, Ces caractéristiques des mouvements les rendent très plausibles pourêtrepourêtre des mouvements optimaux et en physiologie, ce paradigme d'optimalité est l'une des hypothèses dominantes (pour des explications plus rigoureuses dans le contexte de la physiologie, voir [1, 2]). Par conséquent, le bon cadre mathématique pour les mouvements est le cadre de contrêle optimal, c'est-` a-dire que les mouvements réalisés par le système mécanique minimisent certaines fonctions de coût. Cependant, même si on sait qu'un mouvement est optimal, les critères optimisés sont cachés. Ainsi, pour modéliser les mouvements humains, nous devons d'abord résoudre unprobì eme de contrôle optimal inverse: compte tenu des données des mouvements réalisés et de la dynamique du système mécanique, trouvez la fonction de coût par rapportàrapport`rapportà laquelle les mouvements sont optimaux, c'est-` a-dire, les solutions duprobì eme de contrôle optimal correspondant, Leprobì eme du contrôle optimal inverse fait l'objet d'une attentionparticulì ere au cours desdernì eres décennies. Le regain d'intérêt est dû au nombre croissant d'applications. En particulier dans la modélisation des mouvements humains en physiologie, ce qui a conduitàconduit`conduità une nouvelle approche dans le domaine de la robotique humanoide

, Dans cette perspective, des mouvements différents ontétéontété mis en oeuvre, par exemple la locomotion humaine [5]. Le même schéma est appliqué aux robots censés agir comme des systèmes biologiques autres que l'homme, par exemple un quadrotor se déplacant comme un insecte volant

, Une autre application en robotique concerne les robots autonomes, les voitures autonomes en Bibliography

E. Todorov and M. I. Jordan, Optimal feedback control as a theory of motor coordination, Nat Neurosci, vol.5, pp.1226-1235, 2002.

E. Todorov, Optimal control theory, Bayesian Brain: Probabilistic Approaches to Neural Coding, pp.269-298, 2006.

Y. Chitour, F. Jean, M. , and P. , Optimal Control Models of Goal-oriented Human Locomotion, SIAM Journal on Control and Optimization, vol.50, issue.1, pp.147-170, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00493444

B. Berret, C. Darlot, F. Jean, T. Pozzo, C. Papaxanthis et al., The Inactivation Principle: Mathematical Solutions Minimizing the Absolute Work and Biological Implications for the Planning of Arm Movements, PLOS Computational Biology, vol.4, issue.10, pp.1-25, 2008.
URL : https://hal.archives-ouvertes.fr/inserm-00705805

J. Arechavaleta, H. H. Laumond, and A. Berthoz, An optimality principle governing human walking, IEEE Transactions on Robotics, vol.24, issue.1, pp.5-14, 2008.
DOI : 10.1109/tro.2008.915449
URL : https://hal.archives-ouvertes.fr/tel-00260990

I. A. Faruque, F. T. Muijres, K. M. Macfarlane, A. Kehlenbeck, H. et al., Identification of optimal feedback control rules from microquadrotor and insect flight trajectories, Biological Cybernetics, vol.112, issue.3, pp.165-179, 2018.

M. Kuderer, S. Gulati, and W. Burgard, Learning driving styles for autonomous vehicles from demonstration, IEEE International Conference on Robotics and Automation (ICRA), pp.2641-2646, 2015.

J. Mainprice, R. Hayne, and D. Berenson, Predicting human reaching motion in collaborative tasks using Inverse Optimal Control and iterative re-planning, IEEE International Conference on Robotics and Automation (ICRA), pp.885-892, 2015.

J. Mainprice, R. Hayne, and D. Berenson, Goal Set Inverse Optimal Control and Iterative Replanning for Predicting Human Reaching Motions in Shared Workspaces, IEEE Transactions on Robotics, vol.32, issue.4, pp.897-908, 2016.

T. Nielsen and F. Jensen, Learning a decision maker's utility function from (possibly) inconsistent behavior, Artificial Intelligence, vol.160, pp.53-78, 2004.

A. Ajami, J. Gauthier, T. Maillot, and U. Serres, ESAIM: Control, Optimisation and Calculus of Variations, vol.19, pp.1030-1054, 2013.

