Polynomial growth of reachable neighborhoods for analytic control systems
Let $n \ge 3$ and let $X_0, \dots, X_m$ be real-analytic vector fields near $0 \in \mathbb{R}^n$, with $X_0(0) = 0$. For measurable controls $u(t) \in [-1,1]^m$ consider $$\dot{x} = X_0(x) + \sum_{j=1}^{m} u_j(t) X_j(x), \qquad x(0) = 0,$$ and let $R_{\le t}(0)$ be the set of endpoints reached in time at most $t$ by trajectories that stay in the domain of the vector fields. Assume that $0 \in \operatorname{int} R_{\le t}(0)$ for every sufficiently small $t > 0$. Jafarpour recalls that an inclusion as below is equivalent to local Hölder continuity of the minimum-time function, and that such an inclusion holds for every small-time locally controllable analytic system in dimension two. He proved that if an analytic system satisfies such an inclusion with exponent $N$, then every analytic system whose vector fields agree with its own up to terms of order higher than $N$ also satisfies the assumption above.
Problem. Must there exist $C, T > 0$ and an integer $N \ge 1$, depending on the system, such that $$B(0, C t^N) \subseteq R_{\le t}(0) \qquad (0 < t < T)?$$
Source: Jafarpour (2020), Conjecture 1.2, with reference to Agrachev (1999); stated for n ≥ 3 and controls in [−1, 1]^m by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, analytic control systems, reachable sets, small-time local controllability
References
@article{jafarpour2020,
author = {Jafarpour, Saber},
title = {On small-time local controllability},
journal = {SIAM J. Control Optim.},
volume = {58},
number = {1},
year = {2020},
pages = {425--446},
doi = {10.1137/16M1068797},
eprint = {1604.02432},
archivePrefix = {arXiv},
}
@incollection{agrachev1999,
author = {Agrachev, Andrei A.},
title = {Is it possible to recognize local controllability in a finite number of differentiations?},
booktitle = {Open Problems in Mathematical Systems and Control Theory},
editor = {Blondel, Vincent D. and Sontag, Eduardo D. and Vidyasagar, M. and Willems, Jan C.},
series = {Communications and Control Engineering},
publisher = {Springer},
address = {London},
year = {1999},
pages = {15--18},
doi = {10.1007/978-1-4471-0807-8_4},
}
@misc{aim-analytic-reachable-set-polynomial-growth,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Polynomial growth of reachable neighborhoods for analytic control systems}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/417-analytic-reachable-set-polynomial-growth.md},
note = {Entry 417 at commit a153f46, 2026-10-13},
}
Resolution claims
We construct a polynomial control-affine system in five dimensions, with two controls in $[-1,1]$, that is small-time locally controllable at the origin but admits no polynomial lower bound for its reachable neighborhoods. Every point on the negative fourth axis reachable in time at most $t$ has distance less than $t^{11}e^{-1/(6t)}/4$ from the origin. Nevertheless, exact positive and negative fourth-coordinate loops, …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00740, "Polynomial growth of reachable neighborhoods for analytic control systems", clopen.solutions.