Control theory problems and the Rashevskii-Chow theorem on a Cartan group
type: Item
instance of: scholarly article
title: Control theory problems and the Rashevskii-Chow theorem on a Cartan group
author: A. V. Greshnov
author: Roman Ivanovich Zhukov
published in: Siberian Mathematical Journal
publication date: 2024-11-01T00:00:00Z
review text: The canonical Cartan group \(\mathbb{K}\) is a certain 5-dimensional Lie group, diffeomorphic to \(\mathbb{R}^5\), whose group structure depends on three positive constants. It is equipped with a rank-2 distribution defined by two left-invariant vector fields on \(\mathbb{K}\). This distribution is bracket-generating, ensuring that any two points in \(\mathbb{K}\) can be connected by a horizontal curve, as guaranteed by the Rashevskii-Chow Theorem. In this context, horizontal curves correspond to the trajectories of a ball rolling on a plane without slipping.\N\NThe authors address the problem of finding the minimum number \(k\) such that, locally, any two points in \(\mathbb{K}\) can be connected by a horizontal curve composed of \(k\) segments, each segment being an integral curve of \(X\) or \(Y\) (referred to as a horizontal \(k\)-broken line). They prove that the minimum \(k\) for this connection is \(4\). Furthermore, they establish that the minimum \(k\) for which a closed horizontal \(k\)-broken line exists is \(6\).
reviewed by: Paolo Piccione
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 58A30
zbMATH DE Number: 7937481
zbMATH Keywords: Cartan group
zbMATH Keywords: Rashevskii-Chow theorem
MaRDI profile type: Publication
cites work: Hypoelliptic second order differential equations
cites work: Q4718347
cites work: A Comprehensive Introduction to Sub-Riemannian Geometry
cites work: Stratified Lie Groups and Potential Theory for their Sub-Laplacians
cites work: Carnot-Carathéodory metrics and quasiisometries of symmetric spaces of rank 1
cites work: Q4824176
cites work: The geometry of \(cc\)-balls and the constants in the ball-box theorem on Heisenberg group algebras
cites work: Universal methods of the search of normal geodesics on Lie groups with left-invariant sub-Riemannian metric
cites work: Extremals of a left-invariant sub-Finsler metric on the Engel group
cites work: Abnormal extremals of left-invariant sub-Finsler quasimetrics on four-dimensional Lie groups
cites work: Abnormal extremals of left-invariant sub-Finsler quasimetrics on four-dimensional Lie groups with three-dimensional generating distributions
cites work: A sub-Finsler problem on the Cartan group
cites work: Maxwell strata and cut locus in the sub-Riemannian problem on the Engel group
cites work: Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group
cites work: An abnormal set for the \((2,3,5,8)\)-distribution
cites work: Sub-Riemannian Cartan sphere
cites work: Existence of tangent lines to Carnot-Carathéodory geodesics
cites work: Q4797475
cites work: Horizontal joinability in canonical 3-step Carnot groups with corank 2 horizontal distributions
cites work: Optimal horizontal joinability on the Engel group
cites work: The Agrachev-Barilari-Boscain method and estimates for the number of segments of horizontal broken lines joining points in the canonical Carnot group \(G_{3,3}\)
cites work: Horizontal joinability on 5-dimensional 2-step Carnot groups with a codimension 2 horizontal distribution
cites work: Q3945260
cites work: Q4870772
cites work: Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for Carnot-Carathéodory metrics
DOI: 10.1134/S0037446624050100
description: scientific article; zbMATH DE number 7937481