. Definitions and .. Of-the-results, 118 5.2.1 Definition of the control problem, 118 5.2.2 Previous 1D results, difficulties of the 2D and 3D generalizations . . . . 120 5.2.3 Statement of the main results, p.123

2. Spectral-controllability-in and 3. , 125 5.3.1 Haraux and Jaffard 's result, p.129

.. Generic-spectral-controllability-for-the-quantum-box, 133 5.5.1 Reduction of the problem 133 5.5.2 Proof strategy for the genericity of (B k ), p.140

S. Agmon, A representation theorem for solutions of the helmholtz equation and resolvent estimates for the laplacian Analysis, et cetera : research papers published in honor of Jürgen Moser's 60th birthday, pp.39-76, 1990.

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, pp.31-86, 2004.
DOI : 10.1007/978-3-662-06404-7

J. H. Albert, Genericity of simple eigenvalues for elliptic PDE's. Proceedings of the, pp.413-418, 1975.

E. L. Allgower and K. Georg, Continuation and path following, Acta Numerica, vol.26, issue.87, pp.1-64, 2008.
DOI : 10.1007/BF01390054

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.7009

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, vol.1846, issue.154, p.156, 2004.
DOI : 10.1007/b98245

S. Avdonin and S. Ivanov, Families of exponentials : the method of moments in controllability problems for distributed parameter systems, p.121, 1995.

C. Baiocchi, V. Komornik, and P. Loreti, Ingham-Beurling type theorem with weakened gap conditions, Acta Mathematica Hungarica, vol.97, issue.1/2, pp.55-95, 2002.
DOI : 10.1023/A:1020806811956

J. M. Ball, J. E. Marsden, and M. Slemrod, Controllability for Distributed Bilinear Systems, SIAM Journal on Control and Optimization, vol.20, issue.4, p.114, 1982.
DOI : 10.1137/0320042

URL : http://authors.library.caltech.edu/4635/1/BALsiamjco82.pdf

K. Beauchard, Local controllability of a 1-D Schr??dinger equation, Journal de Math??matiques Pures et Appliqu??es, vol.84, issue.7, pp.851-956, 2005.
DOI : 10.1016/j.matpur.2005.02.005

K. Beauchard and J. Coron, Controllability of a quantum particle in a moving potential well, Journal of Functional Analysis, vol.232, issue.2, pp.328-389, 2006.
DOI : 10.1016/j.jfa.2005.03.021

URL : https://hal.archives-ouvertes.fr/hal-00825517

A. Bellaïche, The tangent space in sub-riemannian geometry, Sub-Riemannian Geometry, pp.1-78, 1996.

A. Bellaïche, F. Jean, and J. Risler, Geometry of nonholonomic systems, Robot Motion Planning and Control, chapter, p.28, 1998.
DOI : 10.1007/BFb0036071

M. Berger and B. Gostiaux, Differential Geometry : Manifolds, Curves, and Surfaces, volume 115 of Graduate Texts in Mathematics, p.97, 1988.
DOI : 10.1007/978-1-4612-1033-7

A. Beurling, Harmonic analysis, volume 2 of The collected works of Arne Beurling, Birkhäuser, p.121, 1989.

A. Bicchi and A. Marigo, Dexterous Grippers: Putting Nonholonomy to Work for Fine Manipulation, The International Journal of Robotics Research, vol.59, issue.7, pp.427-442, 2002.
DOI : 10.1177/027836402321261968

A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, vol.24, issue.8, 2003.
DOI : 10.1007/b97376

J. F. Bonnans, J. Ch, C. Gilbert, C. Lemaréchal, and . Sagastizábal, Numerical Optimization-Theoretical and Practical Aspects, p.58, 2006.

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Mathématiques et Applications SMAI, vol.40, p.93, 2002.

