In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done by means of arbitrarily small body deformations that can be superimposed to any preassigned sequence of macro shape changes. Our second contribution is to prove that, when no macro deformations are prescribed, tracking is generically possible by means of shape changes obtained as a suitable combination of only four elementary deformations. Eventually, still considering finite dimensional deformations, we state results about the existence of optimal swimming strategies on short time intervals, for a wide class of cost functionals.

Keywords: locomotion, biomechanics, stokes fluid, geometric control theory

@article{COCV_2014__20_1_236_0, author = {Loh\'eac, J\'er\^ome and Munnier, Alexandre}, title = {Controllability of {3D} low {Reynolds} number swimmers}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {236--268}, publisher = {EDP-Sciences}, volume = {20}, number = {1}, year = {2014}, doi = {10.1051/cocv/2013063}, mrnumber = {3182699}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2013063/} }

TY - JOUR AU - Lohéac, Jérôme AU - Munnier, Alexandre TI - Controllability of 3D low Reynolds number swimmers JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 236 EP - 268 VL - 20 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2013063/ DO - 10.1051/cocv/2013063 LA - en ID - COCV_2014__20_1_236_0 ER -

%0 Journal Article %A Lohéac, Jérôme %A Munnier, Alexandre %T Controllability of 3D low Reynolds number swimmers %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 236-268 %V 20 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2013063/ %R 10.1051/cocv/2013063 %G en %F COCV_2014__20_1_236_0

Lohéac, Jérôme; Munnier, Alexandre. Controllability of 3D low Reynolds number swimmers. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 1, pp. 236-268. doi : 10.1051/cocv/2013063. http://archive.numdam.org/articles/10.1051/cocv/2013063/

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