in Banach spaces. Proc. Am. Math. Soc. 16 (1965) 1077–1083. | DOI | MR | Zbl

[19] T.Y. Wu, Mathematical biofluiddynamic and mechanophysiology of fish locotion. Math. Method Appl. Sci. 24 (2001) 1541–1564. | DOI | MR | Zbl

Cité par Sources :