N. Bourbaki, Groupe et Algèbre de Lie, 1972.

R. W. Brockett, Control Theory and Singular Riemannian Geometry, New Directions in Applied Mathematics, 1981.
DOI : 10.1007/978-1-4612-5651-9_2

F. Bullo and A. Lewis, Geometric Control of Mechanical Systems, Texts in Applied Mathematics, vol.49, p.22, 2004.
DOI : 10.1007/978-1-4899-7276-7

F. Bullo, A. Lewis, and N. Leonard, Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups, IEEE Transactions on Automatic Control, vol.45, issue.8, p.26, 2000.
DOI : 10.1109/9.871753

A. Chelouah and Y. Chitour, On the motion planning of rolling surfaces, Forum Math, vol.15, issue.108, pp.727-758, 2003.

Y. Chitour, Path planning on compact Lie groups using a homotopy method, Systems & Control Letters, vol.47, issue.5, pp.383-392, 2002.
DOI : 10.1016/S0167-6911(02)00235-9

Y. Chitour, A continuation method for motion planning problems ESAIM : Control, Optimisation and Calculus of Variations, pp.139-168, 2006.

Y. Chitour, J. Coron, and M. Garavello, On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete and Continuous Dynamical Systems, pp.643-672, 2006.

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

Y. Chitour, F. Jean, and E. Trélat, Singular Trajectories of Control-Affine Systems, SIAM Journal on Control and Optimization, vol.47, issue.2, p.93, 2008.
DOI : 10.1137/060663003

URL : https://hal.archives-ouvertes.fr/hal-00086397

Y. Chitour and H. J. Sussmann, Line-Integral Estimates and Motion Planning Using the Continuation Method, Essays on methematical robotics, pp.91-125, 1998.
DOI : 10.1007/978-1-4612-1710-7_4

W. L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann, vol.117, issue.26, pp.98-115, 1940.
DOI : 10.1142/9789812776921_0005

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences, vol.93, issue.154, p.156, 1998.
DOI : 10.1007/978-1-4614-4942-3

J. Coron, Global asymptotic stabilization for controllable systems without drift, Mathematics of Control, Signals, and Systems, vol.2, issue.3, pp.295-312, 1992.
DOI : 10.1007/BF01211563

J. Coron, On the controllability of 2-D incompressible perfect fluids, Journal de Mathématique Pures et Appliquées, vol.75, issue.3, pp.155-188, 1996.

R. Dautray and J. Lions, Analyse mathématique et calcul scientifique pour les sciences et les techniques, p.161, 1986.

A. Divelbiss and J. Wen, A path space approach to nonholonomic motion planning in the presence of obstacles, IEEE Transactions on Robotics and Automation, vol.13, issue.3, pp.443-451, 1997.
DOI : 10.1109/70.585905

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol.19, p.157, 1999.

M. Fliess, J. Lévine, P. Martin, and P. Rouchon, Flatness and defect of non-linear systems: introductory theory and examples, International Journal of Control, vol.4, issue.6, pp.1327-1361, 1995.
DOI : 10.1109/9.73561

J. Gauthier, B. Jakubczyk, and V. Zakalyukin, Motion Planning and Fastly Oscillating Controls, SIAM Journal on Control and Optimization, vol.48, issue.5, p.22, 2010.
DOI : 10.1137/090761884

N. Goodman, Nilpotent Lie groups, Springer Lecture Notes in Mathematics, vol.562, p.28, 1976.
DOI : 10.1007/BFb0087594

M. Grayson and R. Grossman, 4. Vector Fields and Nilpotent Lie Algebras, Symbolic Computation : Applications to Scientific Computing, pp.77-96, 1989.
DOI : 10.1137/1.9781611971033.ch4

M. Grayson and R. Grossman, Models for free nilpotent Lie algebras, Journal of Algebra, vol.135, issue.1, pp.177-191, 1991.
DOI : 10.1016/0021-8693(90)90156-I

A. Haraux, Séries lacunaires et contôle semi-interne des vibrations d'une plaque rectangulaire, Journal de Mathématique Pures et Appliquées, vol.69, pp.457-465, 1986.

