Small sub-Riemannian balls onR 3

El-Alaoui, El-H. Ch.; Gauthier, Jean-Paul; Kupka, Ivan

doi:10.1007/bf02269424

Cited by 87 publications

(35 citation statements)

References 12 publications

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“…When a Lie group structure is not available there are also some results: the optimal synthesis was obtained for a neighborhood of the starting point in the 3D contact case in [6,7,15] and in the 4D quasi-contact case in [14]. The optimal synthesis was obtained in the important Martinet nilpotent case, where abnormal minimizers can be optimal (see [5]).…”

Section: Historymentioning

confidence: 99%

On 2-Step, Corank 2, Nilpotent Sub-Riemannian Metrics

Barilari¹,

Boscain²,

Gauthier³

2012

SIAM J. Control Optim.

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In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics that are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2 sub-Riemannian metric.

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Section: Historymentioning

confidence: 99%

On 2-Step, Corank 2, Nilpotent Sub-Riemannian Metrics

Barilari¹,

Boscain²,

Gauthier³

2012

SIAM J. Control Optim.

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“…We expect that results of this work can be applied to these domains. On the other hand, problem (1.1)-(1.5) is the first completely studied sub-Riemannian problem without rotational symmetry; this problem has the local structure of generic contact sub-Riemannian problems as described in works [1,6]. This is an immediate continuation of the previous work [9].…”

Section: Introductionmentioning

confidence: 62%

Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane

Sachkov¹

2009

ESAIM: COCV

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Abstract.The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.Mathematics Subject Classification. 49J15, 93B29, 93C10, 53C17, 22E30.

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“…In the 3D case, it is proven in [4,18] that for an initial covector (cos θ, sin θ, h 0 ) ∈ C q (1/2), the conjugate time at q satisfies as h 0 → ±∞ t c (cos θ, sin θ, h 0 ) = 2π…”

Section: Approximation Of Short Geodesicsmentioning

confidence: 99%

Short Geodesics Losing Optimality in Contact Sub-Riemannian Manifolds and Stability of the 5-Dimensional Caustic

Sacchelli¹

2019

SIAM J. Control Optim.

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We study the sub-Riemannian exponential for contact distributions on manifolds of dimension greater or equal to 5. We compute an approximation of the sub-Riemannian Hamiltonian flow and show that the conjugate time can have multiplicity 2 in this case. We obtain an approximation of the first conjugate locus for small radii and introduce a geometric invariant to show that the metric for contact distributions typically exhibits an original behavior, different from the classical 3-dimensional case. We apply these methods to the case of 5-dimensional contact manifolds. We provide a stability analysis of the sub-Riemannian caustic from the Lagrangian point of view and classify the singular points of the exponential map.

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Small sub-Riemannian balls onR 3

Cited by 87 publications

References 12 publications

On 2-Step, Corank 2, Nilpotent Sub-Riemannian Metrics

On 2-Step, Corank 2, Nilpotent Sub-Riemannian Metrics

Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane

Short Geodesics Losing Optimality in Contact Sub-Riemannian Manifolds and Stability of the 5-Dimensional Caustic

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