On the role of abnormal minimizers in sub-riemannian geometry
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 10 (2001) no. 3, pp. 405-491.
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     title = {On the role of abnormal minimizers in sub-riemannian geometry},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {405--491},
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Bonnard, B.; Trélat, E. On the role of abnormal minimizers in sub-riemannian geometry. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 10 (2001) no. 3, pp. 405-491. http://archive.numdam.org/item/AFST_2001_6_10_3_405_0/

[1] Agrachev (A.). - Compactness for sub-Riemannian length minimizers and subanalyticity, Rend. Semin. Mat. Torino, Vol. 56, 1999. | MR | Zbl

[2] Agrachev (A.), Bonnard (B.), Chyba (M.), Kupka (I.). - Sub-Riemannian sphere in the Martinet flat case, ESAIM/COCV, Vol. 2, 377-448, 1997. | Numdam | MR | Zbl

[3] Agrachev (A.), El Alaoui, Gauthier (J.P.). - Sub-Riemannian metrics on R3, Canadian Math. Cont. Proc., Vol. 25, 29-76, 1998. | MR | Zbl

[4] Agrachev (A.), El Alaoui, Gauthier (J.P.), Kupka (I.). - Generic singularities of sub-Riemannian on IR,3, C.R.A.S., Paris, 377-384, 1996. | MR | Zbl

[5] Agrachev (A.), Sarychev (A.). - On abnormal extremals for Lagrange variational problems, J. Math. Systems, Estimation and Control, Vol. 8, No. 1, 1998, 87-118. | MR | Zbl

[6] Agrachev (A.), Sarychev (A.). - Strong minimality of abnormal geodesics for 2-distributions in SR-geometry, J. of Dynamical and Control Systems, Vol. 1, No. 2, 139-176, 1995. | MR | Zbl

[7] Bellaïche (A.), Risler (J.J.) (Editors). - Sub-Riemannian geometry, Birkhäuser, 1996, Progress in Mathematics, Vol. 144. | MR | Zbl

[8] Bliss (G.A.). - , Lectures on the calculus of variations, University of Chicago Press, Chicago, 1946. | MR | Zbl

[9] Bonnard (B.). - Feedback equivalence for nonlinear systems and the time optimal control problem, SIAM J. on control and optimization, 29, 1300-1321, 1991. | MR | Zbl

[10] Bonnard (B.), Chyba (M.). - Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie SR dans le cas Martinet, ESAIM/COCV, Vol. 4, 245-334, 1999. | Numdam | MR | Zbl

[11] Bonnard (B.), Heutte (H.). - La propriété de stricte anormalité est générique, Preprint Labo. de Topologie Dijon, 1995.

[12] Bonnard (B.), Kupka (I.). - Théorie des singularités et optimalité des trajectoires singulières dans le problème du temps minimal, Forum Math. 5, 111-159, 1991. | MR | Zbl

[13] Bonnard (B.), Kupka (I.). - Generic properties of singular trajectories, Annales de l'IHP, Analyse non linéaire, Vol. 14, No. 2, 167-186, 1997. | Numdam | MR | Zbl

[14] Bonnard (B.), Launay (G.), Trélat (E.). - The transcendence we need to compute the sphere and wave front in Martinet SR-geometry, Contemporary Mathematics and its Applications, Vol. 64, VINITI, Moscow, 1999, pp 82-117. English version in Journal of Mathematical Sciences (Kluwer), Vol. 103 (6), 2001, pp 688-708. | MR | Zbl

[15] Bruno (A.D.). - Local methods in nonlinear differential equations, Springer Verlag, 1989. | MR | Zbl

[16] Chaperon (M.). - Géométrie différentielle et singularités des systèmes dynamiques, Astérisque 138-139, SMF, 1986. | Numdam | MR | Zbl

[17] Chyba (M.). - Le cas Martinet en géométrie sous-Riemannienne, Thèse de l'Université de Bourgogne, 1997.

[18] Davis (H.). - Introduction to nonlinear differential and integral equations, Dover, 1962. | MR | Zbl

[19] Van Den Dries (L.), Macintyre (A.), Marker (D.). - The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics, 140, 183-205, 1994. | MR | Zbl

[20] Van Den Dries (L.), Miller (C.). - Geometric categories and o-minimal structures, Duke Math. Journal, Vol. 84, No. 2, 1996. | MR | Zbl

[21] Heutte (H.). - Propriétés génériques des extrémales singulières dans le cas multi-entrée, Preprint Labo. de Topologie Dijon, 1995.

