. Dans-le-quatrième-chapitre, Quel modèle existe t-il pour les algèbres de Lie associées ? Est ce que le système adjoint de Lie-Poisson associé est intégrable ? Si oui, de quel type sont les solutions ? trigonométriques ? elliptiques ? ou autres ? Nous avons étudié l'espace sous-riemannien symétrique SO(5)/ SO(3) et la SR-algèbre de Lie correspondante (où g = so(5)) et les dicultés se manifestent dans les deux points ; d'un coté nous n'avons pas réussit à exprimer les solutions du système de Lie-Poisson en terme de fonctions elliptiques, nous pouvons nous intéresser aux autres espaces sous-riemanniens symétriques classiés dans, p.178

]. M. Bibliographie, P. Adler, and . Van-moerbeke, The algebraic integrability of geodesic ow on so(4), Invent. Math, vol.67, issue.1, p.297326, 1982.

A. Agrachev, D. Barilari, and U. Boscain, On the Hausdor volume in sub- Riemannian geometry. Accepté dans Calculus of Variations and PDE's, 526, 2011.

A. Agrachev and J. P. Gauthier, Sub-Riemannian metrics and isoperimetric problems in the contact case, honor of L. Pontriaguine, 90 th birthday commemoration, p.548, 1999.

A. Agrachev and J. P. Gauthier, Sub-Riemannian metrics and isoperimetric problems, Journal of Mathematical Sciences, vol.103, issue.6, p.639663, 2001.

A. Agrachev, J. P. Gauthier, and H. Chakir, Subriemannian metrics on R 3 . Geometric control and nonholonomic mechanics, Proceedings of Canad, p.2976, 1998.

A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, Local invariants of smooth control systems, Acta Applicandae Mathematicae, vol.16, issue.155, 1989.
DOI : 10.1007/BF01307214

A. Agrachev and Y. L. Sachkov, Lectures on Geometric Control Theory, 2004.

A. Anzaldo-meneses and F. Monroy-pérez, Optimal control in the Heisenberg group, J. Dynam. Control Systems, vol.5, p.473499, 1999.

V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol.60, 1989.

V. I. Arnold and S. P. Novikov, Dynamical Systems IV : Symplectic Geometry and its Applications, 2001.
DOI : 10.1007/978-3-662-06791-8

M. Audin, Symplectic Geometry of Integrable Hamiltonian Systems, Birkhäuser, 2003.
DOI : 10.1007/978-3-0348-8071-8

A. Bellaiche, The tangent space in sub-Rimannian geometry, Sub- Riemannian Geometry, p.484, 1996.

A. Bellaiche and J. Risler, Sub-Riemannian Geometry, Birkhäuser, 1996.
DOI : 10.1007/978-3-0348-9210-0

R. M. Bianchini and G. Stefani, Controllability Along a Trajectory: A Variational Approach, SIAM Journal on Control and Optimization, vol.31, issue.4, p.900927, 1993.
DOI : 10.1137/0331039

A. M. Bloch, Nonholonomic Mechanics and Control, 2003.
DOI : 10.1007/b97376

A. V. Bocharov and A. M. Verbovetsky, Symmetries and Conservation Laws for Dierential Equations of Mathematical Physics, 1999.

B. Bonnard, M. Chyba, and E. Trélat, Sub-Riemannian geometry : oneparameter deformation of the Martinet at case, J. Dyn. Control Systems, vol.4, issue.1, p.5976, 1998.

U. Boscain and F. Rossi, Invariant Carnot???Caratheodory Metrics on $S^3$, $SO(3)$, $SL(2)$, and Lens Spaces, SIAM Journal on Control and Optimization, vol.47, issue.4, p.18511878, 2008.
DOI : 10.1137/070703727

T. Bountis, M. Bier, and J. Hijmans, On the integrability of some generalized Lotka-Volterra systems, Physics Letters A, vol.97, issue.1-2, p.1114, 1983.
DOI : 10.1016/0375-9601(83)90088-9

J. B. Boyling, An Axiomatic Approach to Classical Thermodynamics, Proc. R. Soc, p.3570, 1972.
DOI : 10.1098/rspa.1972.0100

R. Brockett and L. Dai, Nonholonomic kinematics and the role of elliptic functions in constructive controllability, Nonholonomic Motion Planning, p.121, 1993.

