Finite-jet determination of small-time local controllability
Let $n \ge 3$ and let $X_0, \dots, X_m$ be real-analytic vector fields near $0 \in \mathbb{R}^n$ with $X_0(0) = 0$. Consider the control-affine system $$\dot x = X_0(x) + \sum_{j=1}^m u_j(t) X_j(x)$$ with measurable controls $u = (u_1, \dots, u_m)$ taking values in $[-1,1]^m$. It is small-time locally controllable at $0$ if, for every sufficiently small $t > 0$, the set of points reachable from $0$ in time at most $t$ contains a neighbourhood of $0$.
Results of Petrov give a positive answer to the question below in the plane, as Agrachev and Jafarpour note. For families of vector fields, Agrachev observed that a positive answer would follow if a number of switchings independent of $t$ always sufficed, and that this fails for some pairs of polynomial vector fields in $\mathbb{R}^4$, by an example of Kawski. Jafarpour proved that if, in addition, the reachable sets contain balls $B(0, C t^N)$ for all small $t > 0$, then every system whose vector fields have the same Taylor polynomials of order $N$ at $0$ as the $X_j$ again has this property, and so is small-time locally controllable (under a completeness assumption on the vector fields).
Problem. If the system is small-time locally controllable at $0$, must there be an integer $N$, which may depend on the system, such that every real-analytic tuple $Y_0, \dots, Y_m$ with $$D^\alpha Y_j(0) = D^\alpha X_j(0) \qquad (0 \le j \le m,\ |\alpha| \le N)$$ is also small-time locally controllable at $0$, with the same control set $[-1,1]^m$?
Source: Agrachev (1999), question (a), for families of real-analytic vector fields; for control-affine systems, Jafarpour (2020), Conjecture 1.1. Stated for n ≥ 3 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, control-affine systems, nonlinear control, small-time local controllability
References
@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},
}
@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},
}
@article{petrov1976,
author = {Petrov, N. N.},
title = {Local controllability},
journal = {Differentsial'nye Uravneniya},
volume = {12},
number = {12},
year = {1976},
pages = {2214--2222},
}
@article{kawski1988,
author = {Kawski, Matthias},
title = {Control variations with an increasing number of switchings},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {18},
number = {2},
year = {1988},
pages = {149--152},
doi = {10.1090/S0273-0979-1988-15630-3},
}
@misc{aim-finite-jet-small-time-control,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Finite-jet determination of small-time local controllability}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/141-finite-jet-small-time-control.md},
note = {Entry 141 at commit a153f46, 2026-10-13},
}
Resolution claims
Small-time local controllability asks whether a system can reach a neighborhood of an equilibrium in arbitrarily short time. We give a negative answer to the finite-jet determination question for real-analytic systems with bounded measurable controls. Starting from a previously constructed five-dimensional polynomial system with two controls, we show that adding a positive polynomial term of arbitrarily high order destroys this …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | Failure of finite-jet determination for small-time local controllability v1 recorded from AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00519, "Finite-jet determination of small-time local controllability", clopen.solutions.