A. Haraux and S. Jaffard, Pointwise and Spectral Control of Plate Vibrations, Revista Matem??tica Iberoamericana, vol.7, issue.125, pp.1-24, 1991.
DOI : 10.4171/RMI/103

A. Henrot and M. Pierre, Variation et optimisation de forme, Mathématiques et Applications SMAI, vol.48, p.136, 2005.
DOI : 10.1007/3-540-37689-5

URL : https://hal.archives-ouvertes.fr/hal-00013871

D. Henry, Perturbation of the boundary in Boundary-Value Problems of Partial Differential Equations, p.134, 2005.
DOI : 10.1017/CBO9780511546730

H. Hermes, Nilpotent and High-Order Approximations of Vector Field Systems, SIAM Review, vol.33, issue.2, pp.238-264, 1991.
DOI : 10.1137/1033050

J. Hiriart-urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithm I, volume 305 of A series of Comprehensive Studies in Mathematics, p.108, 1993.

A. Ingham, Some trigonometrical inequalities with applications to the theory of series, Mathematische Zeitschrift, vol.3, issue.2, pp.367-369, 1936.
DOI : 10.1112/plms/s2-38.1.458

S. Jaffard and S. Micu, Estimates of the constants in generalized Ingham's inequality and applications to the control of the wave equation, Asymptotic Analysis, vol.28, pp.3-4181, 2001.

S. Jaffard, M. Tucsnak, and E. Zuazua, On a theorem of Ingham, The Journal of Fourier Analysis and Applications, vol.41, issue.5, pp.577-582, 1997.
DOI : 10.1007/BF02648885

S. Jaffard, M. Tucsnak, and E. Zuazua, Singular Internal Stabilization of the Wave Equation, Journal of Differential Equations, vol.145, issue.1, pp.184-215, 1998.
DOI : 10.1006/jdeq.1997.3385

F. Jean, Uniform estimation of sub-riemannian balls, Journal of Dynamical and Control Systems, vol.7, issue.4, pp.473-500, 2001.
DOI : 10.1023/A:1013154500463

URL : https://hal.archives-ouvertes.fr/hal-01010757

F. Jean, R. Long, G. Oriolo, and M. Vendittelli, An approximate algorithm for nonholonomic motion planning, p.38, 2008.

F. Jean, G. Oriolo, and M. Vendittelli, A global convergent steering algorithm for regular nonholonomic systems, Proceedings of 44th IEEE Conference on Decision and Control, pp.29-30, 2005.

V. Jurdjevic, Geometric Control Theory, volume 51 of Cambridge Studies in Advanced Mathematics, p.31, 1997.

J. Kahane, Pseudo-périodicité et séries de Fourier lacunaires Annales Scientifiques de l'Ecole Normale Supérieure, pp.93-150, 1962.
DOI : 10.24033/asens.1108

URL : http://archive.numdam.org/article/ASENS_1962_3_79_2_93_0.pdf

T. Kato, Perturbation theory for linear operators, p.160, 1976.

W. Klingenberg, Riemannian geometry, Studies in Mathematics. de Gruyter, vol.1, p.108, 1982.
DOI : 10.1515/9783110905120

V. Komornik and P. Loreti, A further note on a theorem of Ingham and simultaneous observability in critical time, Inverse Problems, vol.20, issue.5, pp.1649-1661, 2004.
DOI : 10.1088/0266-5611/20/5/020

URL : https://hal.archives-ouvertes.fr/hal-00084027

V. Komornik and P. Loreti, Fourier series in control theory, Monographs in Mathematics, p.121, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00086863

W. Krabs, On moment theory and controllability of one-dimensional vibrating systems and heating processes, p.121, 1992.
DOI : 10.1007/BFb0039513

G. Lafferriere, A general strategy for computing steering controls of systems without drift, [1991] Proceedings of the 30th IEEE Conference on Decision and Control, pp.29-63, 1991.
DOI : 10.1109/CDC.1991.261506

G. Lafferriere and H. J. Sussmann, A differential geometry approach to motion planning, Nonholonomic Motion Planning, pp.235-270

N. Leonard and P. S. Krishnaprasad, Motion control of drift-free, left-invariant systems on Lie groups, IEEE Transactions on Automatic Control, vol.40, issue.9, pp.1539-1554, 1995.
DOI : 10.1109/9.412625

B. and Y. Levin, Lecture notes on entire functions, volume 150 of Translations of mathematical monographs, p.129, 1996.

Z. Li and J. Canny, Motion of two rigid bodies with rolling constraint, IEEE Transactions on Robotics and Automation, vol.6, issue.1, pp.62-72, 1990.
DOI : 10.1109/70.88118

J. Lions and E. Magenes, Problèmes aux limites non homogènes et application, volume 1 of Travaux et recherches mathématiques, p.146, 1968.

