In Takagi [Phys. Rev. E 92 (2015) 023020], based on copepod observations, Takagi proposed a model to interpret the swimming behaviour of these microorganisms using sinusoidal paddling or sequential paddling followed by a recovery stroke in unison, and compares them invoking the concept of efficiency. Our aim is to provide an interpretation of Takagi’s results in the frame of optimal control theory and sub-Riemannian geometry. The maximum principle is used to select two types of periodic control candidates as minimizers: sinusoidal up to time reparameterization and the sequential paddling, interpreted as an abnormal stroke in sub-Riemannian geometry. Geometric analysis combined with numerical simulations are decisive tools to compute the optimal solutions, refining Takagi computations. A family of simple strokes with small amplitudes emanating from a center is characterized as an invariant of SR-geometry and allows to identify the metric used by the swimmer. The notion of efficiency is discussed in detail and related with normality properties of minimizers.
Accepté le :
DOI : 10.1051/cocv/2017071
Mots-clés : Stokes flow, optimal control theory, sub-Riemannian geometry, abnormal closed geodesics
@article{COCV_2019__25__A9_0,
author = {Bettiol, P. and Bonnard, B. and Nolot, A. and Rouot, J.},
title = {Sub-Riemannian geometry and swimming at low {Reynolds} number: the {Copepod} case},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2017071},
zbl = {1434.70061},
mrnumber = {3943362},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2017071/}
}
TY - JOUR AU - Bettiol, P. AU - Bonnard, B. AU - Nolot, A. AU - Rouot, J. TI - Sub-Riemannian geometry and swimming at low Reynolds number: the Copepod case JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2017071/ DO - 10.1051/cocv/2017071 LA - en ID - COCV_2019__25__A9_0 ER -
%0 Journal Article %A Bettiol, P. %A Bonnard, B. %A Nolot, A. %A Rouot, J. %T Sub-Riemannian geometry and swimming at low Reynolds number: the Copepod case %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2017071/ %R 10.1051/cocv/2017071 %G en %F COCV_2019__25__A9_0
Bettiol, P.; Bonnard, B.; Nolot, A.; Rouot, J. Sub-Riemannian geometry and swimming at low Reynolds number: the Copepod case. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 9. doi : 10.1051/cocv/2017071. http://www.numdam.org/articles/10.1051/cocv/2017071/
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