Normal Curves in Sub-Finsler Lie Groups: Branching for Strongly Convex Norms and Face Stability for Polyhedral Norms

TL;DR

This paper investigates normal curves in sub-Finsler Lie groups, analyzing their properties, branching behavior, and face stability under polyhedral norms using convex analysis and control theory tools.

Contribution

It introduces a convex analysis approach to the Pontryagin Maximum Principle for sub-Finsler structures and studies the face stability of normal curves under polyhedral norms.

Findings

Normal curves can branch and exhibit local optimality.

Under polyhedral norms, controls are confined to a single face of the norm sphere.

The paper provides conditions for existence and properties of normal curves in sub-Finsler Lie groups.

Abstract

We consider Lie groups equipped with left-invariant subbundles of their tangent bundles and norms on them. On these sub-Finsler structures, we study the normal curves in the sense of control theory. We revisit the Pontryagin Maximum Principle using tools from convex analysis, expressing the normal equation as a differential inclusion involving the subdifferential of the dual norm. In addition to several properties of normal curves, we discuss their existence, the possibility of branching, and local optimality. Finally, we focus on polyhedral norms and show that normal curves have controls that locally take values in a single face of a sphere with respect to the norm.