Minimum-time eradication of an expanding set inside a geographic barrier
Let $V \subset \mathbb{R}^2$ be a bounded connected Lipschitz domain and $M > 0$. For $T > 0$, a motion is a set $E \subset (0,T) \times V$ of finite perimeter; write $\partial^* E$ for its reduced boundary, $\nu = (\nu_0, \nu_1, \nu_2)$ for its generalized inner unit normal and $\mathcal{H}^2$ for two-dimensional Hausdorff measure. It is admissible if $$\int_{\partial^* E \cap ((a,b) \times V)} \max\Bigl\{-\nu_0 + \sqrt{\nu_1^2 + \nu_2^2},\, 0\Bigr\}\, d\mathcal{H}^2 \le M(b - a) \qquad (0 \le a < b \le T),$$ and it eradicates in time $T$ if its slices $\Omega(t) = \{x : (t,x) \in E\}$ satisfy $\mathbf{1}_{\Omega(t)} \to \mathbf{1}_V$ as $t \downarrow 0$ and $\mathbf{1}_{\Omega(t)} \to 0$ as $t \uparrow T$, in $L^1(V)$. This is Bressan, Marchini and Staicu's weak formulation of a contaminated region $\Omega(t)$ that expands with unit speed where no control acts, and from which an area $M$ can be cleared per unit time: at a smooth front with inward normal speed $\beta$ the effort at time $t$ is $\int_{\partial\Omega(t) \cap V} \max\{1 + \beta, 0\}\, ds$, and the boundary of $V$ is a barrier that costs nothing. They proved that if some admissible motion eradicates in finite time, then one eradicates in minimum time, and that an admissible motion from $V$ to $\emptyset$ whose area strictly decreases is optimal if $\frac{d}{dt}|\Omega(t)| = \mathcal{H}^1(\partial\Omega(t) \cap V) - M$ and every $\Omega(t)$ has the least relative perimeter in $V$ among subsets of $V$ of the same area, as happens for a disc. Bressan observed that for many domains, a triangle for instance, no motion satisfies these conditions.
Problem. For every such $V$ and $M$ for which some admissible motion eradicates in finite time, determine the least such time $T$ and characterize the motions that attain it.
Source: Bressan (2025), Problem 3, for the constrained eradication problem of Bressan, Marchini and Staicu (2025). Stated by AIM — Open Applied Problems in their BV formulation, for Lipschitz domains.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, eradication strategies, moving sets, sets of finite perimeter
References
@article{bressan2025,
author = {Bressan, Alberto},
title = {Results and open questions on the boundary control of moving sets},
journal = {J. Math. Sci.},
volume = {300},
number = {2},
year = {2026},
pages = {122--150},
doi = {10.1007/s10958-026-08509-w},
eprint = {2502.06050},
archivePrefix = {arXiv},
}
@article{bressan2025bms,
author = {Bressan, Alberto and Marchini, Elsa M. and Staicu, Vasile},
title = {Optimally controlled moving sets with geographical constraints},
journal = {Milan J. Math.},
volume = {93},
number = {1},
year = {2025},
pages = {263--329},
doi = {10.1007/s00032-025-00419-x},
eprint = {2502.05968},
archivePrefix = {arXiv},
}
@misc{aim-minimum-time-constrained-eradication,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Minimum-time eradication of an expanding set inside a geographic barrier}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/259-minimum-time-constrained-eradication.md},
note = {Entry 259 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-00536, "Minimum-time eradication of an expanding set inside a geographic barrier", clopen.solutions.