Relation between Euler’s elasticae and sub-Riemannian geodesics on SE(2)

Mashtakov, Alexey; Ardentov, A. A.; Sachkov, Yu. L.

doi:10.1134/s1560354716070066

Cited by 2 publications

(1 citation statement)

References 18 publications

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“…Subsequently, Euler's problem of elasticae was considered in [18], [111], [129]- [131], [133], [136], and [137], and our presentation in this section is based on these papers. 2.6.2.…”

Section: 411mentioning

confidence: 99%

Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions

Sachkov

2023

Russian Math. Surveys

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Left-invariant optimal control problems on Lie groups are an important class of problems with a large symmetry group. They are theoretically interesting because they can often be investigated in full and general laws can be studied by using these model problems. In particular, problems on nilpotent Lie groups provide a fundamental nilpotent approximation to general problems. Also, left-invariant problems often arise in applications such as classical and quantum mechanics, geometry, robotics, visual perception models, and image processing. The aim of this paper is to present a survey of the main concepts, methods, and results pertaining to left-invariant optimal control problems on Lie groups that can be integrated by elliptic functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Bibliography: 162 titles.

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“…Subsequently, Euler's problem of elasticae was considered in [18], [111], [129]- [131], [133], [136], and [137], and our presentation in this section is based on these papers. 2.6.2.…”

Section: 411mentioning

confidence: 99%

Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions

Sachkov

2023

Russian Math. Surveys

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Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions

Сачков¹,

Sachkov

2023

Uspekhi Mat. Nauk

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Левоинвариантные задачи оптимального управления на группах Ли образуют важный класс задач с большой группой симметрий. Они интересны в теоретическом плане, так как часто допускают полное исследование и на этих модельных задачах можно изучить общие закономерности. В частности, задачи на нильпотентных группах Ли доставляют фундаментальную нильпотентную аппроксимацию общих задач. Левоинвариантные задачи также часто возникают в приложениях: в классической и квантовой механике, геометрии, робототехнике, моделях зрения и обработке изображений. Цель данной работы - дать обзор основных понятий, методов и результатов, относящихся к левоинвариантным задачам оптимального управления на группах Ли, интегрируемым в эллиптических функциях. Основное внимание уделено описанию экстремальных траекторий и их оптимальности, времени разреза и множества разреза, оптимального синтеза. Библиография: 162 названия.

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Relation between Euler’s elasticae and sub-Riemannian geodesics on SE(2)

Abstract: In this note we describe a relation between Euler's elasticae and sub-Riemannian geodesics on SE(2). Analyzing the Hamiltonian system of Pontryagin maximum principle we show that these two curves coincide only in the case when they are segments of a straight line.

Cited by 2 publications

References 18 publications

Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions

Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions

Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions

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