In this paper we study a mathematical model of one-dimensional swimmers performing a planar motion while fully immersed in a viscous fluid. The swimmers are assumed to be of small size, and all inertial effects are neglected. Hydrodynamic interactions are treated in a simplified way, using the local drag approximation of resistive force theory. We prove existence and uniqueness of the solution of the equations of motion driven by shape changes of the swimmer. Moreover, we prove a controllability result showing that given any pair of initial and final states, there exists a history of shape changes such that the resulting motion takes the swimmer from the initial to the final state. We give a constructive proof, based on the composition of elementary maneuvers (straightening and its inverse, rotation, translation), each of which represents the solution of an interesting motion planning problem. Finally, we prove the existence of solutions for the optimal control problem of finding, among the histories of shape changes taking the swimmer from an initial to a final state, the one of minimal energetic cost.
DOI : 10.1051/cocv/2014023
Keywords: Motion in viscous fluids, fluid-solid interaction, micro-swimmers, resistive force theory, controllability, optimal control
Maso, Gianni Dal 1 ; DeSimone, Antonio 1 ; Morandotti, Marco 2
@article{COCV_2015__21_1_190_0,
author = {Maso, Gianni Dal and DeSimone, Antonio and Morandotti, Marco},
title = {One-dimensional swimmers in viscous fluids: dynamics, controllability, and existence of optimal controls},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {190--216},
year = {2015},
publisher = {EDP-Sciences},
volume = {21},
number = {1},
doi = {10.1051/cocv/2014023},
zbl = {1308.76348},
mrnumber = {3348420},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2014023/}
}
TY - JOUR AU - Maso, Gianni Dal AU - DeSimone, Antonio AU - Morandotti, Marco TI - One-dimensional swimmers in viscous fluids: dynamics, controllability, and existence of optimal controls JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 190 EP - 216 VL - 21 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2014023/ DO - 10.1051/cocv/2014023 LA - en ID - COCV_2015__21_1_190_0 ER -
%0 Journal Article %A Maso, Gianni Dal %A DeSimone, Antonio %A Morandotti, Marco %T One-dimensional swimmers in viscous fluids: dynamics, controllability, and existence of optimal controls %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 190-216 %V 21 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2014023/ %R 10.1051/cocv/2014023 %G en %F COCV_2015__21_1_190_0
Maso, Gianni Dal; DeSimone, Antonio; Morandotti, Marco. One-dimensional swimmers in viscous fluids: dynamics, controllability, and existence of optimal controls. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 190-216. doi: 10.1051/cocv/2014023
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