R. Freeman and P. Kokotovic, Inverse Optimality in Robust Stabilization, SIAM Journal on Control and Optimization, vol.34, issue.4, pp.1365-1391, 1996.
DOI : 10.1137/s0363012993258732

M. Krstic and Z. Li, Inverse optimal design of input-to-state stabilizing nonlinear controllers, IEEE Transactions on Automatic Control, vol.43, issue.3, pp.336-350, 1998.

K. Sugimoto and Y. Yamamoto, On successive pole assignment by linearquadratic optimal feedbacks, Linear Algebra and its Applications, vol.122, pp.697-723, 1989.
DOI : 10.1016/0024-3795(89)90673-3
URL : https://doi.org/10.1016/0024-3795(89)90673-3

R. E. Kalman, When Is a Linear Control System Optimal?, Journal of Basic Engineering, vol.86, issue.1, pp.51-60, 1964.
DOI : 10.1115/1.3653115

E. Kreindler and A. Jameson, Optimality of linear control systems, IEEE Transactions on Automatic Control, vol.17, issue.3, pp.349-351, 1972.

E. Kreindler and J. K. Hedrick, On equivalence of quadratic loss functions, International Journal of Control, vol.11, issue.2, pp.213-222, 1970.

B. Anderson, The Inverse Problem of Linear Optimal Control, 1966.

T. Fujii and M. Narazaki, A Complete Optimality Condition in the Inverse Problem of Optimal Control, SIAM Journal on Control and Optimization, vol.22, issue.2, pp.327-341, 1984.

K. Sugimoto and Y. Yamamoto, New solution to the inverse regulator problem by the polynomial matrix method, International Journal of Control, vol.45, issue.5, pp.1627-1640, 1987.

H. Kong, G. Goodwin, and M. Seron, A revisit to inverse optimality of linear systems, International Journal of Control, vol.85, issue.10, pp.1506-1514, 2012.

H. Helmholtz, Uber die physikalische Bedeutung des Prinzips der kleinsten Wirkung, J. fur die Reine Angewandte Math, vol.100, pp.137-166, 1887.

A. Mayer, Die Existenz bedingungen eines kinetischen Potentiales, Berich. Verh. Konig. Sach. Gesell. wissen. Leipzig, Math. Phys. Kl, vol.84, pp.519-529, 1896.

I. M. Anderson and T. Duchamp, On the Existence of Global Variational Principles, American Journal of Mathematics, vol.102, issue.5, pp.781-868, 1980.

N. Sonin, Varshavskie Universitetskie Izvestiya, Russian, No. 1-2, 1886, pp.1-68, 2012.

J. Douglas, Solution of the Inverse Problem of the Calculus of Variations, Transactions of the American Mathematical Society, vol.50, issue.1, pp.71-128, 1941.

O. Krupková, The Geometry of Ordinary Variational Equations, Lecture notes in Mathematics, vol.1678, 1997.

D. Saunders, Thirty years of the inverse problem in the calculus of variations, Reports on Mathematical Physics, vol.66, issue.1, pp.43-53, 2010.

V. Volterra, Leçons sur les fonctions de lignes, professéesprofessées`professéesà la Sorbonne en 1912, par Vito Volterra, recueillies et rédigées par Joseph Pérès, 1913.

M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Transl. from Russian, 1959.

E. Tonti, Variational formulation of nonlinear differential equations I, II, Académie royale de Belgique, vol.55, issue.4, pp.262-278, 1969.

D. V. Zenkov, The inverse problem of the calculus of variations. Local and global theory, vol.2, 2015.

I. M. Anderson and J. Thompson, The Inverse problem of the calculus of variations for ordinary differential equations, Memoirs of the American Mathematical Society, vol.98, 1992.

F. Thau, On the inverse optimum control problem for a class of nonlinear autonomous systems, IEEE Transactions on Automatic Control, vol.12, issue.6, pp.674-681, 1967.

P. Moylan and D. B. Anderson, Nonlinear regulator and an inverse optimal control problem, IEEE Transactions on Automatic Control, vol.18, pp.460-465, 1973.
DOI : 10.1109/tac.1973.1100365

J. G. Park and K. Y. Lee, An Inverse Optimal Control Problem and Its Application to the Choice of Performance Index for Economic Stabilization Policy, IEEE Transactions on Systems, Man, and Cybernetics, issue.1, pp.64-76, 1975.

