Mollifier smoothing of left-invariant strongly convex $C^0$-Finsler structures on Lie groups and convergence of extremals

TL;DR

This paper introduces a mollifier smoothing for left-invariant strongly convex $C^0$-Finsler structures on Lie groups and proves the convergence of extremals obtained via the Pontryagin maximum principle.

Contribution

The authors develop a smoothing technique for $C^0$-Finsler structures on Lie groups and establish the convergence of extremals under this smoothing.

Findings

Existence and uniqueness of Pontryagin extremals for the smoothed and original structures.

Uniform convergence of extremals from the smoothed structures to the original extremals.

Extension of mollifier smoothing techniques to strongly convex $C^0$-Finsler structures on Lie groups.

Abstract

Let $M$ be a smooth manifold and $TM$ its tangent bundle. A $C^0$-Finsler structure of $M$ is a continuous function $F:TM \rightarrow \mathbb{R}$ such that $F$ restricted to each tangent space $T_xM$ of $M$ is an asymmetric norm. $F$ is strongly convex if $F\vert_{T_xM}$ is a strongly convex asymmetric norm for every $x \in M$. Let $G$ be a Lie group endowed with a left-invariant strongly convex $C^0$-Finsler structure $F$. We introduce a smoothing $F_{\varepsilon}$ of $F$, which is a left-invariant version of the mollifier smoothing presented previously by the same authors. We study extremals $x(t)$ on $(G,F)$ using the Pontryagin maximum principle. Given $(x_0,\alpha_0)$ in the cotangent bundle $T^\ast G$ of $G$, we prove that there exist a unique Pontryagin extremal $t\in \mathbb{R} \mapsto (x(t), \alpha(t))$ such that $(x(0),\alpha(0))=(x_0,\alpha_0)$. Moreover, if $t \in \mathbb{R} \mapsto (x_\varepsilon(t), \alpha_{\varepsilon}(t))$ is the unique Pontryagin extremal on $(G,F_\varepsilon)$ such that $(x_\varepsilon(0), \alpha_{\varepsilon}(0))=(x_0, \alpha_0)$, then we prove that $(x_{\varepsilon}(t),\alpha_\varepsilon(t))$ converges uniformly to $(x(t),\alpha(t))$ on compact intervals of $\mathbb{R}$.