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 $F:TM\to \mathbb{R}$ such that $F$ restricted to each tangent space $T_{x}M$ of $M$ is an asymmetric norm. $F$ is strongly convex if $F{|}_{T_{x}M}$ 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 $F_{\epsilon }$ 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 $t\in \mathbb{R}\mapsto (x_{\epsilon }(t),{\alpha }_{\epsilon }(t))$ is the unique Pontryagin extremal on $(G,F_{\epsilon })$ such that $(x_{\epsilon }(0),{\alpha }_{\epsilon }(0))=(x_0,{\alpha }_0)$, then we prove that $(x_{\epsilon }(t),{\alpha }_{\epsilon }(t))$ converges uniformly to $(x(t),\alpha (t))$ on compact intervals of $\mathbb{R}$.

51 pages