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arXiv · 2608.28582

Some quartic control results for scalar-input systems

Karine Beauchard·Frédéric Marbach
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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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Karine Beauchard, Frédéric Marbach. 2026-08-28. Some quartic control results for scalar-input systems. https://arxiv.org/abs/2608.28582

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