A1 Journal article (refereed)
Lipschitz Carnot-Carathéodory Structures and their Limits (2023)

Antonelli, G., Le Donne, E., & Nicolussi Golo, S. (2023). Lipschitz Carnot-Carathéodory Structures and their Limits. Journal of Dynamical and Control Systems, 29, 805-854. https://doi.org/10.1007/s10883-022-09613-1

JYU authors or editors

Publication details

All authors or editors: Antonelli, Gioacchino; Le Donne, Enrico; Nicolussi Golo, Sebastiano

Journal or series: Journal of Dynamical and Control Systems

ISSN: 1079-2724

eISSN: 1573-8698

Publication year: 2023

Publication date: 25/11/2022

Volume: 29

Pages range: 805-854

Publisher: Springer Science and Business Media LLC

Publication country: United States

Publication language: English

DOI: https://doi.org/10.1007/s10883-022-09613-1

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/84937

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2111.06789

Abstract

In this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not boundedly compact and the convergence is not uniform on compact sets. We discuss several examples in which our convergence result can be applied. Among them, we prove a subFinsler Mitchell’s Theorem with continuously varying norms, and a general convergence result for Carnot-Carathéodory distances associated to subspaces and norms on the Lie algebra of a connected Lie group.

Keywords: differential geometry; measure theory; control theory

Free keywords: sub-Finsler geometry; sub-Riemannian geometry; Lipschitz vector fields; Mitchell’s theorem

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Ministry reporting: Yes

Reporting Year: 2022

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