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Abnormal sub-Riemannian geodesics: Morse index and rigidity. (English) Zbl 0866.58023
Summary: Considering a smooth manifold \(M\) provided with a sub-Riemannian structure, i.e., with Riemannian metric and nonintegrable vector distribution, we set a problem of finding for two given points \(q^0\), \(q^1\in M\) a length minimizer among Lipschitzian paths tangent to the vector distribution (admissible) and connecting these points. Extremals of this variational problem are called sub-Riemannian geodesics and we single out the abnormal sub-Riemannian geodesics, which correspond to the vanishing Lagrange multiplier for the length functional. These abnormal geodesics are not related to the Riemannian structure but only to the vector distribution and, in fact, are singular points in the set of admissible paths connecting \(q^0\) and \(q^1\). Developing the Legendre-Jacobi-Morse-type theory of second variation for abnormal geodesics we investigate some of their specific properties such as weak minimality and rigidity-isolatedness in the space of admissible paths connecting the two given points.

MSC:
58E30 Variational principles in infinite-dimensional spaces
93B29 Differential-geometric methods in systems theory (MSC2000)
53C22 Geodesics in global differential geometry
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[1] Agrachev, A. A., Quadratic mappings in geometric control theory, Itogi Nauki i Tekhniki, J. Soviet Math., Vol. 51, 2667-2734, (1990), English transl · Zbl 1267.93035
[2] Agrachev, A. A., Topology of quadratic mappings and hessians of smooth mappings, in: itogi nauki i tekhniki, Acad. Nauk SSSR, Vol. 26, 85-124, (1988), Algebra, Topologia, Geometria; VINITI
[3] Agrachev, A. A., The second-order optimality condition in the general nonlinear case, Matem Sbornik, Math. USSR Sbornik, Vol. 31, 551-568, (1977), English transl · Zbl 0386.49012
[4] Agrachev, A. A.; Gamkrelidze, R. V., Second-order optimality condition for the time-optimal problem, Matem. Sbornik, Math. USSR Sbornik, Vol. 29, 547-576, (1976), English transl · Zbl 0377.49014
[5] Agrachev, A. A.; Gamkrelidze, R. V., Exponential representation of flows and chronological calculus, Matem. Sbornik, Math. USSR Sbornik, Vol. 35, 727-785, (1979), English transl · Zbl 0429.34044
[6] Agrachev, A. A.; Gamkrelidze, R. V.; Sarychev, A. V., Local invariants of smooth control systems, Acta Applicandae Mathematicae, Vol. 14, 191-237, (1989) · 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] Agrachev, A. A.; Sarychev, A. V., On abnormal extremals for Lagrange variational problems (summary), J. Mathematical Systems, Estimation and Control, Vol. 5, 127-130, (1995) · Zbl 0826.49012
[9] Arnol’d, V. I., Mathematical methods of classical mechanics, (1978), Springer-Verlag New York-Berlin · Zbl 0407.57025
[10] Arnol’d, V. I.; Varchenko, A. N.; Gusein-Zade, S. M., Singularities of differentiable mappings, Vol. 1, (1985), Birkhäuser Boston · Zbl 0554.58001
[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] Goh, B. S., Necessary conditions for singular extremals involving multiple control variables, SIAM J. Control, Vol. 4, 716-731, (1966) · Zbl 0161.29004
[14] V. Guillemin and S. Sternberg, Geometric Asymptotics, Amer. Math. Soc., Providence, Rhode Island, 1977.
[15] Haynes, G. W.; Hermes, H., Nonlinear controllability via Lie theory, SIAM J. on Control, Vol. 8, 450-460, (1970) · Zbl 0229.93012
[16] Kelley, H. J.; Kopp, R.; Moyer, H. G., Singular extremals, (Leitman, G., Topics in Optimization, (1967), Academic Press New York, N. Y.), 63-101
[17] Krener, A. J., The high-order maximum principle and its applications to singular extremals, SIAM J. on Control and Optimiz., Vol. 15, 256-293, (1977) · Zbl 0354.49008
[18] Lion, G.; Vergne, M., The Weyl representation, Maslov index and theta-series, (1980), Birkhäuser Boston · 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
[21] Sarychev, A. V., Integral representation for the trajectories of control system with generalized right-hand side (in Russian), Differentsialnye Uravnenia, Differential Equations, Vol. 24, 1021-1031, (1988), English transl. · Zbl 0674.34064
[22] Sarychev, A. V., The index of the second variation of the control system, Matem. Sbornik, Math. USSR Sbornik, Vol. 41, 383-401, (1982), English transl · 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.
[24] H. J. Sussmann and W. Liu, Shortest Paths for Sub-Riemannian Metrics on Rank-2 Distributions, Preprint, 1993.
[25] Young, L. C., Lectures on the calculus of variations and optimal control theory, (1980), Chelsea New York · Zbl 0177.37801
[26] Agrachev, A. A.; Sarychev, V., Strong minimality of abnormal geodesics for 2-distributions, J. of Dynamical and Control Systems, Vol. 1, 139-176, (1995) · Zbl 0951.53029
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