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

Enrico Le Donne, Sebastiano Nicolussi Golo, Nicola Paddeu

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.

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

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.

Paper Structure

This paper contains 21 sections, 21 theorems, 81 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The key idea
  3. Outline of the paper
  4. Acknowledgements
  5. Preliminaries in convex analysis
  6. Elements from convex analysis
  7. Strict and strongly convex norms
  8. Sub-Finsler Lie groups
  9. Normal curves in sub-Finsler Lie groups
  10. Normal curves for strongly convex norms
  11. Lifts of normal curves via submetries
  12. Branching of normal curves
  13. Absence of branching
  14. Presence of branching
  15. Example: branching in a $2D$-group with strongly convex norm
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let G be a Lie group with polarization $V$. Assume that there exists $X,Y\in V$ such that the set $\{ \mathop{\mathrm{Ad}}\nolimits_{\exp(t Y)}X : t\in\mathbb{R}\}$ is unbounded. Then, there exists a strongly convex norm $\|\cdot\|$ on $V$ for which there is branching of normal geodesics in $G$ for

Figures (3)

  • Figure 1: The unit sphere of the norm $\|\cdot\|$ is the non-smooth strongly convex set $\{(x,y)\in\mathbb{R}^2 : |x| = \frac{y^2}{2} - \frac{1}{2}\}$. The blue and red lines correspond to two different covectors $\lambda$ and $\lambda'$ defining supporting hyperplanes of the sphere.
  • Figure 2: The numerical integration of the normal equation for the covectors $\lambda$ and $\lambda'$. In blue, the normal curve with associated covector $\lambda$ starting at $(0,1)$. In red, the normal curve with associated covector $\lambda'$ starting at $(0,1)$.
  • Figure 3: The planar projection of the curve $\gamma$. It encloses area $\beta^2$ in the plane and it has length $4\beta$, like $\gamma$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2: Fenchel conjugates, Clarke-book
  • Proposition 2.3
  • Proposition 2.4: Properties of Fenchel conjugates referenza-C11-Fenchel-conjugate
  • Definition 2.5: Subdifferential, Clarke-book
  • Remark 2.6: Subdifferential inversion, Clarke-book
  • ...and 46 more