Continuous selection of small-time null controls for analytic systems
Let $f_0, f_1 : \mathbb{R}^n \to \mathbb{R}^n$ be real analytic with $f_0(0) = 0$, and consider the scalar-input system $$\dot y = f_0(y) + u(t) f_1(y), \qquad y(0) = y_0.$$ Coron identified continuous dependence of the controls on the initial state as key to passing from controllability to stabilisation, for instance by continuous time-varying feedback laws. As Marbach notes, all known sufficient conditions for small-time local null controllability, the hypothesis below, also give a continuous selection of controls as in the conclusion.
Problem. Assume that for every $T > 0$ some $\delta > 0$ has the following property: every $y_0$ with $|y_0| < \delta$ admits $u \in L^1(0,T)$ with $\|u\|_{L^1} \le T$ whose solution exists on $[0,T]$ and satisfies $y(T) = 0$. Must it follow that for every $T > 0$ there are $\delta_T > 0$ and a continuous map $$U_T : B_{\delta_T}(0) \longrightarrow L^1(0,T), \qquad U_T(0) = 0,$$ such that, for every $y_0 \in B_{\delta_T}(0)$, the trajectory driven by $U_T(y_0)$ exists on $[0,T]$ and ends at $0$? Continuity refers to the norm topology of $L^1(0,T)$.
Source: Marbach (2026), Open Problem 4.17, with Definitions 4.1 and 4.16; stated in this form, requiring the trajectories to exist on [0, T], 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, feedback stabilization, small-time local controllability
References
@misc{marbach2026,
author = {Marbach, Fr{\'e}d{\'e}ric},
title = {Time-iteration methods for controllability},
year = {2026},
eprint = {2602.19272},
archivePrefix = {arXiv},
note = {Lecture notes},
}
@article{coron1992,
author = {Coron, Jean-Michel},
title = {Links between local controllability and local continuous stabilization},
journal = {IFAC Proceedings Volumes},
volume = {25},
number = {13},
year = {1992},
pages = {165--171},
doi = {10.1016/S1474-6670(17)52276-4},
}
@misc{aim-analytic-controls-continuous-selection,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Continuous selection of small-time null controls for analytic systems}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/409-analytic-controls-continuous-selection.md},
note = {Entry 409 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00499, "Continuous selection of small-time null controls for analytic systems", clopen.solutions.