. For-every-?-?-?-r, ?. The-quantities-?-x-s, T. ). , ?. , T. et al., ? ? ) are either all of them infinite or all of them finite

?. ?. If-s, ?. , T. ). , ?. , T. et al., ? ? ) are all of them finite, for every T ? 0

?. If, ?. , T. ). , ?. , T. et al., (? ? ) are finite, then there exist minimizers or maximizers to every

?. U. Any-minimizer-u, ?. , and T. , (a) is such that u(t) R p is constant at every moment

A. A. Agrachev, Introduction to optimal control theory, Summer School on Mathematical Control Theory, 2001.

A. A. Agrachev, Non linear and optimal control theory, 2008.

A. A. Agrachev, D. Barilari, and U. Boscain, Introduction to Riemannan and sub- Riemannian geometry, p.2012, 2012.

F. Alouges, A. Desimone, L. Giraldi, and M. Zoppello, Self-propulsion of slender microswimmers by curvature control : N-link swimmers, 2013.

F. Alouges, A. Desimone, and L. Heltai, NUMERICAL STRATEGIES FOR STROKE OPTIMIZATION OF AXISYMMETRIC MICROSWIMMERS, Mathematical Models and Methods in Applied Sciences, vol.21, issue.02, 2011.
DOI : 10.1142/S0218202511005088

F. Alouges, A. Desimone, L. Heltai, A. Lefebvre, and B. Merlet, Optimally swimming Stokesian robots. Discrete and Continuous Dynamical Systems Series B, p.2013

F. Alouges, A. Desimone, and A. Lefebvre, Optimal Strokes for Low Reynolds Number Swimmers: An Example, Journal of Nonlinear Science, vol.209, issue.3, pp.277-302, 2008.
DOI : 10.1007/s00332-007-9013-7

F. Alouges, A. Desimone, and A. Lefebvre, Biological fluid dynamics, nonlinear partial differential equations. Encyclopedia of Complexity and Systems Science, 2009.

F. Alouges, A. Desimone, and A. Lefebvre, Optimal strokes for axisymmetric microswimmers, The European Physical Journal E, vol.28, issue.3, pp.279-284, 2009.
DOI : 10.1140/epje/i2008-10406-4

F. Alouges and L. Giraldi, Enhanced Controllability of Low Reynolds Number Swimmers in the Presence of a Wall, Acta Applicandae Mathematicae, vol.80, issue.6, 2013.
DOI : 10.1007/s10440-013-9824-5

URL : https://hal.archives-ouvertes.fr/hal-00735919

P. R. Amestoy, I. S. Duff, J. Koster, and J. Y. Excellent, A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling, SIAM Journal on Matrix Analysis and Applications, vol.23, issue.1, pp.15-41, 2001.
DOI : 10.1137/S0895479899358194

URL : https://hal.archives-ouvertes.fr/hal-00808293

M. Arroyo, D. Milan, L. Heltai, and A. Desimone, Reverse engineering the euglenoid movement, Proc. Nat. Acad. Sciences USA, pp.17874-17879, 2012.

D. Barilari, Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry, Journal of Mathematical Sciences, vol.136, issue.2
DOI : 10.1007/s10958-013-1585-1

URL : https://hal.archives-ouvertes.fr/hal-00672262

L. E. Becker, S. A. Koehler, and H. A. Stone, On self-propulsion of micro-machines at low Reynolds number: Purcells three-link swimmer, Journal of Fluid Mechanics, vol.490, 2003.
DOI : 10.1017/S0022112003005184

H. B. Belgacem, S. Conti, A. Desimone, and S. Müller, Rigorous Bounds for the F??ppl???von K??rm??n Theory of Isotropically Compressed Plates, Journal of Nonlinear Science, vol.10, issue.6, pp.661-683, 2000.
DOI : 10.1007/s003320010007

A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, Hydrodynamic Attraction of Swimming Microorganisms by Surfaces, Physical Review Letters, vol.101, issue.3, 2008.
DOI : 10.1103/PhysRevLett.101.038102

L. V. Berlyand, S. D. Ryan, B. M. Haines, and D. A. Karpeev, A kinetic model for semi-dilute bacterial suspensions, 2012.