K. Sugimoto and Y. Yamamoto, Solution to the inverse regulator problem for discrete-time systems, International Journal of Control, vol.48, issue.3, pp.1285-1300, 1988.

A. Y. Ng and S. Russell, Algorithms for Inverse Reinforcement Learning, Proc. 17th International Conf. on Machine Learning, pp.663-670, 2000.

N. D. Ratliff, J. A. Bagnell, and M. A. Zinkevich, Maximum Margin Planning, Proceedings of the 23rd International Conference on Machine Learning, ICML '06, pp.729-736, 2006.
DOI : 10.1145/1143844.1143936

J. Casti, On the general inverse problem of optimal control theory, Journal of Optimization Theory and Applications, vol.32, issue.4, pp.491-497, 1980.

E. Pauwels, D. H. Lasserre, and J. , Inverse optimal control with polynomial optimization, IEEE Conference on Decision and Control ( CDC ), pp.5581-5586, 2014.
DOI : 10.1109/cdc.2014.7040262
URL : https://hal.archives-ouvertes.fr/hal-00961722

K. Mombaur, A. Truong, and J. Laumond, From human to humanoid locomotionnan inverse optimal control approach, Autonomous Robots, vol.28, issue.3, pp.369-383, 2010.
DOI : 10.1007/s10514-009-9170-7

A. S. Puydupin-jamin, M. Johnson, and T. Bretl, A convex approach to inverse optimal control and its application to modeling human locomotion, IEEE International Conference on Robotics and Automation, pp.531-536, 2012.
DOI : 10.1109/icra.2012.6225317

B. Berret, E. Chiovetto, F. Nori, P. , and T. , Evidence for Composite Cost Functions in Arm Movement Planning: An Inverse Optimal Control Approach, PLOS Computational Biology, vol.7, issue.10, pp.1-18, 2011.
URL : https://hal.archives-ouvertes.fr/inserm-00704789

J. Gauthier, B. Berret, J. , and F. , A biomechanical inactivation principle, Proceedings of the Steklov Institute of Mathematics, vol.268, pp.93-116, 2010.
DOI : 10.1134/s0081543810010098
URL : https://hal.archives-ouvertes.fr/hal-00974994

F. C. Chittaro, F. Jean, M. , and P. , On Inverse Optimal Control Problems of Human Locomotion: Stability and Robustness of the Minimizers, Journal of Mathematical Sciences, vol.195, issue.3, pp.269-287, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00774720

E. Beltrami, Teoria fondamentale degli spazii di curvatura costante, vol.2, pp.232-255, 1868.
DOI : 10.1007/bf02419615
URL : https://zenodo.org/record/2243105/files/article.pdf

T. Levi-civita, Sulle trasformazioni delle equazioni dinamiche, vol.24, pp.255-300, 1896.
DOI : 10.1007/bf02419530

P. Topalov and V. S. Matveev, Geodesic Equivalence via Integrability, Geometriae Dedicata, vol.96, issue.1, pp.91-115, 2003.

I. Zelenko, On geodesic equivalence of Riemannian metrics and subRiemannian metrics on distributions of corank 1, Journal of Mathematical Sciences, vol.135, issue.4, pp.79-105, 2004.

V. S. Matveev, Three-dimensional manifolds having metrics with the same geodesics, Topology, vol.42, issue.6, pp.1371-1395, 2003.

B. Bonnard, J. Caillau, and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control, vol.13, pp.207-236, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00086380

V. Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics, vol.52, 1997.

A. A. Agrachev and Y. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences, vol.87, 2004.

D. Barilari and F. Boarotto, On the Set of Points of Smoothness for the Value Function of Affine Optimal Control Problems, SIAM Journal on Control and Optimization, vol.56, issue.2, pp.649-671, 2018.

E. Trélat, Contrôle optimal: théorie & applications, Vuibert Paris, 2005.

A. Agrachev, D. Barilari, R. , and L. , Curvature: a variational approach, Memoirs of the AMS
URL : https://hal.archives-ouvertes.fr/hal-00838195

A. A. Agrachev, Any sub-Riemannian metric has points of smoothness, Doklady Mathematics, vol.79, issue.1, pp.45-47, 2009.

