Asymptotics of accessibility sets along an abnormal trajectory

Emmanuel Trélat

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 6, page 387-414
  • ISSN: 1292-8119

Abstract

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We describe precisely, under generic conditions, the contact of the accessibility set at time T with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer γ into two sectors, bordered by the first Pontryagin's cone along γ, called the L∞-sector and the L2-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.

How to cite

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Trélat, Emmanuel. "Asymptotics of accessibility sets along an abnormal trajectory." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 387-414. <http://eudml.org/doc/197282>.

@article{Trélat2010,
abstract = { We describe precisely, under generic conditions, the contact of the accessibility set at time T with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer γ into two sectors, bordered by the first Pontryagin's cone along γ, called the L∞-sector and the L2-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem. },
author = {Trélat, Emmanuel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Accessibility set; abnormal trajectory; end-point mapping; single-input affine control system; sub-Riemannian geometry.; accessibilty set; abnormal trajectory; single-input affine control system; sub-Riemannian geometry; bounded control; normal form; spectral analytic tools},
language = {eng},
month = {3},
pages = {387-414},
publisher = {EDP Sciences},
title = {Asymptotics of accessibility sets along an abnormal trajectory},
url = {http://eudml.org/doc/197282},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Trélat, Emmanuel
TI - Asymptotics of accessibility sets along an abnormal trajectory
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 387
EP - 414
AB - We describe precisely, under generic conditions, the contact of the accessibility set at time T with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer γ into two sectors, bordered by the first Pontryagin's cone along γ, called the L∞-sector and the L2-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.
LA - eng
KW - Accessibility set; abnormal trajectory; end-point mapping; single-input affine control system; sub-Riemannian geometry.; accessibilty set; abnormal trajectory; single-input affine control system; sub-Riemannian geometry; bounded control; normal form; spectral analytic tools
UR - http://eudml.org/doc/197282
ER -

References

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