L. V. Berlyand, S. D. Ryan, A. Sokolov, and I. S. Aranson, Collective dynamics in semidilute bacterial suspensions, 2013.

J. R. Blake, A note on the image system for a stokeslet in a no-slip boundary, Proc. Gamb, p.303, 1971.
DOI : 10.1017/S0022112070000745

F. Bonnans, P. Martinon, and V. Grélard, Bocop -A collection of examples, pp.2012-8053
URL : https://hal.archives-ouvertes.fr/hal-00726992

M. Bonnivard, ON THE STABILITY OF SELF-PROPELLED BODIES WITH RESPECT TO THEIR SHAPE MOTION, Mathematical Models and Methods in Applied Sciences, 2011.
DOI : 10.1142/S0218202511005179

URL : https://hal.archives-ouvertes.fr/hal-00565408

C. Brennen and H. Winet, Fluid Mechanics of Propulsion by Cilia and Flagella, Annual Review of Fluid Mechanics, vol.9, issue.1, pp.339-398, 1977.
DOI : 10.1146/annurev.fl.09.010177.002011

H. Brenner and J. Happel, Low Reynolds number hydrodynamics : with special applications to particulate media, 1965.

C. J. Brokaw, Bending moments in free-swimming flagella, J. Exp. Biol, vol.53, pp.445-464, 1970.

T. Chambrion and A. Munnier, Locomotion and Control of a Self-Propelled Shape-Changing Body in a Fluid, Journal of Nonlinear Science, vol.468, issue.17???18, pp.325-385, 2011.
DOI : 10.1007/s00332-010-9084-8

URL : https://hal.archives-ouvertes.fr/hal-00422429

T. Chambrion and A. Munnier, Generic Controllability of 3D Swimmers in a Perfect Fluid, SIAM Journal on Control and Optimization, vol.50, issue.5, pp.2814-2835, 2012.
DOI : 10.1137/110828654

URL : https://hal.archives-ouvertes.fr/hal-00579986

J. M. Coron, Control and Nonlinearity, 2007.
DOI : 10.1090/surv/136

S. Court, Probì emes d'interactions entre une structure déformable et un fluide visqueux et incompressible, 2012.

R. G. Cox, The motion of long slender bodies in a viscous fluid Part 1. General theory, Journal of Fluid Mechanics, vol.32, issue.04, pp.791-810, 1970.
DOI : 10.1016/0009-2509(64)85005-3

G. Dal-maso, A. Desimone, and M. Morandotti, An Existence and Uniqueness Result for the Motion of Self-Propelled Microswimmers, SIAM Journal on Mathematical Analysis, vol.43, issue.3, pp.1345-1368, 2011.
DOI : 10.1137/10080083X

P. Degond and S. Motch, Large Scale Dynamics of the Persistent Turning Walker Model of Fish Behavior, Journal of Statistical Physics, vol.75, issue.1/2, 2008.
DOI : 10.1007/s10955-008-9529-8

URL : https://hal.archives-ouvertes.fr/hal-00182431

A. Desimone, Hysteresis and imperfection sensitivity in small ferromagnetic particles, Meccanica, vol.2, issue.5, pp.591-603, 1995.
DOI : 10.1007/BF01557087

A. Desimone and L. Teresi, Elastic energies for nematic elastomers, The European Physical Journal E, vol.29, issue.2, pp.191-204, 2009.
DOI : 10.1140/epje/i2009-10467-9

R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone et al., Microscopic artificial swimmers, Nature, vol.437, issue.7060, pp.862-865, 2005.
DOI : 10.1038/nature04090

URL : https://hal.archives-ouvertes.fr/hal-00015847

B. M. Friedrich, I. H. Riedel-kruse, J. Howard, and F. Jülicher, High-precision tracking of sperm swimming fine structure provides strong test of resistive force theory, Journal of Experimental Biology, vol.213, issue.8, 2010.
DOI : 10.1242/jeb.039800

E. A. Gaffney, H. Gadêlha, D. J. Smith, J. R. Blake, and J. C. Kirkman-brown, Mammalian Sperm Motility: Observation and Theory, Annual Review of Fluid Mechanics, vol.43, issue.1, pp.501-528, 2011.
DOI : 10.1146/annurev-fluid-121108-145442

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stockes Equations, 2011.