W. Liu, An Approximation Algorithm for Nonholonomic Systems, SIAM Journal on Control and Optimization, vol.35, issue.4, pp.1328-1365, 1997.
DOI : 10.1137/S0363012993260501

A. Marigo and A. Bicchi, Rolling bodies with regular surface: controllability theory and applications, IEEE Transactions on Automatic Control, vol.45, issue.9, pp.1586-1599, 2000.
DOI : 10.1109/9.880610

P. Martin, R. M. Murray, and P. Rouchon, Flat systems: open problems, infinite dimensional extension, symmetries and catalog, Advances in the Control of Nonlinear Systems, Lecture Notes in Control and Information Sciences, pp.33-57, 2001.
DOI : 10.1007/BFb0110378

P. Martin, R. M. Murray, and P. Rouchon, Flat systems, equivalence and trajectory generation, 2003.
URL : https://hal.archives-ouvertes.fr/cel-00392180

R. Montgomery, Abnormal Minimizers, SIAM Journal on Control and Optimization, vol.32, issue.6, p.93, 1994.
DOI : 10.1137/S0363012993244945

R. Montgomery, A tour of subriemannian geometries, their geodesics, and applications, volume 91 of Mathematical Surveys and Monographs, pp.31-96, 2002.

R. M. Murray, Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems, Mathematics of Control, Signals, and Systems, vol.38, issue.5, pp.58-75, 1994.
DOI : 10.1007/BF01211485

R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC, vol.8, issue.86, pp.12-85, 1994.

R. M. Murray and S. S. Sastry, Nonholonomic motion planning: steering using sinusoids, IEEE Transactions on Automatic Control, vol.38, issue.5, pp.700-716, 1993.
DOI : 10.1109/9.277235

URL : http://authors.library.caltech.edu/7315/1/MURieeetac93.pdf

A. Nachman, Reconstructions From Boundary Measurements, The Annals of Mathematics, vol.128, issue.3, pp.531-576, 1988.
DOI : 10.2307/1971435

J. Nedelec, Acoustic and Electromagnetic Equations -Integral Representations for Harmonic Problems, Applied Mathematical Sciences, vol.144, issue.156, p.157, 2001.

G. Oriolo and M. Vendittelli, A framework for the stabilization of general nonholonomic systems with an application to the plate-ball mechanism, IEEE Transactions on Robotics, vol.21, issue.2, pp.162-175, 2005.
DOI : 10.1109/TRO.2004.839231

J. H. Ortega and E. Zuazua, Generic simplicity of the eigenvalues of the Stokes system in two space dimensions Advances in Differential Equations, pp.987-1023, 2001.

J. H. Ortega and E. Zuazua, On a Constrained Approximate Controllability Problem for the Heat Equation: Addendum, Journal of Optimization Theory and Applications, vol.2, issue.5, pp.183-190, 2003.
DOI : 10.1023/A:1024747710420

P. K. Rashevsky, Any two points of a totally nonholonomic space may be connected by an admissible line, Uch. Zap. Ped. Inst. im. Liebknechta, issue.2, p.26, 1938.

R. M. Redheffer, Remarks on incompleteness of {e i?nt }, non averaging sets, and entire functions, Proceedings of the American Mathematical Society, pp.365-369, 1951.

L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Mathematica, vol.137, issue.0, pp.247-320, 1976.
DOI : 10.1007/BF02392419

P. Rouchon, Control of a quantum particle in a moving potential well, 2nd IFAC Workshop on Lagrangian and Hamiltonian Methods for nonlinear Control, p.114, 2003.

L. Schwartz, Etude des sommes d'exponentielles, Actualités Sci. Ind. Hermann, p.121, 1959.

L. Schwartz, Méthodes mathématiques pour les sciences physiques, Collection Enseignement des Sciences. Hermann, vol.162, p.163, 1998.

