Abstract:
This paper is a continuation of the work by the same authors on the
Cartan group equipped with the sub-Finsler ℓ∞ℓ∞ norm.
We start by giving a detailed presentation of the structure of bang-bang extremal trajectories.
Then we prove upper bounds on the number of switchings on bang-bang minimizers.
We prove that any normal extremal is either bang-bang, or singular, or mixed.
Consequently, we study mixed extremals.
In particular, we prove that every two points can be connected by a piecewise smooth
minimizer, and we give a uniform bound on the number of such pieces.
The work of A. Ardentov and Yu. Sachkov was supported by the Russian Science Foundation under grant 17-11-
01387 and performed at the Ailamazyan Program Systems Institute of the Russian Academy of Sciences. E. Le Donne was partially supported by the Academy of Finland (grant 288501 “Geometry of sub-Riemannian groups”) and by the European Research Council (ERC Starting Grant 713998 GeoMeG “Geometry of Metric Groups”).
\Bibitem{ArdLe Sac19}
\by Andrei A. Ardentov, Enrico Le Donne, Yuri L. Sachkov
\paper Sub-Finsler Geodesics on the Cartan Group
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 1
\pages 36--60
\mathnet{http://mi.mathnet.ru/rcd388}
\crossref{https://doi.org/10.1134/S1560354719010027}
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This publication is cited in the following 11 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)
Alexey Podobryaev, 2022 16th International Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy's Conference), 2022, 1
Podobryaev A.V., “Casimir Functions of Free Nilpotent Lie Groups of Steps 3 and 4”, J. Dyn. Control Syst., 27:4 (2021), 625–644
A. Montanari, D. Morbidelli, “Multiexponential maps in Carnot groups with applications to convexity and differentiability”, Ann. Mat. Pura Appl., 200:1 (2021), 253–272
Pozuelo J., Ritore M., “Pansu-Wulff Shapes in H-1”, Adv. Calc. Var., 2021
Yuri L. Sachkov, “Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems”, Regul. Chaotic Dyn., 25:1 (2020), 33–39
Yu. L. Sachkov, “Coadjoint Orbits and Time-Optimal Problems for Step-2 Free Nilpotent Lie Groups”, Math. Notes, 108:6 (2020), 867–876
Yu. L. Sachkov, “Periodic time-optimal controls on two-step free-nilpotent Lie groups”, Dokl. Math., 101:3 (2020), 262–264
E. Hakavuori, “Infinite geodesics and isometric embeddings in Carnot groups of step 2”, SIAM J. Control Optim., 58:1 (2020), 447–461
M. Sigalotti, “Bounds on time-optimal concatenations of arcs for two-input driftless 3D systems”, IFAC-PapersOnLine, 53:2 (2020), 6863
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