Abnormal Trajectories in the Sub-Riemannian $(2,3,5,8)$ Problem

Yu. L. Sachkov ; E. F. Sachkova

Informatics and Automation, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 252-285.

Résumé

Abnormal trajectories are of particular interest for sub-Riemannian geometry, because the most complicated singularities of the sub-Riemannian metric are located just near such trajectories. Important open questions in sub-Riemannian geometry are to establish whether the abnormal length minimizers are smooth and to describe the set filled with abnormal trajectories starting from a fixed point. For example, the Sard conjecture in sub-Riemannian geometry states that this set has measure zero. In this paper, we consider this and other related properties of such a set for the left-invariant sub-Riemannian problem with growth vector $(2,3,5,8)$. We also study the global and local optimality of abnormal trajectories and obtain their explicit parametrization.

Détail

Keywords: sub-Riemannian geometry, abnormal trajectories, abnormal set, local and global optimality.

Bibliographie

[1] Agrachev A.A., “Geometry of optimal control problems and Hamiltonian systems”, Nonlinear and optimal control theory, Lect. CIME Summer Sch. (Cetraro, Italy, 2004), Lect. Notes Math., 1932, Springer, Berlin, 2008, 1–59 | DOI | MR | Zbl

[2] Agrachev A., Barilari D., Boscain U., A comprehensive introduction to sub-Riemannian geometry: From the Hamiltonian viewpoint, Cambridge Stud. Adv. Math., 181, Cambridge Univ. Press, Cambridge, 2020 | MR | Zbl

[3] A. A. Agrachev and Yu. L. Sachkov, Control Theory from the Geometric Viewpoint, Encycl. Math. Sci., 87, Springer, Berlin, 2004 | MR | Zbl

[4] A. A. Agrachev and A. V. Sarychev, “Filtrations of a Lie algebra of vector fields and nilpoint approximation of controlled systems”, Sov. Math., Dokl., 36:1 (1988), 104–108 | MR | Zbl

[5] Ardentov A., Hakavuori E., “Cut time in the sub-Riemannian problem on the Cartan group”, ESAIM. Control Optim. Calc. Var., 28 (2022), 12 | DOI | MR | Zbl

[6] A. A. Ardentov and Yu. L. Sachkov, “Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group”, Sb. Math., 202:11 (2011), 1593–1615 | DOI | DOI | MR | Zbl

[7] Ardentov A.A., Sachkov Yu.L., “Cut time in sub-Riemannian problem on Engel group”, ESAIM. Control Optim. Calc. Var., 21:4 (2015), 958–988 | DOI | MR | Zbl

[8] Bierstone E., Milman P.D., “Semianalytic and subanalytic sets”, Publ. math. Inst. hautes études sci., 67 (1988), 5–42 | DOI | MR | Zbl

[9] Boarotto F., Vittone D., “A dynamical approach to the Sard problem in Carnot groups”, J. Diff. Eqns., 269:6 (2020), 4998–5033 | DOI | MR | Zbl

[10] M. Goresky and R. MacPherson, Stratified Morse Theory, Springer, Berlin, 1988 | MR | Zbl

[11] Hakavuori E., “Infinite geodesics and isometric embeddings in Carnot groups of step 2”, SIAM J. Control Optim., 58:1 (2020), 447–461 | DOI | MR | Zbl

[12] Hakavuori E., Le Donne E., “Non-minimality of corners in subriemannian geometry”, Invent. math., 206:3 (2016), 693–704 | DOI | MR | Zbl

[13] Hakavuori E., Le Donne E., “Blowups and blowdowns of geodesics in Carnot groups”, J. Diff. Geom., 123:2 (2023), 267–310 | MR

[14] Hironaka H., “Subanalytic sets”, Number theory, algebraic geometry and commutative algebra: In honor of Y. Akizuki, Kinokuniya, Tokyo, 1973, 453–493 | MR

[15] Le Donne E., Leonardi G.P., Monti R., Vittone D., “Extremal polynomials in stratified groups”, Commun. Anal. Geom., 26:4 (2018), 723–757 | DOI | MR | Zbl

[16] Liu W., Sussmann H.J., Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. AMS, 118, no. 564, Amer. Math. Soc., Providence, RI, 1995 | MR

[17] L. V. Lokutsievskiy and Yu. L. Sachkov, “Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater”, Sb. Math., 209:5 (2018), 672–713 | DOI | DOI | MR | Zbl

[18] Montgomery R., “Abnormal minimizers”, SIAM J. Control Optim., 32:6 (1994), 1605–1620 | DOI | MR | Zbl

[19] Montgomery R., “Survey of singular geodesics”, Sub-Riemannian geometry, Prog. Math., 144, Birkhäuser, Basel, 1996, 325–339 | MR | Zbl

[20] Montgomery R., A tour of subriemannian geometries, their geodesics and applications, Math. Surv. Monogr., 91, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[21] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Pergamon, Oxford, 1964 | MR | MR | Zbl

[22] Yu. L. Sachkov, “Exponential map in the generalized Dido problem”, Sb. Math., 194:9 (2003), 1331–1359 | DOI | DOI | MR | Zbl

[23] Yu. L. Sachkov, “Discrete symmetries in the generalized Dido problem”, Sb. Math., 197:2 (2006), 235–257 | DOI | DOI | MR | Zbl

[24] Yu. L. Sachkov, “The Maxwell set in the generalized Dido problem”, Sb. Math., 197:4 (2006), 595–621 | DOI | DOI | MR | Zbl

[25] Yu. L. Sachkov, “Complete description of the Maxwell strata in the generalized Dido problem”, Sb. Math., 197:6 (2006), 901–950 | DOI | DOI | MR | Zbl

[26] Yu. L. Sachkov, Introduction to Geometric Control, Springer Optim. Appl., 192, Springer, Cham, 2022 | MR | Zbl

[27] 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

[28] Yu. L. Sachkov, “Left-invariant optimal control problems on Lie groups: Classification and problems integrable by elementary functions”, Russ. Math. Surv., 77:1 (2022), 99–163 | DOI | DOI | MR | Zbl

[29] Yu. L. Sachkov and E. F. Sachkova, “Degenerate abnormal trajectories in a sub-Riemannian problem with growth vector $(2,3,5,8)$”, Diff. Eqns., 53:3 (2017), 352–365 | DOI | MR | Zbl

[30] Sachkov Yu.L., Sachkova E.F., “Symmetries and parameterization of abnormal extremals in the sub-Riemannian problem with the growth vector $(2,3,5,8)$”, Russ. J. Nonlinear Dyn., 15:4 (2019), 579–587 | MR

[31] Yu. L. Sachkov and E. F. Sachkova, “The structure of abnormal extremals in a sub-Riemannian problem with growth vector $(2,3,5,8)$”, Sb. Math., 211:10 (2020), 1460–1485 | DOI | DOI | MR | Zbl