Volume 27, 2021
|Number of page(s)||52|
|Published online||30 April 2021|
Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry*,**,***,****
Ailamazyan Program Systems Institute, Russian Academy of Sciences,
2 Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia.
3 Sirius University of Science and Technology, Sochi, Russia.
***** Corresponding author: firstname.lastname@example.org
Accepted: 25 February 2021
We consider a series of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set Ω. The considered problems are well studied for the case when Ω is a unit disc, but barely studied for arbitrary Ω. We derive extremals to these problems in general case by using machinery of convex trigonometry, which allows us to do this identically and independently on the shape of Ω. The paper describes geodesics in (i) the Finsler problem on the Lobachevsky hyperbolic plane; (ii) left-invariant sub-Finsler problems on all unimodular 3D Lie groups (SU(2), SL(2), SE(2), SH(2)); (iii) the problem of rolling ball on a plane with distance function given by Ω; (iv) a series of “yacht problems” generalizing Euler’s elastic problem, Markov-Dubins problem, Reeds-Shepp problem and a new sub-Riemannian problem on SE(2); and (v) the plane dynamic motion problem.
Mathematics Subject Classification: 49Q99
Key words: Sub-Finsler geometry / convex trigonometry / optimal control problem / Lobachevsky hyperbolic plane / unimodular 3D Lie group / yacht problems / rolling ball
Section 6 was written by A.A. Ardentov. Sections 1–3, 5, 7 were written by L.V. Lokutsievskiy. Section 4 was written by Yu.L. Sachkov. All results in this paper are products of authors collaborative work.
The work of A.A. Ardentov is supported by the Russian Science Foundation under grant 17-11-01387-p and performed in Ailamazyan Program Systems Institute of Russian Academy of Sciences.
The work of L.V. Lokutsievskiy is supported by the Russian Science Foundation under grant 20-11-20169 and performed in Steklov Mathematical Institute of Russian Academy of Sciences.
© EDP Sciences, SMAI 2021
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