Antonelli Gioacchino, Le Donne Enrico, Nicolussi Golo Sebastiano
Courant Institute of Mathematical Sciences, NYU, 251 Mercer Street, New York, NY 10012 USA.
Department of Mathematics, University of Fribourg, Chemin du Musée 23, 1700 Fribourg, Switzerland.
J Dyn Control Syst. 2023;29(3):805-854. doi: 10.1007/s10883-022-09613-1. Epub 2022 Nov 25.
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.
在本文中，我们讨论了与光滑流形上Lipschitz向量场的收敛结构和连续变化的范数相关的距离的收敛性。我们证明，在对极限向量场结构的一个温和可控性假设下，与在紧子集上一致收敛的等Lipschitz向量场结构以及在紧子集上一致收敛的范数相关的距离，局部一致收敛到极限卡诺 - 卡拉西奥多里距离。在极限距离是有界紧的情况下，我们表明距离的收敛在紧集上是一致的。我们给出一个例子，其中极限距离不是有界紧的，并且收敛在紧集上不是一致的。我们讨论了几个可以应用我们收敛结果的例子。其中，我们证明了一个具有连续变化范数的次芬斯勒米切尔定理，以及一个关于连通李群的李代数上子空间和范数相关的卡诺 - 卡拉西奥多里距离的一般收敛结果。
