PreprintCC BY-SA 4.0HAL
Factor-parity Hall sets and controllability
Abstract
We introduce a class of Hall sets, which we call factor-parity Hall sets, whose elements split into good and bad brackets. We prove that the good brackets can be steered simultaneously and arbitrarily in small time, which yields sufficient conditions for the small-time local controllability of control-affine systems. This positive result draws on constructions of Kawski, Agrachev-Gamkrelidze and Krastanov. Conversely, we prove that each bad bracket generates an obstruction to controllability, hence a family of necessary conditions.
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BibTeX
@misc{beauchard2026factorparity,
title = {{Factor-parity Hall sets and controllability}},
author = {Karine Beauchard and Frédéric Marbach},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04056},
eprint = {2609.04056},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05769671v1},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Factor-parity Hall sets and controllability
AU - Karine Beauchard
AU - Frédéric Marbach
PY - 2026
DA - 2026/09/03
PB - HAL
DO - 10.48550/arxiv.2609.04056
UR - https://hal.science/hal-05769671v1
AB - We introduce a class of Hall sets, which we call factor-parity Hall sets, whose elements split into good and bad brackets. We prove that the good brackets can be steered simultaneously and arbitrarily in small time, which yields sufficient conditions for the small-time local controllability of control-affine systems. This positive result draws on constructions of Kawski, Agrachev-Gamkrelidze and Krastanov. Conversely, we prove that each bad bracket generates an obstruction to controllability, hence a family of necessary conditions.
ER -