]. R. Brockett, Control Theory and Singular Riemannian Geometry
DOI : 10.1007/978-1-4612-5651-9_2

O. Calin and D. C. Chang, Sub-Riemannian Geometry : General Theory and Examples, 2009.
DOI : 10.1017/CBO9781139195966

W. L. Chow, Über system von linearen partiellen dierentialgleichungen erster ord-nung, Math. Ann, vol.117, p.98105, 1939.

P. Eberlein, Geometry of $2$-step nilpotent groups with a left invariant metric, Annales scientifiques de l'??cole normale sup??rieure, vol.27, issue.5
DOI : 10.24033/asens.1702

P. Eberlein, Geometry of $2$-step nilpotent groups with a left invariant metric, Annales scientifiques de l'??cole normale sup??rieure, vol.27, issue.5, p.805828, 1994.
DOI : 10.24033/asens.1702

R. Assoudi and A. Gerber, Geodesic ow on 4-dimensional subhomogeneous spaces, integrability and elliptic functions

L. Euler, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimitrici Latissimo Sensu Accepti, 1744.

E. Falbel and C. Gorodski, On contact sub-riemannian symmetric spaces, Annales scientifiques de l'??cole normale sup??rieure, vol.28, issue.5
DOI : 10.24033/asens.1726

E. Falbel, C. Gorodski, and P. Bieliavsky, The classication of simplyconnected contact sub-Riemannian symmetric spaces, Pacic Journal of Mathematics, vol.188, p.6582, 1999.

E. Falbel, C. Gorodski, and M. Rumin, Holonomy of Sub-Riemannian Manifolds, International Journal of Mathematics, vol.08, issue.03, p.317344, 1997.
DOI : 10.1142/S0129167X97000159

F. R. Gantmakher, The Theory of Matrices, 1977.

J. P. Gauthier, Nonholonomic Interpolation for Kinematic Problems, Entropy and Complexity, Mathematical Control Theory and Finance, issue.5, p.48187210, 2008.
DOI : 10.1007/978-3-540-69532-5_11

J. P. Gauthier, U. Boscain, and T. Chambrion, On the K+P problem for a three-level quantum system : Optimality implies resonance, Journal of Dynamical and Control Systems, vol.8, p.547572, 2002.

J. P. Gauthier, U. Boscain, and T. Chambrion, Optimal control on a nlevel quantum system, Lagrangian and Hamiltonian Methods for Nonlinear Control, p.129134, 2003.

J. P. Gauthier, U. Boscain, G. Charlot, S. Guérin, and H. Jauslin, Optimal control in laser-induced population transfer for two and three level quantum systems, Journal of Mathematical Physics, vol.43, p.21072132, 2002.

J. P. Gauthier, B. Jakubczyk, and V. Zakalyukin, Motion Planning and Fastly Oscillating Controls, SIAM Journal on Control and Optimization, vol.48, issue.5, p.34333448, 2010.
DOI : 10.1137/090761884

J. P. Gauthier and V. Zakalyukin, On the Motion Planning Problem, Complexity, Entropy, and Nonholonomic Interpolation, Journal of Dynamical and Control Systems, vol.91, issue.8, p.34333448, 2006.
DOI : 10.1007/s10450-006-0005-y

B. Gaveau, Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents, Acta Mathematica, vol.139, issue.0, p.94153, 1977.
DOI : 10.1007/BF02392235

N. Goodman, Nilpotent Lie groups, 1976.
DOI : 10.1007/BFb0087594

V. N. Gorbuzov and A. F. Pranevich, First integrals of linear dierential systems, Classical Analysis and ODEs, Mathematics -Dynamical Systems, 2008.

