Exact null control of a harmonically confined kinetic equation from a wedge
For $0 < \theta < \pi/4$ let $\omega_\theta = \{(x, v) \in \mathbb{R}^2 : v = \alpha x \text{ for some } 0 < \alpha < \tan\theta\}$, and consider the Kolmogorov equation with a harmonic external potential, $$\partial_t f + v \partial_x f - x \partial_v f - \partial_v^2 f = \mathbf{1}_{\omega_\theta} h, \qquad f(0) = f_0 \in L^2(\mathbb{R}^2).$$ The flow of its transport part rotates the $(x, v)$ plane. Beauchard, Egidi and Pravda-Starov proved that $\omega_\theta$ satisfies their integral thickness condition for this equation in time $T$ when $T > \pi - \theta$ and not when $T < \pi - \theta$, and that the equation is not null-controllable from $\omega_\theta$ in time $T < \pi - \theta$ (2020). Alphonse and Martin proved that the condition also fails for $T = \pi - \theta$, and that the equation is approximately null-controllable with uniform cost from $\omega_\theta$ in time $T$ if and only if $T > \pi - \theta$: for every $\varepsilon > 0$ and every $f_0$ there is a control $h$ with $\|f(T)\|_{L^2} \le \varepsilon \|f_0\|_{L^2}$ and $\|h\|_{L^2((0,T) \times \omega_\theta)}^2 \le C_{\varepsilon,T} \|f_0\|_{L^2}^2$ (2023).
Problem. Is it true that for every $T > \pi - \theta$ and every $f_0$ there is $h \in L^2((0,T) \times \omega_\theta)$ for which the mild solution satisfies $f(T) = 0$?
Source: Beauchard, Egidi and Pravda-Starov (2020), Proposition 2.4(iii)(c), for T ≥ π − θ; Alphonse and Martin (2023), Example 2.7, excluded T = π − θ.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, hypoelliptic equations, null controllability, Ornstein–Uhlenbeck operators
References
@article{beauchard2020,
author = {Beauchard, Karine and Egidi, Michela and Pravda-Starov, Karel},
title = {Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports},
journal = {C. R. Math. Acad. Sci. Paris},
volume = {358},
number = {6},
year = {2020},
pages = {651--700},
doi = {10.5802/crmath.79},
eprint = {1908.10603},
archivePrefix = {arXiv},
}
@article{alphonse2023,
author = {Alphonse, Paul and Martin, J{\'e}r{\'e}my},
title = {Approximate null-controllability with uniform cost for the hypoelliptic {O}rnstein--{U}hlenbeck equations},
journal = {SIAM J. Control Optim.},
volume = {61},
number = {3},
year = {2023},
pages = {1679--1711},
doi = {10.1137/22M1487412},
eprint = {2201.01516},
archivePrefix = {arXiv},
}
@misc{aim-harmonic-kolmogorov-wedge-null-control,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Exact null control of a harmonically confined kinetic equation from a wedge}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/395-harmonic-kolmogorov-wedge-null-control.md},
note = {Entry 395 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-00529, "Exact null control of a harmonically confined kinetic equation from a wedge", clopen.solutions.