Abnormal extremals of left-invariant sub-Finsler quasimetrics on four-dimensional Lie groups

Extremals of a left-invariant sub-Finsler quasimetric on the Cartan group

Using the Pontryagin Maximum Principle for the time-optimal problem in coordinates of the first kind, we find extremals of abitrary left-invariant sub-Finsler quasimetric on the Cartan group defined by a distribution of rank two.

Abnormal extremals of left-invariant sub-Finsler quasimetrics on four-dimensional Lie groups with three-dimensional generating distributions

We find three-dimensional subspaces of four-dimensional connected Lie algebras, generating these algebras, and abnormal extremals on connected Lie groups with these Lie algebras and with left-invariant sub-Finsler quasimetrics defined by seminorms on such subspaces. In terms of the structure constants of Lie algebras and dual seminorms, we establish a criterion for the strong abnormality of these extremals.

A sub-Finsler problem on the Cartan group

In this paper we study a sub-Finsler geometric problem on the free-nilpotent group of rank 2 and step 3. Such a group is also called Cartan group and has a natural structure of Carnot group, which we metrize considering the $\ell_\infty$ norm on its first layer. We adopt the point of view of time-optimal control theory. We characterize extremal curves via Pontryagin maximum principle. We describe abnormal and singular arcs, and construct the bang-bang flow.

Sub-Finsler几何与非完整力学

Sub-Finsler geometry and nonholonomic mechanics

本文探讨了sub-Finsler流形上长度泛函的变分方法，引入非完整sub-Finsler结构，证明分布关于Barthel非线性联络测地不变，并给出极值曲线存在的充要条件及sub-Laplacian与曲率的关系。