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

Ryuichi Fukuoka, Anderson Macedo Setti

TL;DR

This work develops a left-invariant mollifier smoothing $F_\varepsilon$ for strongly convex $C^0$-Finsler structures on Lie groups and analyzes Pontryagin extremals via the extended geodesic field. It proves existence and uniqueness of Pontryagin extremals for the base structure, shows that the smoothing preserves strong convexity with a fixed modulus, and establishes uniform convergence of the smoothed structures, dual norms, and extreme curves to their unsmoothed counterparts on compact time intervals. By coupling mollifier smoothing with PMP, the authors demonstrate uniform convergence of extremals not only on the group but also at the Lie-algebra level and in the cotangent bundle, providing a robust framework for approximation in $C^0$-Finsler geometry on Lie groups. These results bridge mollification techniques with Hamiltonian methods to enable stable analysis and approximation of left-invariant geodesic-type extremals in non-smooth Finsler settings.

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 $(x_0,α_0)$ in the cotangent bundle $T^\ast G$ of $G$, we prove that there exist a unique Pontryagin extremal $t\in \mathbb{R} \mapsto (x(t), α(t))$ such that $(x(0),α(0))=(x_0,α_0)$. Moreover, if $t \in \mathbb{R} \mapsto (x_\varepsilon(t), α_{\varepsilon}(t))$ is the unique Pontryagin extremal on $(G,F_\varepsilon)$ such that $(x_\varepsilon(0), α_{\varepsilon}(0))=(x_0, α_0)$, then we prove that $(x_{\varepsilon}(t),α_\varepsilon(t))$ converges uniformly to $(x(t),α(t))$ on compact intervals of $\mathbb{R}$.

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

TL;DR

This work develops a left-invariant mollifier smoothing for strongly convex -Finsler structures on Lie groups and analyzes Pontryagin extremals via the extended geodesic field. It proves existence and uniqueness of Pontryagin extremals for the base structure, shows that the smoothing preserves strong convexity with a fixed modulus, and establishes uniform convergence of the smoothed structures, dual norms, and extreme curves to their unsmoothed counterparts on compact time intervals. By coupling mollifier smoothing with PMP, the authors demonstrate uniform convergence of extremals not only on the group but also at the Lie-algebra level and in the cotangent bundle, providing a robust framework for approximation in -Finsler geometry on Lie groups. These results bridge mollification techniques with Hamiltonian methods to enable stable analysis and approximation of left-invariant geodesic-type extremals in non-smooth Finsler settings.

Abstract

Let be a smooth manifold and its tangent bundle. A -Finsler structure of is a continuous function such that restricted to each tangent space of is an asymmetric norm. is strongly convex if is a strongly convex asymmetric norm for every . Let be a Lie group endowed with a left-invariant strongly convex -Finsler structure . We introduce a smoothing of , which is a left-invariant version of the mollifier smoothing presented previously by the same authors. We study extremals on using the Pontryagin maximum principle. Given in the cotangent bundle of , we prove that there exist a unique Pontryagin extremal such that . Moreover, if is the unique Pontryagin extremal on such that , then we prove that converges uniformly to on compact intervals of .

Paper Structure

This paper contains 16 sections, 51 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex functions and asymmetric norms
  4. The maximizer $u(\alpha)$ of $\alpha \in \mathbb{V}^\ast\backslash \{0\}$ on $S_F[0,1]$
  5. Almost radial growth rates of $F^2$ for asymmetric norms $F$ on $\mathbb{V}$
  6. The extended geodesic field
  7. Mollifier smoothing of convex functions
  8. Distance between parallel hyperplanes
  9. Strongly convex asymmetric norms and strongly convex functions
  10. Pontryagin extremals
  11. Mollifier smoothing of strongly convex asymmetric norms
  12. Convergence of left-invariant mollifier smoothings on Lie groups
  13. The uniform convergences $u_\varepsilon \rightarrow u$ and $\tilde{\mathcal{E}}_\varepsilon \rightarrow \tilde{\mathcal{E}}$ on compact subsets of $\mathfrak{g}^\ast \backslash \{0\}$
  14. Uniform convergence of Pontryagin extremals
  15. Pontryagin extremals on $(\mathfrak{g}\backslash \{0\}) \times (\mathfrak{g}^\ast\backslash \{0\}$)
  16. ...and 1 more sections

Key Result

Theorem 2.4

Let $S\subset \mathbb{V}$ be a non-empty convex set and $f:S \rightarrow \mathbb{R}$. Then $f$ is a convex function iff $\mathop{\mathrm{epi}}\nolimits f$ is a convex subset of $\mathbb{V} \times \mathbb{R}$. In particular, if $r\in \mathbb{R}$ and $f$ is a convex function, then are convex sets.

Theorems & Definitions (115)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • ...and 105 more