G. P. Galdi and M. Kyied, Steady Flow of a Navier???Stokes Liquid Past an Elastic Body, Archive for Rational Mechanics and Analysis, vol.86, issue.2, 2011.
DOI : 10.1007/s00205-009-0224-y

G. P. Galdi and A. L. Sylvestre, On the motion of a rigid body in a Navier-Stokes liquid under the action of a time-periodic force, Indiana University Mathematics Journal, vol.58, issue.6, 2009.
DOI : 10.1512/iumj.2009.58.3758

A. Gebremedhin, A. Pothen, and A. Walther, Exploiting Sparsity in Jacobian Computation via Coloring and Automatic Differentiation: A Case Study in a Simulated Moving Bed Process, Proceedings of the Fifth International Conference on Automatic Differentiation, pp.339-349, 2008.
DOI : 10.1007/978-3-540-68942-3_29

D. , G. Varet, and M. Hillairet, Computation of the drag force on a rough sphere close to a wall, pp.1201-1224, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00573435

L. Giraldi, P. Martinon, and M. Zoppello, Controllability and optimal strokes for N-link micro-swimmer, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00798363

O. Glass and T. Horsin, Approximate Lagrangian controllability for the 2D Euler equation. Application to the control of the shape of vortex patches, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00660790

O. Glass and T. Horsin, Prescribing the motion of a set of particles in a 3D perfect fluid, SIAM Journal on Control and Optimization, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00616872

R. Golestanian and A. Ajdari, Analytic results for the three-sphere swimmer at low Reynolds number, Physical Review E, vol.77, issue.3, p.36308, 2008.
DOI : 10.1103/PhysRevE.77.036308

R. Golestanian and A. Ajdari, Stochastic low Reynolds number swimmers, Journal of Physics: Condensed Matter, vol.21, issue.20, 2009.
DOI : 10.1088/0953-8984/21/20/204104

URL : http://arxiv.org/abs/0901.1624

J. Gray and J. Hancock, The propulsion of sea-urchin spermatozoa, Journal of Experimental Biology, 1955.

R. E. Johnson and C. J. Brokaw, Flagellar hydrodynamics. A comparison between resistive-force theory and slender-body theory, Biophysical Journal, vol.25, issue.1, pp.113-127, 1979.
DOI : 10.1016/S0006-3495(79)85281-9

V. Jurdjevic, Geometric control theory, 1997.
DOI : 10.1017/CBO9780511530036

J. B. Keller and S. I. Rubinow, Swimming of flagellated microorganisms, Biophysical Journal, vol.16, issue.2, pp.151-170, 1976.
DOI : 10.1016/S0006-3495(76)85672-X

E. Lauga and T. Powers, The hydrodynamics of swimming micro-organisms, Rep. Prog. Phys, vol.72, issue.09660, 2009.

A. M. Lishansky and O. Kenneth, Surface tank treading: Propulsion of Purcell???s toroidal swimmer, Physics of Fluids, vol.20, issue.6, 2008.
DOI : 10.1063/1.2939069

J. Lohéac, Contrôle en temps optimal et nagè a bas nombre de Reynolds, 2012.

J. Lohéac and A. Munnier, Controllability of 3D low Reynolds number swimmers, ESAIM: Control, Optimisation and Calculus of Variations, vol.20, issue.1, 2013.
DOI : 10.1051/cocv/2013063

J. Lohéac, J. F. Scheid, and M. Tucsnak, Controllability and Time Optimal Control for Low Reynolds Numbers Swimmers, Acta Applicandae Mathematicae, vol.209, issue.1, 2013.
DOI : 10.1007/s10440-012-9760-9

J. , S. Mart-`-mart-`-in, T. Takahashi, and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms, Quart. Appl. Math, vol.65, issue.3, 2007.

