Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups

Bizyaev, Ivan A.; Borisov, Alexey V.; Kilin, Alexander A.; Mamaev, Ivan S.

doi:10.1134/s1560354716060125

Cited by 20 publications

(9 citation statements)

References 28 publications

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“…It would be interesting to characterize similarly optimal controls in the cases s = 3, k ≥ 3 and s ≥ 4. In these cases, if U = { k i=1 u 2 i ≤ 1}, the normal Hamiltonian system of Pontryagin maximum principle is not Liouville integrable [19,20].…”

Section: General Free-nilpotent Lie Groupsmentioning

confidence: 99%

Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems

Sachkov

2020

Regul. Chaot. Dyn.

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We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with strictly convex set of control parameters (in particular, sub-Finsler problems).We describe all linear-in-momenta Casimirs on the dual of the Lie algebra.In the case of rank 3 Lie groups we describe the symplectic foliation on the dual of the Lie algebra. On this basis we show that extremal controls are either constant or periodic.Some related results for other Carnot groups are presented.

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Section: General Free-nilpotent Lie Groupsmentioning

confidence: 99%

Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems

Sachkov

2020

Regul. Chaot. Dyn.

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“…Remark 1. For free nilpotent Lie coalgebras of rank ≥ 3 and step ≥ 3, and of rank ≥ 2 and step ≥ 4, typical symplectic leaves have dimension at least 4, and the corresponding Hamiltonian systems are not Liouville integrable [15,25].…”

Section: Integrabilitymentioning

confidence: 99%

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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“…The sub-Riemannian Cartan group C is the nilpotent model for all sub-Riemannian problems with growth vector (2,3,5). As a consequence of the non-integrability results of [9,14], it is the only free nilpotent group with step three or greater for which the Hamiltonian system of the Pontryagin Maximum Principle (PMP) [17] is Liouville integrable.…”

Section: Introductionmentioning

confidence: 99%

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

Ardentov

Hakavuori

2022

ESAIM: COCV

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We study the sub-Riemannian structure determined by a left-invariant distribution of rank 2 on a step 3 Carnot group of dimension 5. We prove the conjectured cut times of Yu. Sachkov for the sub-Riemannian Cartan problem. Along the proof, we obtain a comparison with the known cut times in the sub-Riemannian Engel group, and a sufficient (generic) condition for the uniqueness of the length minimizer between two points. Hence we reduce the optimal synthesis to solving a certain system of equations in elliptic functions.

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Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups

Cited by 20 publications

References 28 publications

Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems

Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems

Conjugate time in sub-Riemannian problem on Cartan group

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

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