Asymptotics of accessibility sets along an abnormal trajectory

Emmanuel Trélat

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • 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 L 2 -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 (2001): 387-414. <http://eudml.org/doc/90599>.

@article{Trélat2001,
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 $\gamma $ into two sectors, bordered by the first Pontryagin’s cone along $\gamma $, called the $\mathrm \{L\}^\{\infty \}$-sector and the $\mathrm \{L\}^\{2\}$-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; sub-Riemannian geometry; bounded control; normal form; spectral analytic tools},
language = {eng},
pages = {387-414},
publisher = {EDP-Sciences},
title = {Asymptotics of accessibility sets along an abnormal trajectory},
url = {http://eudml.org/doc/90599},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Trélat, Emmanuel
TI - Asymptotics of accessibility sets along an abnormal trajectory
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
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 $\gamma $ into two sectors, bordered by the first Pontryagin’s cone along $\gamma $, called the $\mathrm {L}^{\infty }$-sector and the $\mathrm {L}^{2}$-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; sub-Riemannian geometry; bounded control; normal form; spectral analytic tools
UR - http://eudml.org/doc/90599
ER -

References

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