S. Michelin and E. Lauga, Efficiency optimization and symmetry-breaking in a model of ciliary locomotion, Physics of Fluids, vol.22, issue.11, 2010.
DOI : 10.1063/1.3507951

URL : https://hal.archives-ouvertes.fr/hal-01020661

R. Montgomery, A tour of subriemannian geometries, theirs geodesics and applications, 2002.

A. Munnier, Locomotion of Deformable Bodies in an Ideal Fluid: Newtonian versus Lagrangian Formalisms, Journal of Nonlinear Science, vol.468, issue.17???18, pp.665-715, 2009.
DOI : 10.1007/s00332-009-9047-0

URL : https://hal.archives-ouvertes.fr/hal-00276252

A. Munnier, Passive and Self-Propelled Locomotion of an Elastic Swimmer in a Perfect Fluid, SIAM Journal on Applied Dynamical Systems, vol.10, issue.4, pp.1363-1403, 2011.
DOI : 10.1137/100805455

URL : https://hal.archives-ouvertes.fr/hal-00509657

A. Najafi and R. Golestanian, Simple swimmer at low Reynolds number: Three linked spheres, Physical Review E, vol.69, issue.6, p.62901, 2004.
DOI : 10.1103/PhysRevE.69.062901

A. Najafi and R. Zargar, Two-sphere low-Reynolds-number propeller, Physical Review E, vol.81, issue.6, p.81, 2010.
DOI : 10.1103/PhysRevE.81.067301

Y. Or and M. Murray, Dynamics and stability of a class of low Reynolds number swimmers near a wall, Physical Review E, vol.79, issue.4, p.45302, 2009.
DOI : 10.1103/PhysRevE.79.045302

E. Passov and Y. Or, Dynamics of Purcell???s three-link microswimmer with a passive elastic tail, The European Physical Journal E, vol.705, issue.8, pp.1-9, 2012.
DOI : 10.1140/epje/i2012-12078-9

E. M. Purcell, Life at low Reynolds number, American Journal of Physics, vol.45, issue.1, pp.3-11, 1977.
DOI : 10.1119/1.10903

S. H. Rad and A. Najafi, Hydrodynamic interactions of spherical particles in a fluid confined by a rough no-slip wall, Physical Review E, vol.82, issue.3, 2010.
DOI : 10.1103/PhysRevE.82.036305

I. H. Riedel-kruse, A. Hilfinger, J. Howard, and F. Jülicher, How molecular motors shape the flagellar beat, HFSP Journal, vol.1, issue.3, pp.192-208, 2007.
DOI : 10.2976/1.2773861

R. Rikmenspoel, G. Van-herpen, and P. Eijkout, Cinematographic Observations of the Movements of Bull Sperm Cells, Physics in Medicine and Biology, vol.5, issue.2, pp.167-181, 1960.
DOI : 10.1088/0031-9155/5/2/306

L. Rothschild, Non-random Distribution of Bull Spermatozoa in a Drop of Sperm Suspension, Nature, vol.200, issue.4904, 1963.
DOI : 10.1038/200381b0

J. , S. Mart-`-mart-`-in, J. F. Scheid, T. Takahashi, and M. Tucsnak, An initial and boundary value problem modeling of fish-like swimming, Arch. Ration. Mech. Anal, 2008.

J. P. Sauvage, Molecular Machines and Motors, 2001.
DOI : 10.1007/3-540-44421-1

Y. Sawa, K. Urayama, T. Tokigawa, A. Desimone, and L. Teresi, Thermally Driven Giant Bending of Liquid Crystal Elastomer Films with Hybrid Alignment, Macromolecules, vol.43, issue.9, pp.4362-4369, 2010.
DOI : 10.1021/ma1003979

W. J. Shack, C. S. Fray, and T. J. Lardner, Observations on the hydrodynamics and swimming motions of mammalian spermatozoa, Bulletin of Mathematical Biology, vol.4, issue.5-6, pp.555-565, 1974.
DOI : 10.1007/BF02463267

A. Shapere and F. Wilczek, Efficiencies of self-propulsion at low Reynolds number, Journal of Fluid Mechanics, vol.3, issue.-1, 1989.
DOI : 10.1017/S002211207100048X

H. Shum, E. A. Gaffney, and D. J. Smith, Modelling bacterial behaviour close to a no-slip plane boundary: the influence of bacterial geometry, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.70, issue.3, pp.1725-1748, 2010.
DOI : 10.1128/MMBR.00001-06

D. J. Smith and J. R. Blake, Surface accumulation of spermatozoa : a fluid dynamic phenomenon. The mathematical scientist, 2009.

