Cut time in the sub-Riemannian problem on the Cartan group

Ardentov, A. A.; Hakavuori, Eero

doi:10.1051/cocv/2022006

Cited by 8 publications

(3 citation statements)

References 22 publications

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“…In a forthcoming paper [7], this conjecture is proved with the use of Th. 2 via a symmetry method [1,10,19,38].…”

Section: Results Of This Papermentioning

confidence: 97%

Conjugate time in sub-Riemannian problem on Cartan group

Sachkov¹

2020

Preprint

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The Cartan group is the free nilpotent Lie group of rank 2 and step 3. We consider the left-invariant sub-Riemannian problem on the Cartan group defined by an inner product in the first layer of its Lie algebra. This problem gives a nilpotent approximation of an arbitrary sub-Riemannian problem with the growth vector (2, 3, 5).In previous works we described a group of symmetries of the sub-Riemannian problem on the Cartan group, and the corresponding Maxwell time -the first time when symmetric geodesics intersect one another. It is known that geodesics are not globally optimal after the Maxwell time.In this work we study local optimality of geodesics on the Cartan group. We prove that the first conjugate time along a geodesic is not less than the Maxwell time corresponding to the group of symmetries. We characterize geodesics for which the first conjugate time is equal to the first Maxwell time. Moreover, we describe continuity of the first conjugate time near infinite values.

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“…In a forthcoming paper [7], this conjecture is proved with the use of Th. 2 via a symmetry method [1,10,19,38].…”

Section: Results Of This Papermentioning

confidence: 97%

Conjugate time in sub-Riemannian problem on Cartan group

Sachkov¹

2020

Preprint

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“…and the image of this map: [127]; § 2.10.5 on [141]; § § 2.10.6 and 2.10.7 on [14]. Also see the recent paper [144].…”

Section: Extremals Consider the Hamiltoniansmentioning

confidence: 94%

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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“…The question of the optimality of abnormal trajectories is quite acute for sub-Riemannian geometry (see [7]), and the structure properties of abnormal sub-Riemannian geodesics (see [8]), connected with the subanalyticity of the corresponding metric (see [9] and [10]), are of great interest. For example, in nilpotent sub-Riemannian problems on the Engel and Cartan groups (see [11] and [12]) -as well as in the not left-invariant three-dimensional sub-Riemannian problem in the Martinet flat case (see [13]) -the subanalyticity condition fails just in a neighbourhood of the points where abnormal trajectories arrive. A long-standing question concerns the optimality of abnormal geodesics (see [14]- [16]).…”

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confidence: 99%

Abnormal extremals in the sub-Riemannian problem for a general model of a robot with a trailer

Ardentov,

Artemova

2023

Sb. Math.

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A symmetric mathematical model of a wheeled robot with a trailer is considered for various types of coupling between the robot and the trailer. It is shown that for fixed coupling parameters and fixed initial position of the robot with trailer there are two symmetric abnormal extremals. In motion along these trajectories the robot and the trailer traverse normal extremal trajectories for the sub-Riemannian problem on the group of motions of the plane; the coupling point always draws an inflectional elastica or a straight line. Bibliography: 33 titles.

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Cut time in the sub-Riemannian problem on the Cartan group

Cited by 8 publications

References 22 publications

Conjugate time in sub-Riemannian problem on Cartan group

Conjugate time in sub-Riemannian problem on Cartan group

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

Abnormal extremals in the sub-Riemannian problem for a general model of a robot with a trailer

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