A. Agrachev, N. N. Chtcherbakova, and I. Zelenko, On Curvatures and Focal Points of Distributions of Dynamical Lagrangian Distributions and their Reductions by First Integrals, Journal of Dynamical and Control Systems, vol.152, issue.2, pp.297-327, 2005.
DOI : 10.1007/s10883-005-6581-4

A. Agrachev and J. P. Gauthier, On the Dido problem and plane isoperimetric problems, Acta Applicandae Mathematicae, vol.57, issue.3, pp.287-338, 1999.
DOI : 10.1023/A:1006237201915

A. Agrachev and A. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.13, issue.6, pp.635-690, 1996.
DOI : 10.1016/S0294-1449(16)30118-4
URL : https://doi.org/10.1016/s0294-1449(16)30118-4

F. Alouges, A. Desimone, L. Giraldi, and M. Zoppello, Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers, International Journal of Non-Linear Mechanics, vol.56, pp.132-141, 2013.
DOI : 10.1016/j.ijnonlinmec.2013.04.012

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 : 10.1007/978-3-642-61529-0

M. Arcostanzo, M. Arnaud, P. Bolle, and M. Zavidovique, Tonelli Hamiltonians without conjugate points and $$C^0$$ C 0 integrability, Mathematische Zeitschrift, vol.29, issue.6, pp.165-194, 2015.
DOI : 10.1002/cpa.3160290104
URL : https://hal.archives-ouvertes.fr/hal-00865723

V. I. Arnol?d, S. M. Guse?-in-zade, and A. N. Varchenko, Singularities of differentiable maps, 1985.

J. E. Avron, R. , and O. , A geometric theory of swimming: Purcell's swimmer and its symmetrized cousin, New Journal of Physics, vol.10, issue.6, p.63016, 2008.
DOI : 10.1088/1367-2630/10/6/063016

G. K. Batchelor, Slender-body theory for particles of arbitrary cross-section in Stokes flow, Journal of Fluid Mechanics, vol.16, issue.03, pp.419-440, 1970.
DOI : 10.1063/1.1730995

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, pp.15-35, 2003.
DOI : 10.1017/S0022112003005184

A. Bella¨?chebella¨?che, The tangent space in sub-Riemannian geometry, J. Math. Sci, vol.35, pp.461-476, 1997.

M. Berger, La taxonomie des courbes, Pour la science, vol.297, pp.56-63, 2002.

P. Bettiol, B. Bonnard, L. Giraldi, P. Martinon, and J. Rouot, The three links Purcell swimmer and some geometric problems related to periodic optimal controls, Ser. Comp. App. 18, Variational Methods, 2016.
DOI : 10.1515/9783110430394-010

P. Bettiol, B. Bonnard, A. Nolot, and J. Rouot, Sub-Riemannian geometry and swimming at low Reynolds number: the Copepod case, ESAIM: Control, Optimisation and Calculus of Variations, 2017.
DOI : 10.1051/cocv/2017071
URL : https://hal.archives-ouvertes.fr/hal-01442880

P. Bettiol, B. Bonnard, and J. Rouot, Optimal strokes at low Reynolds number: a geometric and numerical study of Copepod and Purcell swimmers, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01326790

F. Bloch, Nuclear induction, pp.7-8, 1946.
DOI : 10.1016/0031-8914(51)90068-7

F. Bonnans, D. Giorgi, S. Maindrault, P. Martinon, and V. Grélard, Bocop -A collection of examples, Inria Research Report, vol.8053, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00726992

B. Bonnard, Feedback Equivalence for Nonlinear Systems and the Time Optimal Control Problem, SIAM Journal on Control and Optimization, vol.29, issue.6, pp.1300-1321, 1991.
DOI : 10.1137/0329067

B. Bonnard, J. Caillau, and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control, ESAIM: Control, Optimisation and Calculus of Variations, vol.6, issue.2, pp.207-236, 2007.
DOI : 10.1051/cocv:2001115
URL : https://hal.archives-ouvertes.fr/hal-00086380