D. J. Smith, E. A. Gaffney, J. R. Blake, and J. C. Kirkman-brown, Human sperm accumulation near surfaces: a simulation study, Journal of Fluid Mechanics, vol.35, pp.289-320, 2009.
DOI : 10.1529/biophysj.105.069401

D. Tam and A. E. Hosoi, Optimal Stroke Patterns for Purcell???s Three-Link Swimmer, Physical Review Letters, vol.98, issue.6, 2007.
DOI : 10.1103/PhysRevLett.98.068105

G. Taylor, Analysis of the swimming of microscopic organisms, Proc. R. Soc. Lond. A, pp.447-461, 1951.

E. Trelat, Contrôle optimal : théorie and applications, Vuibert, Collection Mathématiques Concrètes, 2005.

A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, pp.25-57, 2006.

A. Walther and A. Griewank, Getting Started with ADOL-C, Combinatorial Scientific Computing. Chapman-Hall CRC Computational Science, 2012.
DOI : 10.1201/b11644-8

B. Watson, J. Friend, and L. Yeo, Piezoelectric ultrasonic resonant motor with stator diameter less than 250 µm : the proteus motor, J. Micromech. Microeng, vol.19, 2009.

E. F. Whittlesey, Analytic functions in Banach spaces, Proc. Amer, pp.1077-1083, 1965.
DOI : 10.1090/S0002-9939-1965-0184092-2

E. F. Whittlesey, Analytic functions in Banach spaces, Proc. Amer, pp.1077-1083, 1965.
DOI : 10.1090/S0002-9939-1965-0184092-2

H. Winet, G. S. Bernstein, and J. Head, Observations on the response of human spermatozoa to gravity, boundaries and fluid shear, Reproduction, vol.70, issue.2, 1984.
DOI : 10.1530/jrf.0.0700511

H. Winet, G. S. Bernstein, and J. Head, Spermatozoon tendency to accumulate at walls is strongest mechanical response, J. Androl, 1984.

T. Y. Wu, Mathematical biofluiddynamic and mechanophysiology of fish locotion, Math. Method Applied Sci, 2001.

C. Ybert, C. Barentin, C. Cottin-bizonne, P. Joseph, and L. Bocquet, Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries, Physics of Fluids, vol.19, issue.12, 2007.
DOI : 10.1063/1.2815730

Y. Yekutieli, R. Sagiv-zohar, R. Aharonov, Y. Engel, B. Hochner et al., Dynamic Model of the Octopus Arm. I. Biomechanics of the Octopus Reaching Movement, Journal of Neurophysiology, vol.94, issue.2, pp.1443-1458, 2005.
DOI : 10.1152/jn.00684.2004

A. P. Yundt, W. J. Shack, and T. J. Lardner, Applicability of hydrodynamic analyses of spermatozoan motion, J. Exp. Biol, vol.62, pp.27-41, 1975.

R. Zargar, A. Najafi, and M. Miri, Three-sphere low-Reynolds-number swimmer near a wall, Physical Review E, vol.80, issue.2, p.26308, 2009.
DOI : 10.1103/PhysRevE.80.026308

S. Zhang, Y. Or, and M. Murray, Experimental demonstration of the dynamics and stability of a low Reynolds number swimmer near a plane wall, Proceedings of the 2010 American Control Conference, 2010.
DOI : 10.1109/ACC.2010.5530846

. Nous-montrons-qu-'un-nageur-qui-est-contrôlable-lorsqu, espace non borné, reste " presque partout " localement contrôlable lorsqu'il nage dans un domaine délimité par un mur plat ou rugueux Au contraire, nous prouvons qu'un nageur qui n'est pas capable d'atteindre toutes les directions lorsqu'il se déplace dans un domaine sans bord peutélargirpeutélargir ses directions accessibles en présence d'un mur (plat ou rugueux) Enfin, ladernì ere partie de la thèse fournit un cadrè a l'´ etude deprobì emes de contrôle optimal associés aux déplacements de nageurs ayant une dynamique sans dérive. Tout d'abord, nousétudionsnousétudions les propriétés mathématiques de plusieursprobì emes de contrôle optimal ayant des coûts fonctionnels différents (existence puis comportement). Ensuite, nous considérons les nageurs ayant deux degrés de liberté. Pour ces modèles particuliers de nageurs, nous présentons un cadre permettant d'en déduire des propriétés géométriques locales pour les solutions de certainsprobì emes de contrôle optimal