Abstract:
We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with a strictly convex set of control parameters (in particular, sub-Finsler problems). We describe all Casimirs linear in momenta on the dual of the Lie algebra.
In the case of rank 3 Lie groups we describe the symplectic foliation on the dual of the Lie algebra. On this basis we show that extremal controls are either constant or periodic. Some related results for other Carnot groups are presented.
Keywords:
optimal control, sub-Finsler geometry, Lie groups, Pontryagin maximum principle.
Sections 1–3 of this work were supported by the Academy of Finland (grant 277923) and by
the European Research Council (ERC Starting Grant 713998 GeoMeG). Sections 4–6 of this work
were supported by the Russian Science Foundation under grant 17-11-01387 and performed at the
Ailamazyan Program Systems Institute of Russian Academy of Sciences.
This publication is cited in the following 5 articles:
Yu. L. Sachkov, “Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions”, Russian Math. Surveys, 78:1 (2023), 65–163
Valentina Franceschi, Roberto Monti, Alberto Righini, Mario Sigalotti, “The Isoperimetric Problem for Regular and Crystalline Norms in H1”, J Geom Anal, 33:1 (2023)
Podobryaev A.V., “Casimir Functions of Free Nilpotent Lie Groups of Steps 3 and 4”, J. Dyn. Control Syst., 27:4 (2021), 625–644
Yu. L. Sachkov, “Coadjoint Orbits and Time-Optimal Problems for Step-2 Free Nilpotent Lie Groups”, Math. Notes, 108:6 (2020), 867–876
A. V. Podobryaev, “Coadjoint orbits of three-step free nilpotent Lie groups and time-optimal control problem”, Dokl. Math., 102:1 (2020), 293–295
Citation in format AMSBIB
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