[22] Il'Yashenko (Yu.S.). - Finiteness theorems for limit cycles, Translations of Mathematical Monographs, Vol. 94, 1991. | MR | Zbl

[23] Krener (A.J.). - The higher order maximal principle and its applications to singular extremals, SIAM J. on Control and Opt., Vol. 15, 256-293, 1977. | MR | Zbl

[24] Kupka (I.). - Abnormal extremals, Preprint, 1992.

[25] Kupka (I.). - Géométrie sous-Riemannienne, Séminaire Bourbaki, 1996. | Numdam | MR | Zbl

[26] Lawden (D.F.). - , Elliptic functions and applications, Springer Verlag, 1980. | MR | Zbl

[27] Lee (E.B.), Markus (L.). - Foundations of optimal control theory, John Wiley, New York, 1967. | MR | Zbl

[28] Lion (J.M.), Rolin (J.P.). - Théorèmes de préparation pour les fonctions logarithmico-exponentielles, Annales de l'Institut Fourier, Tome 47, Fasc. 3, 859-884, 1997. | Numdam | MR | Zbl

[29] Liu (W.S.), Sussmann (H.J.). - Shortest paths for sub-Riemannian metrics of rank two distributions, Memoirs AMS, No. 564, Vol. 118, 1995. | MR | Zbl

[30] Lojasiewicz (S.), Sussmann (H.J.). - Some examples of reachable sets and optimal cost functions that fail to be subanalytic, SIAM J. on Control and Opt., Vol. 23, No. 4, 584-598, 1985. | MR | Zbl

[31] Love (A.E.H.). - A treatise of the mathematica theory of elasticity, Dover, 1944. | MR | Zbl

[32] Mischenko (A.S.) and al. - Lagrangian manifolds and the Maslov operator, Springer Verlag, New York, 1980.

[33] Montgomery (R.). - Abnormal minimizers, SIAM J. on Control and Opt., Vol. 32, No. 6, 1605-1620, 1997. | MR | Zbl

[34] Mourtada (A.), Moussu (R.). - Applications de Dulac et applications pfaffiennes, Bulletin SMF, 125, 1-13, 1997. | Numdam | MR | Zbl

[35] Moussu (R.), Roche (A.). - Théorie de Khovanski et problème de Dulac, Inv. Math., 105, 431-441, 1991. | MR | Zbl

[36] Pelletier (M.). - Communication personnelle.

[37] Pontriaguine (L.) et al. - Théorie mathématique des processus optimaux, Eds Mir, Moscou, 1974. | MR | Zbl

[38] Roussarie (R.). - Bifurcations of planar vector fields and Hilbert's 16th problem, Birkhäuser, Berlin, 1998. | Zbl

[39] Sachkov (Y.L.). - Symmetries of flat rank two distributions and sub-Riemannian structures, Preprint Labo. de Topologie Dijon, 1998. | MR

[40] Trélat (E.). - Some properties of the value function and its level sets for affine control systems with quadratic cost, Journal of Dynamical and Control Systems, Vol. 6, No. 4, Oct. 2000, 511-541. | MR | Zbl

[41] Trélat (E.). - Etude asymptotique et transcendance de la fonction valeur en contrôle optimal ; catégorie log-exp dans le cas sous-Riemannien de Martinet. Phd Thesis, Université de Bourgogne, Dijon, 2000.

[42] Trélat (E.). - Asymptotics of accessibility sets along an abnormal trajectory, ESAIM/COCV, Vol. 6, 387-414, 2001. | Numdam | MR | Zbl

[43] Trélat (E.). - Non subanalyticity of sub-Riemannian Martinet spheres, CRAS, t. 332, Série I, 527-532, 2001. | MR | Zbl

[44] Zelenko (I.), Zhitomirski (M.). - Rigid paths of generic 2-distributions on 3-manifolds, Duke Math. Journal, Vol. 79, No. 2, 281-307, 1995. | MR | Zbl

[45] Zhitomirski (M.). - Typical singularities of differential 1-forms and pfaffian equations, Trans. of Math. Monographs, Vol. 113, AMS, 1992. | MR | Zbl