Abnormal sub-riemannian geodesics : Morse index and rigidity
Annales de l'I.H.P. Analyse non linéaire, Volume 13 (1996) no. 6, p. 635-690
@article{AIHPC_1996__13_6_635_0,
     author = {Agrachev, Andrei A. and Sarychev, A. V.},
     title = {Abnormal sub-riemannian geodesics : Morse index and rigidity},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Gauthier-Villars},
     volume = {13},
     number = {6},
     year = {1996},
     pages = {635-690},
     zbl = {0866.58023},
     mrnumber = {1420493},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_1996__13_6_635_0}
}
Agrachev, A. A.; Sarychev, A. V. Abnormal sub-riemannian geodesics : Morse index and rigidity. Annales de l'I.H.P. Analyse non linéaire, Volume 13 (1996) no. 6, pp. 635-690. http://www.numdam.org/item/AIHPC_1996__13_6_635_0/

[1] A.A. Agrachev, Quadratic Mappings in Geometric Control Theory, in: Itogi Nauki i Tekhniki; Problemy Geometrii, VINITI, Acad. Nauk SSSR, Moscow, Vol. 20, pp. 11-205, 1988. English transl. in J. Soviet Math., Vol. 51, 1990, pp. 2667-2734. | MR 966201

[2] A.A. Agrachev, Topology of Quadratic Mappings and Hessians of Smooth Mappings, in: Itogi Nauki i Tekhniki; Algebra, Topologia, Geometria; VINITI, Acad. Nauk SSSR, Vol. 26, 1988, pp. 85-124. | MR 978394 | Zbl 0719.58006

[3] A.A. Agrachev, The Second-Order Optimality Condition in the General Nonlinear Case, Matem Sbornik, Vol. 102, 1977, pp. 551-568. English transl. in: Math. USSR Sbornik, Vol. 31, 1977. | MR 451084 | Zbl 0362.49016

[4] A.A. Agrachev and R.V. Gamkrelidze, Second-Order Optimality Condition for the Time-Optimal Problem, Matem. Sbornik, Vol. 100, 1976, pp. 610-643. English transl. in: Math. USSR Sbornik, Vol. 29, 1976, pp. 547-576. | MR 425719 | Zbl 0341.49007

[5] A.A. Agrachev and R.V. Gamkrelidze, Exponential Representation of Flows and Chronological Calculus, Matem. Sbornik, Vol. 107, 1978, pp. 467-532. English transl. in: Math. USSR Sbornik, Vol. 35, 1979, pp. 727-785. | MR 524203 | Zbl 0408.34044

[6] A.A. Agrachev, R.V. Gamkrelidze and A.V. Sarychev, Local Invariants of Smooth Control Systems, Acta Applicandae Mathematicae, Vol. 14, 1989, pp. 191-237. | MR 995286 | Zbl 0681.49018

[7] A.A. Agrachev and A.V. Sarychev, On Abnormal Extremals for Lagrange Variational Problems, to appear in J. Mathematical Systems, Estimation and Control. | Zbl 0826.49012

[8] A.A. Agrachev and A.V. Sarychev, On Abnormal Extremals for Lagrange Variational Problems (summary), J. Mathematical Systems, Estimation and Control, Vol. 5, 1995, pp. 127-130. | Zbl 0826.49012

[9] V.I. Arnol'D, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York-Berlin, 1978. | MR 690288 | Zbl 0386.70001

[10] V.I. Arnol'D, A.N. Varchenko and S.M. Gusein-Zade, Singularities of Differentiable Mappings, Vol. 1, Birkhäuser, Boston, 1985. | MR 777682

[11] B. Bonnard and I. Kupka, Théorie de singularités de l'application entree/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, Preprint, 1990.

[12] V. Gershkovich, Engel Structures on Four Dimensional Manifolds, University of Melbourne, Department of Mathematics, Preprint Series No. 10, 1992.

[13] B.S. Goh, Necessary Conditions for Singular Extremals Involving Multiple Control Variables, SIAM J. Control, Vol. 4, 1966, pp. 716-731. | MR 205719 | Zbl 0161.29004

[14] V. Guillemin and S. Sternberg, Geometric Asymptotics, Amer. Math. Soc., Providence, Rhode Island, 1977. | MR 516965 | Zbl 0364.53011

[15] G.W. Haynes and H. Hermes, Nonlinear Controllability via Lie Theory, SIAM J. on Control, Vol. 8, 1970, pp. 450-460. | MR 277859 | Zbl 0229.93012

[16] H.J. Kelley, R. Kopp and H.G. Moyer, Singular Extremals, in: G. Leitman Ed., Topics in Optimization, Academic Press, New York, N. Y., 1967, pp. 63-101. | MR 215153

[17] A.J. Krener, The High-Order Maximum Principle and its Applications to Singular Extremals, SIAM J. on Control and Optimiz., Vol. 15, 1977, pp. 256-293. | MR 433288 | Zbl 0354.49008

[18] G. Lion and M. Vergne, The Weyl Representation, Maslov Index and Theta-series, Birkhäuser, Boston, 1980. | MR 573448 | Zbl 0444.22005

[19] R. Montgomery, Geodesics, Which Do Not Satisfy Geodesic Equations, Preprint, 1991.

[20] M. Morse, The Calculus of variations in the large, Amer. Math. Soc., New York, NY, 1934. | JFM 60.0450.01 | Zbl 0011.02802

[21] A.V. Sarychev, Integral representation for the trajectories of control system with generalized right-hand side (in Russian), Differentsialnye Uravnenia, Vol. 24, 1988, pp. 1551-1564. English transl.: Differential Equations, Vol. 24, 1988, pp. 1021-1031. | MR 965603 | Zbl 0674.34064

[22] A.V. Sarychev, The index of the second variation of the control system, Matem. Sbornik, Vol. 113, 1980, pp. 464-486. English transl. in: Math. USSR Sbornik, Vol. 41, 1982, pp. 383-401. | MR 601890 | Zbl 0484.49012

[23] H.J. Sussmann, A Cornucopia of Abnormal Sub Riemannian Minimizers. Part I: the Four-Dimensional Case, IMA Preprint Series #1073,1992. | MR 1184651

[24] H.J. Sussmann and W. Liu, Shortest Paths for Sub-Riemannian Metrics on Rank-2 Distributions, Preprint, 1993.

[25] L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, Chelsea, New York, 1980.

[26] A.A. Agrachev and A.V. Sarychev, Strong Minimality of Abnormal Geodesics for 2-Distributions, J. of Dynamical and Control Systems, Vol. 1, 1995, pp. 139-176. | MR 1333769 | Zbl 0951.53029