PreprintCC BY-SA 4.0arXiv
Some quartic control results for scalar-input systems
Abstract
We investigate the role of quartic terms in the small-time local controllability of scalar-input systems. First, we prove a new sufficient condition for controllability, which exploits simultaneously more good quartic Lie brackets than previous results. Second, we identify a family of quartic obstructions to controllability, relying on Lie brackets whose coordinates of the second kind are positive-definite functionals of the control. Third, we show that the complementarity of these results can be seen as an answer to Kawski's 1987 open problem concerning the construction of a Hall basis which somehow separates good and bad quartic brackets. We give examples and remarks illustrating some of the intricacies of these notions.
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BibTeX
@misc{beauchard2026some,
title = {{Some quartic control results for scalar-input systems}},
author = {Karine Beauchard and Frédéric Marbach},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28582},
eprint = {2608.28582},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28582},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Some quartic control results for scalar-input systems
AU - Karine Beauchard
AU - Frédéric Marbach
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28582
UR - https://arxiv.org/abs/2608.28582
AB - We investigate the role of quartic terms in the small-time local controllability of scalar-input systems. First, we prove a new sufficient condition for controllability, which exploits simultaneously more good quartic Lie brackets than previous results. Second, we identify a family of quartic obstructions to controllability, relying on Lie brackets whose coordinates of the second kind are positive-definite functionals of the control. Third, we show that the complementarity of these results can be seen as an answer to Kawski's 1987 open problem concerning the construction of a Hall basis which somehow separates good and bad quartic brackets. We give examples and remarks illustrating some of the intricacies of these notions.
ER -