Explicit Solutions for a Series of Optimization Problems
with 2-Dimensional Control via Convex Trigonometry
A. A. Ardentova,*, L. V. Lokutsievskiyb,**, and Yu. L. Sachkova,c,***
a Ailamazyan Program Systems Institute, Russian Academy
of Sciences, Veskovo village, Pereslavl district,
Yaroslavl oblast, 152021 Russia
b Steklov Mathematical Institute, Russian Academy
of Sciences, Moscow, 119991 Russia
c “Sirius” Science and Technology University, Sochi,
354340 Russia
Correspondence to: *e-mail: aaa@pereslavl.ru
Correspondence to: **e-mail: lion.lokut@gmail.com
Correspondence to: ***e-mail: yusachkov@gmail.com
Received 10 June, 2020
Abstract—We consider a number of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set $\Omega $. Solutions to these problems are obtained using methods of convex trigonometry. The paper includes (1) geodesics in the Finsler problem on the Lobachevsky hyperbolic plane; (2) left-invariant sub-Finsler geodesics on all unimodular 3D Lie groups (${\text{SU}}(2)$, ${\text{SL}}(2)$, ${\text{SE}}(2)$, ${\text{SH}}(2)$); (3) the problem of a ball rolling on a plane with a distance function given by $\Omega $; and (4) a series of “yacht problems” generalizing Euler’s elastic problem, the Markov–Dubins problem, the Reeds–Shepp problem, and a new sub-Riemannian problem on SE(2).
Keywords: sub-Finsler geometry, convex trigonometry, optimal control problem, Lobachevsky hyperbolic plane, unimodular 3D Lie groups, rolling ball, Euler’s elastica, yacht problems
DOI: 10.1134/S1064562420050257