Geodesic
Coadjoint Orbits and Time-Optimal Problems for Step-$2$ Free Nilpotent Lie Groups
Matematičeskie zametki, Tome 108 (2020) no. 6, pp. 899-910.

Voir la notice de l'article provenant de la source Math-Net.Ru

The coadjoint orbits and the Casimir functions are described for free nilpotent Lie groups of step $2$. The symplectic foliation consists of affine subspaces in the Lie coalgebra. Left-invariant time-optimal problems are considered on Carnot groups of step $2$ for which the set of admissible velocities is a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. The first integrals of the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle are described. For two-dimensional coadjoint orbits, the constancy and periodicity properties of the solutions of this subsystem, as well as the phase flow, are described.
Mots-clés : coadjoint orbits
Keywords: Casimir functions, time-optimal problem, Pontryagin maximum principle. 
@article{MZM_2020_108_6_a6,
     author = {Yu. L. Sachkov},
     title = {Coadjoint {Orbits} and {Time-Optimal} {Problems} for {Step-}$2$ {Free} {Nilpotent} {Lie} {Groups}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {899--910},
     publisher = {mathdoc},
     volume = {108},
     number = {6},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a6/}
}
TY  - JOUR
AU  - Yu. L. Sachkov
TI  - Coadjoint Orbits and Time-Optimal Problems for Step-$2$ Free Nilpotent Lie Groups
JO  - Matematičeskie zametki
PY  - 2020
SP  - 899
EP  - 910
VL  - 108
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a6/
LA  - ru
ID  - MZM_2020_108_6_a6
ER  - 
%0 Journal Article
%A Yu. L. Sachkov
%T Coadjoint Orbits and Time-Optimal Problems for Step-$2$ Free Nilpotent Lie Groups
%J Matematičeskie zametki
%D 2020
%P 899-910
%V 108
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a6/
%G ru
%F MZM_2020_108_6_a6
Yu. L. Sachkov. Coadjoint Orbits and Time-Optimal Problems for Step-$2$ Free Nilpotent Lie Groups. Matematičeskie zametki, Tome 108 (2020) no. 6, pp. 899-910. http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a6/

[1] R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[2] A. A. Agrachev, Yu. L. Sachkov, Geometricheskaya teoriya upravleniya, Fizmatlit, M., 2005

[3] A. Agrachev, D. Barilari, U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge Univ. Press, Cambridge, 2019 | MR | Zbl

[4] V. N. Berestovskii, “Odnorodnye mnogoobraziya s vnutrennei metrikoi II”, Sib. matem. zhurn., 30:2 (1989), 14–28 | MR

[5] V. N. Berestovskii, “Geodezicheskie negolonomnykh levoinvariantnykh vnutrennikh metrik na gruppe Geizenberga i izoperimetriksy ploskosti Minkovskogo”, Sib. matem. zhurn., 35:1 (1994), 3–11 | MR | Zbl

[6] D. Barilari, U. Boscain, E. Le Donne, M. Sigalotti, “Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions”, J. Dyn. Control Syst., 23:3 (2017), 547–575 | DOI | MR | Zbl

[7] A. A. Ardentov, E. Le Donne, Yu. L. Sachkov, “Sub-Finsler Geodesics on the Cartan Group”, Regul. Chaotic Dyn., 24:1 (2019), 36–60 | DOI | MR | Zbl

[8] E. Le Donne, G. Speight, “Lusin approximation for horizontal curves in step 2 Carnot groups”, Calc. Var. Partial Differential Equations, 55:5 (2016), Art. 111 | MR

[9] E. Le Donne, S. Rigot, “Remarks about Besicovitch covering property in Carnot groups of step 3 and higher”, Proc. Amer. Math. Soc., 144:5 (2016), 2003–2013 | DOI | MR | Zbl

[10] Yu. Sachkov, “Periodic controls in step 2 strictly convex sub-Finsler problems”, Regul. Chaotic Dyn., 25:1 (2020), 33–39 | DOI | MR | Zbl

[11] H. Busemann, “The isoperimetric problem in the Minkowski plane”, Amer. J. Math., 69 (1947), 863–871 | MR | Zbl

[12] R. W. Brockett, “Nonlinear control theory and differential geometry”, Proceedings of the International Congress of Mathematicians, PWN, Warszawa, 1983, 1357–1368 | MR

[13] O. Myasnichenko, “Nilpotent $(3,6)$ sub-Riemannian problem”, J. Dynam. Control Systems, 8:4 (2002), 573–597 | DOI | MR | Zbl

[14] L. Rizzi, U. Serres, “On the cut locus of free, step two Carnot groups”, Proc. Amer. Math. Soc., 145:12 (2017), 5341–5357 | DOI | MR | Zbl

[15] A. A. Kirillov, Lectures on the Orbit Method, Amer. Math. Soc., Providence, RI, 2004 | MR | Zbl

[16] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1970 | MR | Zbl

[17] V. Jurdjevic, Geometric Control Theory, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[18] E. Hakavuori, Infinite Geodesics and Isometric Embeddings in Carnot Groups of Step 2, 2019, arXiv: 1905.03214

[19] Yu. L. Sachkov, “Eksponentsialnoe otobrazhenie v obobschennoi zadache Didony”, Matem. sb., 194:9 (2003), 63–90 | DOI | MR | Zbl

[20] I. A. Bizyaev, A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups”, Regul. Chaotic Dyn., 21:6 (2016), 759–774 | DOI | MR | Zbl

[21] L. V. Lokutsievskii, Yu. L. Sachkov, “Ob integriruemosti po Liuvillyu subrimanovykh zadach na gruppakh Karno glubiny 4 i bolshe”, Matem. sb., 209:5 (2018), 74–119 | DOI | MR

[22] A. V. Podobryaev, “Casimir functions of free nilpotent Lie groups of steps three and four”, J. Dyn. Control Syst. (to appear); 2020, arXiv: 2006.00224 | Zbl