Typical Singularities of Differential 1-Forms
                    and Pfaffian Equations

Zhitomirskii, Michail

doi:10.1090/mmono/113

Cited by 52 publications

(44 citation statements)

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“…[32]) some basic facts about two-dimensional distributions in R 3 (D = span{F 1 , F 2 } associated with the reference control system in our case), which provide useful information on abnormal curves.…”

Section: Abnormal Geodesics For the Copepod Swimmermentioning

confidence: 99%

“…For the normal extremals in the oscillating case and the rotating case presented in Sec. 4.2.3, we take a large number of random initial adjoint vectorsp(0) such that H 1 (q(0),p(0)) 2 + H 2 (q(0),p(0)) 2 = 1 and such that 0 < k(p(0)) < 1 where k is given by (31) for the oscillating case and by (32) for the rotating case. Then we numerically integrate the extremal system.…”

Section: Computations Of Conjugate Pointsmentioning

confidence: 99%

See 1 more Smart Citation

Optimal Strokes at Low Reynolds Number: A Geometric and Numerical Study of Copepod and Purcell Swimmers

Bettiol¹,

Bonnard²,

Rouot³

2018

SIAM J. Control Optim.

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Section: Abnormal Geodesics For the Copepod Swimmermentioning

confidence: 99%

Section: Computations Of Conjugate Pointsmentioning

confidence: 99%

Optimal Strokes at Low Reynolds Number: A Geometric and Numerical Study of Copepod and Purcell Swimmers

Bettiol¹,

Bonnard²,

Rouot³

2018

SIAM J. Control Optim.

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“…Tools from singularity theory can be used to classify the distributions, see [20]. Here we present only the two (stable) models related to our study.…”

Section: Abnormal Curves Are Defined Bymentioning

confidence: 99%

A numerical approach to the optimal control and efficiency of the copepod swimmer

Bonnard

Chyba

Rouot

et al. 2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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Abstract-This article presents a geometric and numerical approach to compute the optimal swimming strokes of a larval copepod. A simplified model of locomotion at low Reynolds number is analyzed in the framework of Sub-Riemannian geometry. Both normal and abnormal geodesics are considered along which the mechanical power dissipated by the swimmer is conserved. Numerical simulations show that, among various periodic strokes, a normal stroke consisting of a simple loop shape is maximizing the efficiency.

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“…We recall the generic classification of rank 2 distributions in IR 3 , see [45], with its interpretation using singular trajectories. Hence we consider a system :…”

Section: Local Classification Of Rank 2 Generic Distributions D In Irmentioning

confidence: 99%

“…From [45], the singularity x = z = 0 is a weak focus and a spiral passing through 0 is with infinite length. Since any minimizer is smooth no piece of abnormal geodesic is a minimizer when computing the distance to 0.…”

Section: Abnormal Geodesicsmentioning

confidence: 99%

On the role of abnormal minimizers in sub-riemannian geometry

Bonnard¹,

Trélat²

2001

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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Consider a sub-Riemannian geometry (U, D, g) where U is a neighborhood at 0 in IR n , D is a rank-2 smooth (C ∞ or C ω ) distribution and g is a smooth metric on D. The objective of this article is to explain the role of abnormal minimizers in SR-geometry. It is based on the analysis of the Martinet SR-geometry.Key words : optimal control, singular trajectories, sub-Riemannian geometry, abnormal minimizers, sphere and wave-front with small radii.Résumé On considère un problème sous-Riemannien (U, D, g) où U est un voisinage de 0 dans IR n , D une distribution lisse de rang 2 et g une métrique lisse sur D. L'objectif de cet article est d'expliquer le rôle des géodésiques anormales minimisantes en géométrie SR. Cette analyse est fondée sur le modèle SR de Martinet.Titre Le rôle des géodésiques anormales minimisantes en géométrie sousRiemannienne.

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Typical Singularities of Differential 1-Forms and Pfaffian Equations

Cited by 52 publications

References 0 publications

Optimal Strokes at Low Reynolds Number: A Geometric and Numerical Study of Copepod and Purcell Swimmers

Optimal Strokes at Low Reynolds Number: A Geometric and Numerical Study of Copepod and Purcell Swimmers

A numerical approach to the optimal control and efficiency of the copepod swimmer

On the role of abnormal minimizers in sub-riemannian geometry

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