J. Simon, Différentiation de problèmes aux limites par rapport au domaine, pp.152-154, 1991.

E. D. Sontag, Control of systems without drift via generic loops, IEEE Transactions on Automatic Control, vol.40, issue.7, pp.1210-1219, 1995.
DOI : 10.1109/9.400490

M. Spivak, A comprehensive introduction to differential geometry III. Publish or Perish, p.88, 1975.

P. Stefan, Accessible Sets, Orbits, and Foliations with Singularities, Proceedings of the London Mathematical Society, vol.3, issue.4, pp.699-713, 1974.
DOI : 10.1112/plms/s3-29.4.699

H. J. Sussmann, Orbits of families of vector fields and integrability of distributions. Transactions of the, pp.171-188, 1973.

H. J. Sussmann, A General Theorem on Local Controllability, SIAM Journal on Control and Optimization, vol.25, issue.1, pp.158-194, 1987.
DOI : 10.1137/0325011

H. J. Sussmann, Two new methods for motion planning for controllable systems without drift, European Control Conference, pp.1501-1506, 1991.

H. J. Sussmann, New Differential Geometric Methods in Nonholonomic Path Finding, Systems, Models, and Feedback : Theory and Applications, pp.365-384, 1992.
DOI : 10.1007/978-1-4757-2204-8_24

H. J. Sussmann, A continuation method for nonholonomic path-finding problems, Proceedings of 32nd IEEE Conference on Decision and Control, pp.2718-2723, 1993.
DOI : 10.1109/CDC.1993.325689

H. J. Sussmann and W. Liu, Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories, [1991] Proceedings of the 30th IEEE Conference on Decision and Control, p.15, 1991.
DOI : 10.1109/CDC.1991.261338

H. J. Sussmann and W. Liu, Lie Bracket Extensions and Averaging: The Single-Bracket Case, Nonholonomic Motion Planning, pp.109-147, 1993.
DOI : 10.1007/978-1-4615-3176-0_4

M. Taylor, Partial Differential Equations I, Applied Mathematical Sciences, vol.115, p.149, 1996.

M. Taylor, Partial Differential Equations II, Applied Mathematical Sciences, vol.116, issue.154, p.156, 1996.

K. Tchon and J. Jakubiak, Endogenous configuration space approach to mobile manipulators: A derivation and performance assessment of Jacobian inverse kinematics algorithms, International Journal of Control, vol.11, issue.14, pp.1387-1419, 2003.
DOI : 10.1109/9.293207

K. Tchon and L. Malek, Singularity robust jacobian inverse kinematics for mobile manipulators Advances in Robot Kinematics : Analysis and Design, pp.155-164, 2008.

E. Trélat, Contôle optimal : théorie et applications, Mathématiques Concrètes. Vuibert, p.22, 2005.

G. Turinici, On the controllability of bilinear quantum systems, Mathematical Models and Methods for Ab Initio Quantum Chemistry, p.114, 2000.
DOI : 10.1007/978-3-642-57237-1_4

URL : https://hal.archives-ouvertes.fr/hal-00536518

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, pp.261-266, 2004.
DOI : 10.1109/TAC.2003.822872

A. M. Vershik, V. Ya, and . Gershkovich, Nonholonomic dynamical systems, geometry of distributions and variational problems, Dynamical Systems VII, volume 16 of Encyclopedia of Mathematical Sciences, p.28, 1994.

T. Wazewski, Sur l'évaluation du domaine d'existence des fonctions implicites réelles ou complexes, Ann. Soc. Polon. Math, vol.20, issue.87, pp.81-120, 1947.

. Résumé, objectif de cette thèse est, d'une part, de fournir des méthodes de planification de mouvements pour les systèmes non-holonomes, et d'autre part

. Mots-clés, Planification de mouvements, systèmes non-holonomes, géométrie sousriemannienne , approximation nilpotente, méthode de continuation, problème de roulement, équation de Schrödinger, contrôlabilité spectrale, minimalité des familles exponentielles, contrôlabilité générique