The sharp generic bound on iterated Fuller singularities
For a positive integer $n$, let $M$ be a smooth connected $n$-manifold and consider $$\dot q = f_0(q) + u f_1(q), \qquad u \in [-1, 1], \qquad f_0, f_1 \in \operatorname{Vec}(M).$$ An extremal triple $(q, u, \lambda)$, with $q : [0, T] \to M$ and $T > 0$, satisfies the Pontryagin maximum principle: $\lambda$ is an absolutely continuous, everywhere nonzero covector lift of $q$ and, almost everywhere, $$H(\lambda, u) = \max_{v \in [-1, 1]} H(\lambda, v), \qquad \dot\lambda = \vec H(\lambda, u), \qquad H(\lambda, v) = \langle \lambda, f_0(q) + v f_1(q) \rangle.$$ Let $O_q \subset [0, T]$ be the largest relatively open set on which some control associated with $q$ is smooth, and put $\Sigma = [0, T] \setminus O_q$. Let $\Sigma_0$ be the set of isolated points of $\Sigma$ and, for $k \ge 1$, let $\Sigma_k$ be the set of isolated points of $\Sigma \setminus \bigcup_{j < k} \Sigma_j$. Boarotto and Sigalotti proved that $\Sigma = \bigcup_{k=0}^{(n-1)^2} \Sigma_k$ for every extremal triple, for all pairs in an open dense subset of $\operatorname{Vec}(M)^2$. For $n = 3$ they proved that generically $\Sigma_k$ is empty for $k > 2$ along time-optimal trajectories.
Problem. Determine the least integer $K(n) \ge 0$, depending only on $n$, such that, for every $M$, an open dense subset of $\operatorname{Vec}(M)^2$, in the Whitney $C^\infty$ topology, satisfies $\Sigma = \bigcup_{k=0}^{K(n)} \Sigma_k$ for every extremal triple.
Source: Stated by AIM — Open Applied Problems, after Boarotto and Sigalotti (2019), the paragraph after Remark 4 (p. 329), who ask for the sharpest bound on the order of Fuller points.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Keywords
- AIM Open Applied Problems, control-affine systems, Fuller singularities
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{boarotto2019,
author = {Boarotto, Francesco and Sigalotti, Mario},
title = {Time-optimal trajectories of generic control-affine systems have at worst iterated {F}uller singularities},
journal = {Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire},
volume = {36},
number = {2},
year = {2019},
pages = {327--346},
doi = {10.1016/j.anihpc.2018.05.005},
eprint = {1705.10055},
archivePrefix = {arXiv},
}
@misc{aim-the-sharp-generic-bound-on-iterated-fuller-singularities,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The sharp generic bound on iterated Fuller singularities}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/730-the-sharp-generic-bound-on-iterated-fuller-singularities.md},
note = {Entry 730 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01126, "The sharp generic bound on iterated Fuller singularities", clopen.solutions.