Sharp characterization of exactly trackable boundary heat fluxes
Let $\Omega \subset \mathbb{R}^d$, $d \ge 2$, be a bounded connected domain with smooth boundary, let $T > 0$, and let $\Gamma, \Sigma \subset \partial\Omega$ be nonempty relatively open sets with disjoint closures. For $u \in L^2((0,T) \times \Gamma)$ let $y_u$ be the solution, in the sense of transposition, of $$\partial_t y_u - \Delta y_u = 0 \ \text{ in } (0,T) \times \Omega, \qquad y_u(0,\cdot) = 0, \qquad y_u|_{\partial\Omega} = \mathbf{1}_\Gamma u,$$ and let $$\mathcal{R}_T = \{\partial_\nu y_u|_{(0,T) \times \Sigma} : u \in L^2((0,T) \times \Gamma)\}$$ be the space of exactly trackable fluxes, where $\nu$ is the outward unit normal. Bárcena-Petisco and Zuazua proved that $\mathcal{R}_T$ is dense in $L^2((0,T) \times \Sigma)$, and noted that its elements are smooth. For the heat equation on an interval, with a homogeneous Dirichlet condition and the flux observed at one end and the control at the other, they showed that every Gevrey function of order less than $2$ in $t$ whose derivatives all vanish at $t = 0$ is exactly trackable. Davron and Lissy characterized, on an interval with a Neumann control, the Dirichlet outputs that can be tracked exactly with square-integrable controls, by Gevrey-type conditions of order $2$.
Problem. Characterize $\mathcal{R}_T$ by necessary and sufficient regularity and compatibility conditions on the target flux itself, including the sharp quantitative restrictions near $t = 0$. Restating membership as the existence of a control, or as an abstract dual inequality, is not the requested function-space characterization.
Source: Bárcena-Petisco and Zuazua (2025), Remark II.5, who ask for the sharp space of trackable targets; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, boundary control, heat equation, tracking controllability
References
@article{barcenapetisco2025,
author = {B{\'a}rcena-Petisco, Jon Asier and Zuazua, Enrique},
title = {Tracking controllability for the heat equation},
journal = {IEEE Trans. Automat. Control},
volume = {70},
number = {3},
year = {2025},
pages = {1935--1940},
doi = {10.1109/TAC.2024.3476174},
eprint = {2310.00314},
archivePrefix = {arXiv},
}
@misc{davron2025,
author = {Davron, Lucas and Lissy, Pierre},
title = {Exact output tracking for the one-dimensional heat equation and applications to the interpolation problem in {G}evrey classes of order 2},
year = {2025},
eprint = {2511.07326},
archivePrefix = {arXiv},
}
@misc{aim-sharp-heat-tracking-range,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp characterization of exactly trackable boundary heat fluxes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/139-sharp-heat-tracking-range.md},
note = {Entry 139 at commit a153f46, 2026-10-13},
}
Resolution claims
We characterize the normal heat fluxes that can be produced on an observation patch by square-integrable Dirichlet data on a disjoint control patch. The domain is any bounded connected smooth Euclidean domain in dimension at least two. The characterization consists of explicit conditions on the target: causal smoothness, weighted Taylor coefficients computed from a Gaussian surface potential, agreement of the …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/heat_tracking.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00770, "Sharp characterization of exactly trackable boundary heat fluxes", clopen.solutions.