Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group

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

doi:10.1070/sm2011v202n11abeh004200

Cited by 49 publications

(93 citation statements)

References 22 publications

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“…The subspaces M x = {q ∈ M | x = 0}, M z = {q ∈ M | z = 0} consist of fixed points of symmetries (1), (2), the cut locus is contained in the union of these subspaces (see further Th. 3). The problem has also continuous symmetries -the one-parameter group of dilations given by the flow of the vector field…”

Section: Previously Obtained Resultsmentioning

confidence: 99%

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Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group

Ardentov¹,

Sachkov²

2017

Regul. Chaot. Dyn.

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We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three.We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics).The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations R + and a discrete group of reflections Z 2 × Z 2 × Z 2 .The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimensional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well.

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Section: Previously Obtained Resultsmentioning

confidence: 99%

“…This paper has the following structure. In Subsection 1.1 we recall results on the problem obtained in previous works [3,4,5]. In Subsection 1.2 we prove some simple preliminary results on the cut locus.…”

Section: Introductionmentioning

confidence: 89%

Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group

Ardentov¹,

Sachkov²

2017

Regul. Chaot. Dyn.

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“…The case of h-type groups is discussed in [AM16]. Analysis in step three examples has been performed in [ABCK97], for the Martinet case, and in [AS11,AS15] for the Engel group. We also mention the very recent paper [BBN16], where a detailed discussion of the cut locus in the biHeisenberg group is performed.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

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

Montanari

Morbidelli

2017

Calc. Var.

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We characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators. We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Finally, by using the Hamiltonian approach, we show that the cut time of strictly normal extremal paths is a smooth explicit function of the initial velocity covector. Finally, using our previous results, we show that at any cut point the distance has a corner-like singularity.Comment: Added Section 6. Final version, to appear on Calc. Va

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“…where g L (τ ) = π(λ L (τ )) and g R (τ ) = π(λ R (τ )). Since G p 0 and G pt are commutative groups, using (14) we have…”

Section: The Intersection γmentioning

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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Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group

Cited by 49 publications

References 22 publications

Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group

Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group

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

Symmetries in left-invariant optimal control problems

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