Geodesic
Abnormal Extremals in the Sub-Riemannian Problem with Growth Vector $(2, 3, 5, 8, 14)$
Russian journal of nonlinear dynamics, Tome 19 (2023) no. 4, pp. 559-573.

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We study the left-invariant sub-Riemannian problem on the free nilpotent Lie group of rank 2 and step 5. We describe some abnormal trajectories and some properties of the set filled by nice abnormal trajectories starting at the identity of the group.
Keywords: sub-Riemannian geometry, abnormal trajectories, geometric control 
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Yu. L. Sachkov; E. F. Sachkova. Abnormal Extremals in the Sub-Riemannian Problem with Growth Vector $(2, 3, 5, 8, 14)$. Russian journal of nonlinear dynamics, Tome 19 (2023) no. 4, pp. 559-573. http://geodesic.mathdoc.fr/item/ND_2023_19_4_a6/

[1] Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesics and Applications, Math. Surveys Monogr., 91, AMS, Providence, R.I., 2002, 259 pp. | Zbl

[2] Agrachev, A. A. and Sachkov, Yu. L., Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci., 87, Springer, Berlin, 2004, xiv, 412 pp. | DOI | Zbl

[3] Agrachev, A., Barilari, D., and Boscain, U., A Comprehensive Introduction to Sub-Riemannian Geometry: From the Hamiltonian Viewpoint, Cambridge Stud. Adv. Math., 181, Cambridge Univ. Press, Cambridge, 2020, xviii, 745 pp. | Zbl

[4] Sachkov, Yu., Introduction to Geometric Control, Springer Optim. Appl., 192, Springer, Cham, 2022, xvii, 162 pp. | MR | Zbl

[5] Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., and Mishchenko, E. F., The Mathematical Theory of Optimal Processes, ed. L. W. Neustadt, Wiley, New York, 1962, viii, 360 pp. | Zbl

[6] Mat. Sb., 211:10 (2020), 112–138 (Russian) | DOI | DOI | MR | Zbl

[7] Mat. Sb., 194:9 (2003), 63–90 (Russian) | DOI | DOI | MR | Zbl

[8] Euler, L., Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimitrici latissimo sensu accepti, Bousquet, Lausanne, 1744, 347 pp.

[9] Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, 4th ed., Dover, New York, 1944, xviii, 643 pp. | Zbl

[10] Hironaka, H., “Subanalytic Sets”, Number Theory, Algebraic Geometry and Commutative Algebra: In Honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, 453–493

[11] Goresky, M. and MacPherson, R., Stratified Morse Theory, Ergeb. Math. Grenzgeb. (3), 14, Springer, Berlin, 1988, xiv, 272 pp. | Zbl

[12] Bierstone, E. and Milman, P. D., “Semianalytic and Subanalytic Sets”, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5–42 | DOI | Zbl

[13] Le Donne, E. and Tripaldi, F., “A Cornucopia of Carnot Groups in Low Dimensions”, Anal. Geom. Metr. Spaces, 10:1 (2022), 155–289 | DOI | MR | Zbl

[14] Mat. Sb., 209:5 (2018), 74–119 (Russian) | DOI | DOI | MR | Zbl

[15] Sachkov, Yu. L., “Conjugate Time in the Sub-Riemannian Problem on the Cartan Group”, J. Dyn. Control Syst., 27:4 (2021), 709–751 | DOI | MR | Zbl

[16] Ardentov, A. and Hakavuori, E., “Cut Time in the Sub-Riemannian Problem on the Cartan Group”, ESAIM Control Optim. Calc. Var., 28 (2022), 12, 19 pp. | DOI | MR | Zbl

[17] Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis, An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, reprint of the 4th ed. (1927), Cambridge Mathematical Library, Cambridge Univ. Press, Cambridge, 1996, vi, 608 pp. | Zbl

[18] Rifford, L. and Trélat, E., “Morse – Sard Type Results in Sub-Riemannian Geometry”, Math. Ann., 332:1 (2005), 145–159 | DOI | MR | Zbl

[19] Belotto da Silva, A. and Rifford, L., “The Sard Conjecture on Martinet Surfaces”, Duke Math. J., 167:8 (2018), 1433–1471 | MR | Zbl