B. Bonnard and M. Chyba, Singular trajectories and their role in control theory, 2003.

B. Bonnard, M. Chyba, A. Jacquemard, and J. Marriott, Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance, Mathematical Control and Related Fields, vol.3, issue.4, pp.397-432, 2013.
DOI : 10.3934/mcrf.2013.3.397
URL : https://hal.archives-ouvertes.fr/hal-00939495

B. Bonnard, M. Chyba, and J. Marriott, Singular Trajectories and the Contrast Imaging Problem in Nuclear Magnetic Resonance, SIAM Journal on Control and Optimization, vol.51, issue.2, pp.1325-1349, 2013.
DOI : 10.1137/110833427
URL : https://hal.archives-ouvertes.fr/hal-00939496

B. Bonnard, M. Claeys, O. Cots, and P. Martinon, Geometric and numerical methods in the contrast imaging problem in nuclear magnetic resonance, Acta Appl. Math, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00867753

B. Bonnard and O. Cots, GEOMETRIC NUMERICAL METHODS AND RESULTS IN THE CONTRAST IMAGING PROBLEM IN NUCLEAR MAGNETIC RESONANCE, Mathematical Models and Methods in Applied Sciences, vol.299, issue.01, pp.187-212, 2014.
DOI : 10.1137/0311048

B. Bonnard, A. Jacquemard, and J. Rouot, Optimal control of an ensemble of Bloch equations with applications in MRI, 2016 IEEE 55th Conference on Decision and Control (CDC), 2016.
DOI : 10.1109/CDC.2016.7798495
URL : https://hal.archives-ouvertes.fr/hal-01287290

B. Bonnard and I. Kupka, Théorie des singularités de l'application entrée/sortie et optimalité des trajectoiressingulì eres dans leprobì eme du temps minimal, Forum Math, vol.5, issue.2, pp.111-159, 1993.

G. A. Bliss, Lectures on the Calculus of Variations, Univ. of Chicago Press, 1946.

R. W. Brockett, Control Theory and Singular Riemannian Geometry, pp.11-27, 1982.
DOI : 10.1007/978-1-4612-5651-9_2

J. Caillau, O. Cots, and J. Gergaud, Differential continuation for regular optimal control problems, Optimization Methods and Software, vol.41, issue.6, pp.177-196, 2012.
DOI : 10.1145/279232.279235

E. Cartan, Les syst??mes de Pfaff, ?? cinq variables et les ??quations aux d??riv??es partielles du second ordre, Annales scientifiques de l'??cole normale sup??rieure, vol.27, pp.109-192, 1910.
DOI : 10.24033/asens.618

T. Chambrion, L. Giraldi, and A. Munnier, Optimal strokes for driftless swimmers: A general geometric approach, ESAIM: Control, Optimisation and Calculus of Variations, 2017.
DOI : 10.1051/cocv/2017012
URL : https://hal.archives-ouvertes.fr/hal-00969259

Y. Chitour, F. Jean, and E. Trélat, Genericity results for singular curves, Journal of Differential Geometry, vol.73, issue.1, pp.45-73, 2006.
DOI : 10.4310/jdg/1146680512
URL : https://hal.archives-ouvertes.fr/hal-00086357

S. Conolly, D. Nishimura, and A. Albert, Optimal Control Solutions to the Magnetic Resonance Selective Excitation Problem, IEEE Transactions on Medical Imaging, vol.5, issue.2, pp.106-115, 1986.
DOI : 10.1109/TMI.1986.4307754

O. Cots, Contrôle optimal géométrique: méthodes homotopiques et applications, 2012.

J. A. Dieudonné and J. B. Carrell, Invariant theory, old and new, Advances in Mathematics, vol.4, issue.1, 1971.
DOI : 10.1016/0001-8708(70)90015-0

