Normal Curves in Sub-Finsler Lie Groups: Branching for Strongly Convex Norms and Face Stability for Polyhedral Norms
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| author | Donne, Enrico Le Golo, Sebastiano Nicolussi Paddeu, Nicola |
| author_facet | Donne, Enrico Le Golo, Sebastiano Nicolussi Paddeu, Nicola |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26261 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normal Curves in Sub-Finsler Lie Groups: Branching for Strongly Convex Norms and Face Stability for Polyhedral Norms Donne, Enrico Le Golo, Sebastiano Nicolussi Paddeu, Nicola Differential Geometry Optimization and Control 53C17, 49J52, 22E30, 37N35, 49K15, 52A41 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. |
| title | Normal Curves in Sub-Finsler Lie Groups: Branching for Strongly Convex Norms and Face Stability for Polyhedral Norms |
| topic | Differential Geometry Optimization and Control 53C17, 49J52, 22E30, 37N35, 49K15, 52A41 |
| url | https://arxiv.org/abs/2510.26261 |