Boundary control to equilibrium for a parabolic obstacle problem
Let $I = (0,1)$ and let $\psi \in C^\infty([0,1])$ satisfy $\psi(0), \psi(1) < 0$ and $\max \psi > 0$. Let $\bar y \in H^1_0(I)$ be the unique minimizer of $\int_I |y_x|^2$ subject to $y \ge \psi$. For $T > 0$, endpoint data $g_0, g_1$ and $y_0 \in H^1_0(I)$ with $y_0 \ge \psi$, consider the parabolic obstacle problem $$\min\{y_t - y_{xx},\, y - \psi\} = 0, \qquad y(t, i) = g_i(t) \ (i = 0, 1), \qquad y(0) = y_0,$$ meaning $y \ge \psi$, $y_t - y_{xx} \ge 0$ and $(y - \psi)(y_t - y_{xx}) = 0$, interpreted through the associated parabolic variational inequality. Its solution can be seen as the limit of solutions of penalised semilinear heat equations (Pighin and Zuazua), and with zero endpoint data $\bar y$ is an attractor of its solutions (Geshkovski).
Problem. For every $y_0 \in H^1_0(I)$ with $y_0 \ge \psi$, do there exist a finite $T > 0$ and endpoint controls $g_0, g_1 \in H^1(0,T)$, with $g_i(0) = g_i(T) = 0$ and $g_i(t) \ge \psi(i)$, such that the variational solution satisfies $y(T) = \bar y$ in $L^2(I)$?
Source: Geshkovski (2021), PhD thesis, §1.5.1, who poses exact controllability of the parabolic obstacle problem to the stationary obstacle solution, for instance by boundary control; stated in this one-dimensional form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, boundary control, obstacle problem, variational inequalities
References
@phdthesis{geshkovski2021,
author = {Geshkovski, Borjan},
title = {Control in moving interfaces and deep learning},
school = {Universidad Aut{\'o}noma de Madrid},
year = {2021},
url = {https://cmc.deusto.eus/wp-content/uploads/PhDThesis-bGeshkovski-may2021.pdf},
}
@article{pighin2018,
author = {Pighin, Dario and Zuazua, Enrique},
title = {Controllability under positivity constraints of semilinear heat equations},
journal = {Math. Control Relat. Fields},
volume = {8},
number = {3-4},
year = {2018},
pages = {935--964},
doi = {10.3934/mcrf.2018041},
eprint = {1711.07678},
archivePrefix = {arXiv},
}
@misc{aim-parabolic-obstacle-boundary-control,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Boundary control to equilibrium for a parabolic obstacle problem}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/398-parabolic-obstacle-boundary-control.md},
note = {Entry 398 at commit a153f46, 2026-10-13},
}
Resolution claims
We give a smooth one-dimensional obstacle and a smooth admissible initial state for which no endpoint controls can steer the parabolic obstacle problem exactly to its stationary energy minimizer in finite time. The obstacle has two interior maxima of equal height. The stationary solution is constant between them, while the obstacle prevents the evolving state from falling below that height …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/obstacle_control_counterexample.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00764, "Boundary control to equilibrium for a parabolic obstacle problem", clopen.solutions.