Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane*

Yuri L. Sachkov

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 2, page 293-321
  • ISSN: 1292-8119

Abstract

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The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal synthesis is constructed.

How to cite

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Sachkov, Yuri L.. "Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane*." ESAIM: Control, Optimisation and Calculus of Variations 17.2 (2011): 293-321. <http://eudml.org/doc/276331>.

@article{Sachkov2011,
abstract = { The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal synthesis is constructed. },
author = {Sachkov, Yuri L.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Optimal control; sub-Riemannian geometry; differential-geometric methods; left-invariant problem; group of motions of a plane; rototranslations; cut locus; optimal synthesis; optimal control},
language = {eng},
month = {5},
number = {2},
pages = {293-321},
publisher = {EDP Sciences},
title = {Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane*},
url = {http://eudml.org/doc/276331},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Sachkov, Yuri L.
TI - Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane*
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/5//
PB - EDP Sciences
VL - 17
IS - 2
SP - 293
EP - 321
AB - The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal synthesis is constructed.
LA - eng
KW - Optimal control; sub-Riemannian geometry; differential-geometric methods; left-invariant problem; group of motions of a plane; rototranslations; cut locus; optimal synthesis; optimal control
UR - http://eudml.org/doc/276331
ER -

References

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  7. I. Moiseev and Yu. L. Sachkov, Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV (2009) DOI: .  Zbl1217.49037DOI10.1051/cocv/2009004
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  10. Yu.L. Sachkov, Conjugate and cut time in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV (2009) DOI: .  DOI10.1051/cocv/2009031
  11. A.M. Vershik and V.Y. Gershkovich, Nonholonomic Dynamical Systems. Geometry of distributions and variational problems, in Itogi Nauki i Tekhniki: Sovremennye Problemy Matematiki, Fundamental'nyje Napravleniya16, VINITI, Moscow (1987) 5–85 [in Russian]. [English translation in Encyclopedia of Math. Sci.16, Dynamical Systems7, Springer Verlag.]  
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  13. S. Wolfram, Mathematica: a system for doing mathematics by computer. Addison-Wesley, Reading, USA (1991).  Zbl0671.65002

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