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
Mots-clés : Motion in viscous fluids, fluid-solid interaction, micro-swimmers, resistive force theory, controllability, optimal control
@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},
publisher = {EDP-Sciences},
volume = {21},
number = {1},
year = {2015},
doi = {10.1051/cocv/2014023},
zbl = {1308.76348},
mrnumber = {3348420},
language = {en},
url = {http://archive.numdam.org/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://archive.numdam.org/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://archive.numdam.org/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. http://archive.numdam.org/articles/10.1051/cocv/2014023/
, and , Numerical strategies for stroke optimization of axisymmetric microswimmers. Math. Model. Meth. Appl. Sci. 21 (2011) 361–397. | DOI | MR | Zbl
, and , Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18 (2008) 277–302. | DOI | MR | Zbl
, and , Optimal strokes for axisymmetric microswimmers. Eur. Phys. J. E 28 (2009) 279–284. | DOI
, , and , Reverse engineering the euglenoid movement. Proc Natl. Acad. Sci. 109 (2012) 17874–17879. | DOI
, Impulsive control of Lagrangian systems and locomotion in fluids. Discrete Contin. Dyn. Syst. 20 (2008) 1–35. | DOI | MR | Zbl
G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Res. Notes Math. Longman, Harlow (1989). | MR | Zbl
and , Locomotion and control of a self-propelled shape-changing body in a fluid. J. Nonlinear Sci. 21 (2011) 325–385. | DOI | MR | Zbl
S. Childress, Mechanics of Swimming and Flying. Vol. 2 of Cambridge Stud. Math. Biol. Cambridge University Press, Cambridge (1981). | MR | Zbl
J.-M. Coron, Control and nonlinearity. Vol. 136 of Math. Surv. Monogr. AMS, Providence, RI, USA (2007). | MR | Zbl
, and , An existence and uniqueness result for the motion of self-propelled micro-swimmers. SIAM J. Math. Anal. 43 1345–1368. | DOI | MR | Zbl
A. DeSimone, L. Heltai, F. Alouges and A. Lefebvre-Lepot, Computing optimal strokes for low Reynolds number swimmers, in Natural locomotion in fluids and on surfaces: swimming, flying, and sliding, edited by S. Childress. IMA Vol. Math. Appl. Springer Verlag (2012).
M.P. Do Carmo, Differential Geometry of Curves and Surfaces. Prentice Hall Inc., Upper Saddle River, New Jersey (1976). | Zbl
, , and , High precision tracking of sperm swimming fine structure provides strong test of resistive force theory. J. Exp. Biol. 213 (2010) 1226–1234. | DOI
, On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Ration. Mech. Anal. 148 (1999) 53–88. | DOI | MR | Zbl
and , The propulsion of sea-urchin spermatozoa. J. Exp. Biol. 32 (1955) 802–814. | DOI
J.K. Hale, Ordinary Differential Equations, 2nd edition. Robert E. Krieger Publishing Co., Huntington, NY (1980). | MR | Zbl
J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with special applications to particulate media. Martinus Nijhoff Publishers, The Hague (1983). | Zbl
and , Flagellar hydrodynamics. A comparison between Resistive-Force Theory and Slender-Body Theory. Biophys. J. 25 (1979) 113–127. | DOI
, and , Problems and progress in microswimming. J. Nonlinear Sci. 6 (1996) 507–541. | DOI | MR | Zbl
, , The hydrodynamics of swimming microorganisms. Rep. Progr. Phys. 72 (2009) 9. | DOI | MR
, 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
, Self-propelled micro-swimmers in a Brinkman fluid. J. Biol. Dyn. 6 Iss. sup1 (2012) 88–103. | DOI | MR | Zbl
and , Optimal swimming of flagellated micro-organisms. J. Fluid Mech. 66 (1974) 391–415. | DOI | Zbl
, Life at low Reynolds number. Amer. J. Phys. 45 (1977) 3–11. | DOI
J. San Martín, , , A control theoretic approach to the swimming of microscopic organisms. Quart. Appl. Math. 65 (2007) 405–424. | DOI | MR | Zbl
G.I. Taylor, Analysis of the swimming of microscopic organisms. In vol. 209 of Proc. Roy. Soc. London, Ser. A (1951) 447–461. | MR | Zbl
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