Geodesic
The structure of abnormal extremals in a sub-Riemannian problem with growth vector $(2, 3, 5, 8)$
Sbornik. Mathematics, Tome 211 (2020) no. 10, pp. 1460-1485 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A left-invariant sub-Riemannian problem on a free nilpotent Lie group of step 4 with two generators is considered. The structure of abnormal extremals is described. These extremals are shown to define an abnormal foliation of the annihilator of the square of the distribution, which is given by the intersections of this annihilator with the leaves of the symplectic foliation of the Lie coalgebra. The question of abnormal trajectories being strictly/nonstrictly abnormal is investigated, their projection onto a plane of the distribution are described, estimates for the corank are given and examples of nonsmooth trajectories are constructed. Bibliography: 14 titles.
Keywords: sub-Riemannian problem, abnormal extremals, abnormal trajectories, strict/nonstrict abnormality. 
@article{SM_2020_211_10_a4,
     author = {Yu. L. Sachkov and E. F. Sachkova},
     title = {The structure of abnormal extremals in {a~sub-Riemannian} problem with growth vector $(2, 3, 5, 8)$},
     journal = {Sbornik. Mathematics},
     pages = {1460--1485},
     year = {2020},
     volume = {211},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2020_211_10_a4/}
}
TY  - JOUR
AU  - Yu. L. Sachkov
AU  - E. F. Sachkova
TI  - The structure of abnormal extremals in a sub-Riemannian problem with growth vector $(2, 3, 5, 8)$
JO  - Sbornik. Mathematics
PY  - 2020
SP  - 1460
EP  - 1485
VL  - 211
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_2020_211_10_a4/
LA  - en
ID  - SM_2020_211_10_a4
ER  - 
%0 Journal Article
%A Yu. L. Sachkov
%A E. F. Sachkova
%T The structure of abnormal extremals in a sub-Riemannian problem with growth vector $(2, 3, 5, 8)$
%J Sbornik. Mathematics
%D 2020
%P 1460-1485
%V 211
%N 10
%U http://geodesic.mathdoc.fr/item/SM_2020_211_10_a4/
%G en
%F SM_2020_211_10_a4
Yu. L. Sachkov; E. F. Sachkova. The structure of abnormal extremals in a sub-Riemannian problem with growth vector $(2, 3, 5, 8)$. Sbornik. Mathematics, Tome 211 (2020) no. 10, pp. 1460-1485. http://geodesic.mathdoc.fr/item/SM_2020_211_10_a4/

[1] A. A. Agrachev, Yu. L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia Math. Sci., 87, Control theory and optimization, II, Springer-Verlag, Berlin, 2004, xiv+412 pp. | DOI | MR | Zbl | Zbl

[2] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math. Surveys Monogr., 91, Amer. Math. Soc., Providence, RI, 2002, xx+259 pp. | MR | Zbl

[3] A. Agrachev, D. Barilari, U. Boscain, A comprehensive introduction to sub-Riemannian geometry, Cambridge Stud. Adv. Math., 181, Cambridge Univ. Press, Cambridge, 2019, xviii+745 pp. | DOI | MR | Zbl

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

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

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

[7] L. V. Lokutsievskiy, 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

[8] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, Intersci. Publ. John Wiley Sons, Inc., New York–London, 1962, viii+360 pp. | MR | MR | Zbl | Zbl

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

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

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

[12] J.-P. Gauthier, Yu. L. Sachkov, “On the free Carnot $(2, 3, 5, 8)$ group”, Programmnye sistemy: teoriya i prilozheniya, 6:2 (2015), 45–61 | DOI

[13] A. A. Kirillov, Lectures on the orbit method, Grad. Stud. Math., 64, Amer. Math. Soc., Providence, RI, 2004, xx+408 pp. | DOI | MR | Zbl

[14] A. I. Markushevich, The theory of analytic functions: a brief course, Mir Publishers, Moscow, 1983, 423 pp. | MR | MR | Zbl | Zbl