Geodesic
Coadjoint orbits of three-step free nilpotent Lie groups and time-optimal control problem
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 493 (2020), pp. 38-41 Cet article a éte moissonné depuis la source Math-Net.Ru

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We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg group and the Engel group. We consider a time-optimal problem on three-step free nilpotent Lie groups with a set of admissible velocities in the first level of the Lie algebra. The behavior of normal extremal trajectories with initial covectors lying in two-dimensional coadjoint orbits is studied. Under some broad conditions on the set of admissible velocities (in particular, in the sub-Riemannian case) the corresponding extremal controls are periodic, constant, or asymptotically constant.
Mots-clés : Carnot group, coadjoint orbits
Keywords: time-optimal control problem, sub-Riemannian geometry, sub-Finsler geometry. 
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     author = {A. V. Podobryaev},
     title = {Coadjoint orbits of three-step free nilpotent {Lie} groups and time-optimal control problem},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {38--41},
     year = {2020},
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     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_493_a7/}
}
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A. V. Podobryaev. Coadjoint orbits of three-step free nilpotent Lie groups and time-optimal control problem. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 493 (2020), pp. 38-41. http://geodesic.mathdoc.fr/item/DANMA_2020_493_a7/

[1] Agrachev A., Barilari D., Boscain U., A comprehensive introduction to sub-Riemannian geometry, Cambridge University Press, Cambridge, 2019, 746 pp. | MR

[2] Barilari D., Boscain U., Le Donne E., Sigalotti M., “Sub-Finsler Structures from the Time-Optimal Control Viewpoint for Some Nilpotent Distributions”, J. Dynamical and Control Systems, 23:3 (2017), 547–575 | DOI | MR | Zbl

[3] Berestovskii V.N., “Odnorodnye mnogoobraziya s vnutrennei metrikoi II”, Sibirskii matematicheskii zhurnal, 30:2 (2017), 14–28

[4] Kirillov A.A., Lectures on the orbit method, Graduate Studies in Mathematics, 64, AMS, Providence, RI, 2004 | DOI | MR | Zbl

[5] Sachkov Yu.L., “Mnozhestvo Maksvella v obobschennoi zadache Didony”, Matem. sb., 197:4 (2006), 123–150 | DOI | Zbl

[6] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F., Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961, 391 pp. | MR

[7] Agrachev A.A., Sachkov Yu.L., Geometricheskaya teoriya upravleniya, Fizmatlit, M., 2005, 392 pp.

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

[9] Sachkov Yu.L., “Periodicheskie optimalnye po bystrodeistviyu upravleniya na dvukhstupennykh svobodnykh nilpotentnykh gruppakh Li”, Doklady RAN. Matematika, informatika, protsessy upravleniya, 492 (2020), 108–111 | DOI