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

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 $\mathbb{R}$.