R. V. Gamkrelidze, Discovery of the Maximum Principle, J. Dynam. Control Systems, vol.5, issue.4, pp.437-451, 1977.
DOI : 10.1007/3-540-29462-7_5

I. M. Gelfand and S. V. Fomin, Calculus of Variations, 1963.

C. Godbillon, Geométrie différentielle et mécanique analytic, 1969.

J. Gregory, Quadratic form theory and differential equations, Mathematics in Science and Engineering, vol.152, 1980.

G. J. Hancock, The Self-Propulsion of Microscopic Organisms through Liquids, Proc. R. Soc. Lond. A, pp.96-121, 1953.
DOI : 10.1098/rspa.1953.0048

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

D. Henrion and J. Lasserre, GloptiPoly, ACM Transactions on Mathematical Software, vol.29, issue.2, pp.165-194, 2003.
DOI : 10.1145/779359.779363
URL : https://hal.archives-ouvertes.fr/hal-00578944

H. Hermes, Lie Algebras of Vector Fields and Local Approximation of Attainable Sets, SIAM Journal on Control and Optimization, vol.16, issue.5, pp.715-727, 1978.
DOI : 10.1137/0316047

F. Jean, Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning, SpringerBriefs in Mathematics, 2014.
DOI : 10.1007/978-3-319-08690-3
URL : https://hal.archives-ouvertes.fr/hal-01137580

F. John, Partial differential equations, reprint of 4th edition, Applied Mathematical Sciences, vol.1, 1991.

V. Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics, vol.52, 1997.
DOI : 10.1017/CBO9780511530036

W. Klingenberg, Riemannian geometry, de Gruyter Studies in Mathematics, 1982.
DOI : 10.1515/9783110905120

A. J. Krener, The High Order Maximal Principle and Its Application to Singular Extremals, SIAM Journal on Control and Optimization, vol.15, issue.2, pp.256-293, 1977.
DOI : 10.1137/0315019

I. Kupka, Geometric theory of extremals in optimal control problems. i. the fold and Maxwell case, Trans. Amer. Math. Soc, vol.299, issue.1, pp.225-243, 1987.

I. Kupka, Géométrie sous-riemannienne. Astérisque, Séminaire Bourbaki, pp.351-380, 1995.

M. Lapert, Développement de nouvelles techniques de contrôle optimal en dynamique quantique : de la Résonance Magnétique Nucléairè a la physique moléculaire, 2011.

M. Lapert, Y. Zhang, S. J. Glaser, and D. Sugny, Towards the time-optimal control of dissipative spin-1/2 particles in nuclear magnetic resonance, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.44, issue.15, p.15, 2011.
DOI : 10.1088/0953-4075/44/15/154014
URL : https://hal.archives-ouvertes.fr/hal-00642391

M. Lapert, Y. Zhang, M. A. Janich, S. J. Glaser, and D. Sugny, Exploring the Physical Limits of Saturation Contrast in Magnetic Resonance Imaging, Scientific Reports, vol.3, issue.1, 2012.
DOI : 10.1002/mrm.1910030217
URL : https://hal.archives-ouvertes.fr/hal-00750055

J. Lasserre, Moments, positive polynomials and their applications. Imperial College Press Optimization Series, p.361, 2010.
DOI : 10.1142/p665

J. Lasserre, D. Henrion, C. Prieur, and E. Trélat, Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations, SIAM Journal on Control and Optimization, vol.47, issue.4, pp.1643-1666, 2008.
DOI : 10.1137/070685051
URL : https://hal.archives-ouvertes.fr/hal-00136032

E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms, Reports on Progress in Physics, vol.72, issue.9, p.9, 2009.
DOI : 10.1088/0034-4885/72/9/096601

D. F. Lawden, Elliptic functions and applications, Applied Mathematical Sciences, vol.80, p.80, 1989.
DOI : 10.1007/978-1-4757-3980-0

E. B. Lee and L. Markus, Foundations of optimal control theory, 1986.

M. H. Levitt, Spin dynamics: basics of nuclear magnetic resonance, 2001.

J. Li and N. Khaneja, Ensemble Control of Bloch Equations, IEEE Transactions on Automatic Control, vol.54, issue.3, pp.528-536, 2009.
DOI : 10.1109/TAC.2009.2012983

D. Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction, 2011.