A. Goriely, Integrability and Nonintegrability of Dynamical Systems Advanced series in nonlinear dynamics, 2001.

B. Grammaticos, J. Moulin-ollagnier, A. Ramani, J. M. Strelcyn, and S. Wojciechowski, Integrals of quadratic ordinary dierential equations in R 3 : The Lokta-Volterra system, Physica A, vol.163, p.8, 1990.

M. Gromov, Carnot-Carath??odory spaces seen from within, Sub- Riemannian Geometry, p.85320, 1996.
DOI : 10.1007/978-3-0348-9210-0_2

URL : http://cds.cern.ch/record/263332/files/P00023376.pdf

S. Helgason, Dierential Geometry, Lie Groups, and Symmetric Spaces, 1978.
DOI : 10.1090/gsm/034

R. Hermann, Dierential Geometry and the Calculus of Variations, 1977.

H. Hermes, Control systems which generate decomposible Lie algebras, J. Di. Eqns, vol.44, p.166187, 1982.

H. Hermes, Nilpotent Approximations of Control Systems and Distributions, SIAM Journal on Control and Optimization, vol.24, issue.4
DOI : 10.1137/0324045

F. Jean, The car with n trailers : Characterisation of the singular congurations . ESAIM : Control, Optimisation and Calculus of Variations, pp.241-266, 1996.

F. Jean, Entropy and complexity of a path in sub-Riemannian geometry, ESAIM: Control, Optimisation and Calculus of Variations, vol.9
DOI : 10.1051/cocv:2003024

URL : https://hal.archives-ouvertes.fr/hal-00849556

F. Jean, Sub-Riemannian geometry Technical report, Lectures given at the Trimester on Dynamical and Control Systems, 2003.

V. Jurdjevic, Geometric Control Theory, 1997.
DOI : 10.1017/CBO9780511530036

V. Jurdjevic, Hamiltonian point of view of non-Euclidean geometry and elliptic functions, Systems & Control Letters, vol.43, issue.1, p.2541, 2001.
DOI : 10.1016/S0167-6911(01)00093-7

V. Jurdjevic, Integrable Hamiltonian Systems on Complex Lie Groups. Memoirs of the, 2005.

M. Kawski, Nonlinear control and combinatorics of words, 1998.

N. Khaneja, S. J. Glaser, and R. Brockett, Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer, Physical Review A, vol.65, issue.3, pp.71-039906, 2005.
DOI : 10.1103/PhysRevA.65.032301

A. Krener, Bilinear and Nonlinear Realizations of Input-Output Maps, SIAM Journal on Control, vol.13, issue.4, p.827834, 1975.
DOI : 10.1137/0313049

I. Kupka, Géométrie sous-riemannienne. Séminaire N. Bourbaky, exp. N817, pp.351380-1995

J. P. Laumond, Controllability of a multibody robot, IEEE Transactions Robotics and Automation, vol.9, p.755763, 1991.

J. P. Laumond, Robot Motion Planning and Control, Lecture Notes on Control and Information Sciences, vol.229, 1997.
DOI : 10.1007/BFb0036069

E. and L. Donne, Lecture notes on sub-Riemannian geometry, Lectures Notes, 2009.

W. Liu and H. Sussmann, Shortest paths for sub-Riemannian metrics on rank-two distributions, Memoirs of the American Mathematical Society, vol.118, issue.564, 1995.
DOI : 10.1090/memo/0564

A. Maciejewski, W. Pasillas-lepine, W. Respondek, and J. Strelcyn, Integrability properties of the geodesic ow for 3-dimensional sub-Riemannian homogeneous spaces

R. Mneimné and F. Testard, Introduction à la Théorie des Groupes de Lie Classiques. Hermann, éditeurs des sciences et des arts, 1986.

