Extremals on Lie groups with asymmetric polyhedral Finsler structures
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arXiv
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| Format: | Preprint |
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2022
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| author | Prudencio, Jéssica B. Fukuoka, Ryuichi |
| author_facet | Prudencio, Jéssica B. Fukuoka, Ryuichi |
| contents | In this work we study extremals on Lie groups $G$ endowed with a left invariant polyhedral Finsler structure. We use the Pontryagin's Maximal Principle (PMP) to find curves on the cotangent bundle of the group, such that its projections on $G$ are extremals. Let $\mathfrak g$ and $\mathfrak g^\ast$ be the Lie algebra of $G$ and its dual space respectively. We represent this problem as a control system $\mathfrak a^\prime (t)= -\mathrm{ad}^\ast(u(t))(\mathfrak a(t))$ of Euler-Arnold type equation, where $u(t)$ is a measurable control in the unit sphere of $\mathfrak g$ and $\mathfrak a(t)$ is an absolutely continuous curve in $\mathfrak g^\ast$. A solution $(u(t), \mathfrak a(t))$ of this control system is a Pontryagin extremal and $\mathfrak a(t)$ is its vertical part. In this work we show that for a fixed vertical part of the Pontryagin extremal $\mathfrak a(t)$, the uniqueness of $u(t)$ such that $(u(t),\mathfrak a(t))$ is a Pontryagin extremal can be studied through an asymptotic curvature of $\mathfrak a(t)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_02465 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Extremals on Lie groups with asymmetric polyhedral Finsler structures Prudencio, Jéssica B. Fukuoka, Ryuichi Differential Geometry 49N99, 53B20, 53B99, 53D25 In this work we study extremals on Lie groups $G$ endowed with a left invariant polyhedral Finsler structure. We use the Pontryagin's Maximal Principle (PMP) to find curves on the cotangent bundle of the group, such that its projections on $G$ are extremals. Let $\mathfrak g$ and $\mathfrak g^\ast$ be the Lie algebra of $G$ and its dual space respectively. We represent this problem as a control system $\mathfrak a^\prime (t)= -\mathrm{ad}^\ast(u(t))(\mathfrak a(t))$ of Euler-Arnold type equation, where $u(t)$ is a measurable control in the unit sphere of $\mathfrak g$ and $\mathfrak a(t)$ is an absolutely continuous curve in $\mathfrak g^\ast$. A solution $(u(t), \mathfrak a(t))$ of this control system is a Pontryagin extremal and $\mathfrak a(t)$ is its vertical part. In this work we show that for a fixed vertical part of the Pontryagin extremal $\mathfrak a(t)$, the uniqueness of $u(t)$ such that $(u(t),\mathfrak a(t))$ is a Pontryagin extremal can be studied through an asymptotic curvature of $\mathfrak a(t)$. |
| title | Extremals on Lie groups with asymmetric polyhedral Finsler structures |
| topic | Differential Geometry 49N99, 53B20, 53B99, 53D25 |
| url | https://arxiv.org/abs/2204.02465 |