A. A. Agrachev and I. Zelenko, Geometry of Jacobi Curves. I, Journal of Dynamical and Control Systems, vol.8, issue.1, pp.93-140, 2002.

I. Zelenko and C. Li, Differential geometry of curves in Lagrange Grassmannians with given Young diagram, Differential Geometry and its Applications, vol.27, pp.723-742, 2009.

F. Nori and R. Frezza, Linear optimal control problems and quadratic cost functions estimation, 12th Mediterranean Conference on Control and Automation, 2004.

A. Agrachev, L. Rizzi, and P. Silveira, On Conjugate Times of LQ Optimal Control Problems, Journal of Dynamical and Control Systems, vol.21, issue.4, pp.625-641, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01096715

A. Ferrante, G. Marro, and L. Ntogramatzidis, A parametrization of the solutions of the finite-horizon LQ problem with general cost and boundary conditions, Automatica, vol.41, issue.8, pp.1359-1366, 2005.

F. Jean, S. Maslovskaya, and I. Zelenko, On projective and affine equivalence of sub-Riemannian metrics, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01683996

F. Jean, S. Maslovskaya, and I. Zelenko, Inverse Optimal Control Problem: the Sub-Riemannian Case, IFAC-PapersOnLine, vol.50, issue.1, pp.500-505, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01452458

G. Hamel, ¨ Uber die Geometrieen, in denen die Geraden die Kürzesten sind, Mathematische Annalen, vol.57, issue.2, pp.231-264, 1903.

U. Dini, Sopra un problema che si presenta nella teoria generale delle rappresentazioni geografiche di una superficie su di un'altra, vol.3, pp.269-293, 1869.

L. P. Eisenhart, Symmetric Tensors of the Second Order Whose First Covariant Derivatives are Zero, Transactions of the American Mathematical Society, vol.25, issue.2, pp.297-306, 1923.

G. De-rham, Sur la réductibilité d'un espace de Riemann, Commentarii Mathematici Helvetici, vol.26, issue.1, pp.328-344, 1952.

S. Benenti, Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation, Journal of Mathematical Physics, vol.38, issue.12, pp.6578-6602, 1997.

S. Benenti, Separability in Riemannian manifolds, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, vol.12, 2016.

I. Zelenko and C. Li, Parametrized curves in Lagrange Grassmannians, Comptes Rendus Mathematique, vol.345, issue.11, pp.647-652, 2007.

B. Kruglikov and V. S. Matveev, The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta, Nonlinearity, vol.29, issue.6, pp.1755-1768, 2016.

A. A. Agrachev and R. V. Gamkrelidze, Feedback-invariant optimal control theory and differential geometryyI. Regular extremals, Journal of Dynamical and Control Systems, vol.3, issue.3, pp.343-389, 1997.

A. Agrachev, D. Barilari, and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry, Lecture notes, 2017.

N. Tanaka, On differential systems, graded Lie algebras and pseudo-groups, Kyoto Journal of Mathematics, vol.10, issue.1, pp.1-82, 1970.

A. Bellä-iche, The tangent space in sub-Riemannian geometry, pp.1-78, 1996.

F. Jean, Control of nonholonomic systems: from sub-Riemannian geometry to motion planning, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01137580

M. A. Gauger, On the Classification of Metabelian Lie Algebras, Transactions of the American Mathematical Society, vol.179, pp.293-329, 1973.

F. Gantmacher, The Theory of Matrices, American Mathematical Society, vol.2, 1959.

H. Weyl, Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffasung, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, vol.1921, pp.99-112, 1921.

R. Montgomery, of Mathematical surveys and monographs, A tour of subriemannian geometries, their geodesics and applications, vol.92, 2002.

R. Bianchini and G. Stefani, Graded Approximations and Controllability Along a Trajectory, SIAM Journal on Control and Optimization, vol.28, issue.4, pp.903-924, 1990.

E. Paoli, Small Time Asymptotics on the Diagonal for Hörmander's Type Hypoelliptic Operators, Journal of Dynamical and Control Systems, vol.23, issue.1, pp.111-143, 2017.

E. C. El-alaoui, J. P. Gauthier, and I. Kupka, Small sub-Riemannian balls on R3, Journal of Dynamical and Control Systems, vol.2, issue.3, pp.359-421, 1996.