M. J. Lighthill, Note on the swimming of slender fish, Journal of Fluid Mechanics, vol.9, issue.02, pp.305-317, 1960.
DOI : 10.1017/S0022112060001110

A. J. Maciejewski and W. Respondek, The nilpotent tangent 3-dimensional sub-Riemannian problem is nonintegrable, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601), 2004.
DOI : 10.1109/CDC.2004.1428669

J. Milnor, Morse theory, Annals of Mathematics Studies, vol.51, 1963.

A. S. Mishchenko, V. E. Shatalov, and B. Y. Sternin, Lagrangian manifolds and the Maslov operator, 1990.
DOI : 10.1007/978-3-642-61259-6

R. Montgomery, Isoholonomic problems and some applications, Communications in Mathematical Physics, vol.65, issue.3, pp.565-592, 1990.
DOI : 10.1007/BF02892134

Y. Or, S. Zhang, and R. M. Murray, Dynamics and Stability of Low-Reynolds-Number Swimming Near a Wall, SIAM Journal on Applied Dynamical Systems, vol.10, issue.3, pp.1013-1041, 2011.
DOI : 10.1137/100808745

E. Passov and Y. Or, Supplementary notes to: Dynamics of Purcells three-link microswimmer with a passive elastic tail, EPJ E, vol.35, pp.1-9, 2012.

L. S. Pontryagin, V. G. Boltyanskii, and R. V. Gamkrelidze, The Mathematical Theory of Optimal Processes, 1962.

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

J. Rouot, P. Bettiol, B. Bonnard, and A. Nolot, Optimal control theory and the efficiency of the swimming mechanism of the Copepod Zooplankton, Proc. 20th IFAC World Congress, 2017.
DOI : 10.1016/j.ifacol.2017.08.100
URL : https://hal.archives-ouvertes.fr/hal-01387423

Y. L. Sachkov, Symmetries of flat rank two distributions and sub-Riemannian structures, Transactions of the American Mathematical Society, vol.356, issue.02, pp.457-494, 2004.
DOI : 10.1090/S0002-9947-03-03342-7

T. E. Skinner, T. Reiss, B. Luy, N. Khaneja, and S. J. Glaser, Application of optimal control theory to the design of broadband excitation pulses for high-resolution NMR, Journal of Magnetic Resonance, vol.163, issue.1, pp.8-15, 2003.
DOI : 10.1016/S1090-7807(03)00153-8

E. D. Sontag, Mathematical control theory Deterministic finite-dimensional systems, second edition, Texts in Applied Mathematics, vol.6, 1998.

H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Transactions of the American Mathematical Society, vol.180, pp.171-188, 1973.
DOI : 10.1090/S0002-9947-1973-0321133-2

H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, Journal of Differential Equations, vol.12, issue.1, pp.95-116, 1972.
DOI : 10.1016/0022-0396(72)90007-1

D. Takagi, Swimming with stiff legs at low Reynolds number, Physical Review E, vol.138, issue.2, p.92, 2015.
DOI : 10.1137/S0036144504445133

R. Vinter, Optimal control. Systems & Control: Foundations & Applications, p.507, 2000.
URL : https://hal.archives-ouvertes.fr/inria-00629428

V. M. Zakaljukin, Lagrangian and Legendre singularities, Funkcional. Anal. i Prilo?en, vol.10, pp.26-36, 1976.

M. Zhitomirski?-i, Typical singularities of differential 1-forms and Pfaffian equations, p.176, 1992.