Numerical Solution of a Left-Invariant Sub-Riemannian Problem on the Group $SO(3)$

D. N. Stepanov, A. V. Podobryaev

A. K. Ailamazyan Program Systems Institute of RAS, ul. Petra-I, s. Veskovo, Pereslavl district, Yaroslavl obl., 152021 Russia

Abstract: We consider a left-invariant sub-Riemannian problem on the Lie group of rotations of a three-dimensional space. We find the cut locus numerically, in fact we construct the optimal synthesis numerically, i. e., the shortest arcs. The software package nutopy designed for the numerical solution of optimal control problems is used. With the help of this package we investigate sub-Riemannian geodesics, conjugate points, Maxwell points and diffeomorphic domains of the exponential map. We describe some operating features of this software package.

Keywords: sub-Riemannian geometry, shortest arcs, caustic, cut time, cut locus, numerical solution

MSC: 53C17, 53C30, 49M05, 65K10

Received: 11.05.2024
Accepted: 22.07.2024

Language: English

DOI: 10.20537/nd241005