R. Montgomery, A tour of Subriemannian Geometries, Their Geodesics and Applications, 2002.
DOI : 10.1090/surv/091

R. Montgomery, M. Shapiro, and A. Stolin, A nonintegrable subriemannian geodesic ow on a Carnot group, Journal of dynamical and control systems, vol.3, p.519530, 1997.

J. Moulin-ollagnier, Polynomial rst integrals of the Lotka-Volterra system, Bull. Sci. math, vol.121, p.463476, 1997.

J. Moulin-ollagnier, Rational integration of the Lotka???Volterra system, Bulletin des Sciences Math??matiques, vol.123, issue.6, p.437466, 1999.
DOI : 10.1016/S0007-4497(99)00111-6

J. Moulin-ollagnier and A. Nowicki, Polynomial rings of constants of the Lotka-Volterra system, Colloquium Mathematicum, vol.81, p.263270, 1999.

R. M. Murray, Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholomic systems, Math. Control Signals Systems, vol.7, p.5875, 1994.

A. Nowicki, On the nonexistence of rational rst integrals for systems of linear dierential equations. Linear algebra and its applications, p.107120, 1996.

A. L. Onishchik and E. B. Vinberg, Lie groups and Lie algebras III : structure of Lie groups and Lie algebras, 1994.
DOI : 10.1007/978-3-662-03066-0

L. S. Pontryagin, V. Boltyanskii, R. Gamkrelidze, and E. Mischenko, The Mathematical Theory of Optimal Processes, 1964.

P. K. Rashevsky, About connecting two points of a completely nonholonomic space by admissible curve, Uch. Zapiski Ped. Inst. Libknechta, vol.2, p.8394, 1938.

L. P. Rothschild and E. M. Stein, Hypoelliptic dierential operators and nilpotent groups, Acta Math, vol.137, p.247320, 1976.
DOI : 10.1007/bf02392419

Y. L. Sachkov, Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane ESAIM : Control, Optimisation and Calculus of Variations, p.10181039, 2010.

Y. L. Sachkov, Symmetries of at rank two distributions and sub-Riemannian structures. Transactions of the American mathematical society, pp.457-494, 2003.

Y. L. Sachkov, Optimality of Euler elasticae, Doklady Mathematics, vol.417, pp.23-25, 2007.

Y. L. Sachkov, Conjugate points in Euler's elastic problem, Journal of Dynamical and Control Systems, vol.14, p.409439, 2008.

Y. L. Sachkov, Maxwell strata in Euler's elastic problem, Journal of Dynamical and Control Systems, vol.14, p.169234, 2008.

J. P. Serre, Lie Algebras and Lie Groups, 1964.

C. L. Siegel, Topics in Complex Function Theory I, Elliptic Functions and Uniformization Theory, 1962.

J. Strelcyn and S. Wojciechowski, A method of nding integrals of 3- dimensional dynamical systems, Phys. Letters, vol.133, 1988.

R. S. Strichartz, Sub-Riemannian geometry, Journal of Differential Geometry, vol.24, issue.2, p.221263, 1986.
DOI : 10.4310/jdg/1214440436

R. S. Strichartz, Corrections to: ``Sub-Riemannian geometry'', Journal of Differential Geometry, vol.30, issue.2, p.595596, 1989.
DOI : 10.4310/jdg/1214443604

E. Trélat, Contrôle Optimal, Théorie et Applications. Vuibert, Mathématiques Concrètes, 2005.

M. Vendittelli, G. Oriolo, F. Jean, and J. Laumond, Nonhomogeneous Nilpotent Approximations for Nonholonomic Systems With Singularities, IEEE Transactions on Automatic Control, vol.49, issue.2, p.261266, 2004.
DOI : 10.1109/TAC.2003.822872

A. M. Vershik, V. Ya, and . Gershkovich, Nonholomic Dynamical Systems, Geometry of Distributions and Variational Problems, 1991.

H. Vogt, Leçons sur la résolution algébrique des équations, Acta Math, vol.30, p.94153, 1977.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 1927.