On the subRiemannian cut locus in a model of free two-step Carnot group

Montanari, Annamaria; Morbidelli, Daniele

doi:10.1007/s00526-017-1149-1

Cited by 20 publications

(14 citation statements)

References 30 publications

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“…Extrapolating from the known results in these cases, in [Mya02,Mya06,MM16a], the authors conjectured precise algebraic formulas for the cut loci of G k , for all k 2. In this note, we disprove these conjectures.…”

Section: Introductionmentioning

confidence: 87%

“…In particular, we exhibit an explicit hierarchy of sets C k ⊂ G k of cut points (see Definition 5), which coincide with the corresponding cut loci for k = 2, 3, but are strictly larger than the conjectured ones for k 4. While the previously conjectured cut loci were, respectively, smooth algebraic sets of codimension Θ(k 2 ) [Mya02,Mya06] and algebraic sets of codimension Θ(k) [MM16a], the set C k is semi-algebraic of codimension 2, for all k (see Theorem 11).…”

Section: Introductionmentioning

confidence: 94%

“…The computation of the cut time and cut locus for the case k = 3 is much harder, and requires a careful manipulation of geodesic equations and symmetries. This has been done independently in [Mya02] and [MM16a]. There, it was proved that, for the step 2 free Carnot groups of rank k = 2, 3, one has…”

Section: The Conjectured Cut Locimentioning

confidence: 99%

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On the cut locus of free, step two Carnot groups

Rizzi

Serres

2017

Proc. Amer. Math. Soc.

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In this note, we study the cut locus of the free, step two Carnot groups G k with k generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Mya02, Mya06] and [MM16a], by exhibiting sets of cut points C k ⊂ G k which, for k 4, are strictly larger than conjectured ones. While the latter were, respectively, smooth semialgebraic sets of codimension Θ(k 2 ) and semi-algebraic sets of codimension Θ(k), the sets C k are semi-algebraic and have codimension 2, yielding the best possible lower bound valid for all k on the size of the cut locus of G k .Furthermore, we study the relation of the cut locus with the so-called abnormal set. In the low dimensional cases, it is known thatFor each k 4, instead, we show that the cut locus always intersects the abnormal set, and there are plenty of abnormal geodesics with finite cut time.Finally, and as a straightforward consequence of our results, we derive an explicit lower bound for the small time heat kernel asymptotics at the points of C k .The question whether C k coincides with the cut locus for k 4 remains open.

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 94%

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On the cut locus of free, step two Carnot groups

Rizzi

Serres

2017

Proc. Amer. Math. Soc.

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“…2. Free nilpotent sub-Riemannian problem with growth vector (3,6) (O. Myasnichenko [8] and independently A. Montanari and D. Morbidelli [9], also some results in the general case of free step two Carnot group achieved by L. Rizzi and U. Serres [10]). 3.…”

Section: Definitionmentioning

confidence: 95%

Symmetric Extremal Trajectories in Left-Invariant Optimal Control Problems

Podobryaev¹

2019

Nelin. Dinam.

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We consider left-invariant optimal control problems on connected Lie groups. We describe the symmetries of the exponential map that are induced by the symmetries of the vertical part of the Hamiltonian system of the Pontryagin maximum principle. These symmetries play a key role in investigation of optimality of extremal trajectories. For connected Lie groups such that the generic coadjoint orbit has codimension not more than 1 and a connected stabilizer we introduce a general construction for such symmetries of the exponential map.

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“…2. Free nilpotent sub-Riemannian problem with growth vector (3,6) (O. Myasnichenko [8] and independently A. Montanari and D. Morbidelli [9], also some results in general case of free step two Carnot group achieved by L. Rizzi and U. Serres [10]).…”

mentioning

confidence: 99%

Symmetries in left-invariant optimal control problems

Podobryaev¹

2020

Sb. Math.

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We consider left-invariant optimal control problems on connected Lie groups such that generic stabilizer of the coadjoint action is connected and has dimension not more than 1. We introduce a construction for symmetries of the exponential map. These symmetries play a key role in investigation of optimality of extremal trajectories.Geometric control theory (see for example [1]) deals with left-invariant optimal control problems on a Lie group G. Consider a family of left-invariant vector fields F u that depend analytically on u ∈ U ⊂ R n . Consider also a left-invariant analytic function ϕ : G × U → R, a point q 1 ∈ G, and a fixed time t 1 > 0. The problem is to find a control u ∈ L ∞ ([0, t 1 ], U ) and a Lipschitz curve q u : [0, t 1 ] → G such that t 1 0 ϕ(q u (t), u(t))dt → min,q u (t) = F u(t) (q u (t)), q u (0) = id, q u (t 1 ) = q 1 ∈ G. (1)

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On the subRiemannian cut locus in a model of free two-step Carnot group

Cited by 20 publications

References 30 publications

On the cut locus of free, step two Carnot groups

On the cut locus of free, step two Carnot groups

Symmetric Extremal Trajectories in Left-Invariant Optimal Control Problems

Symmetries in left-invariant optimal control problems

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