Uniqueness of nonnegative heat controls at the minimum time
Fix constants $a, b > 0$ with $a \ne b$. On the rod $(0,1)$ consider $y_t = y_{xx}$ with $y(0,x) = a$ and nonnegative Dirichlet controls $y(t,0) = u_0(t)$, $y(t,1) = u_1(t)$. Let $T_*$ be the infimum of the times $T$ in which bounded nonnegative controls can produce $y(T,\cdot) = b$. A pair $(\mu_0, \mu_1)$ of nonnegative finite Radon measures on $[0, T_*]$ is admissible if, for every integer $k \ge 1$, $$\int_{[0,T_*]} e^{-k^2\pi^2(T_* - t)} \bigl(d\mu_0(t) - (-1)^k\, d\mu_1(t)\bigr) = \frac{(1 - (-1)^k)\bigl(b - a e^{-k^2\pi^2 T_*}\bigr)}{k^2\pi^2};$$ these are the equations for the sine coefficients of the terminal state. Lohéac, Trélat and Zuazua proved, for nonnegative integrable controls, that the minimal time is positive and finite and is attained by nonnegative Radon measure controls, and that for a symmetric initial state the two controls can be taken equal at every larger time. For finite-dimensional linear systems whose matrix has only real eigenvalues, they later proved that, when the minimal time is positive, the nonnegative control at the minimal time is unique and is a finite sum of Dirac impulses.
Problem. Is the pair of nonnegative finite Radon measures attaining the minimum time unique, that is, is there exactly one admissible pair $(\mu_0, \mu_1)$ at $T_*$, for every $a, b$ as above?
Source: Lohéac, Trélat and Zuazua (2017), §6, "Uniqueness of controls at the minimal time"; stated for constant states, bounded controls and in this moment 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, constrained control, heat equation, minimal control time
References
@article{loheac2017,
author = {Loh{\'e}ac, J{\'e}r{\^o}me and Tr{\'e}lat, Emmanuel and Zuazua, Enrique},
title = {Minimal controllability time for the heat equation under unilateral state or control constraints},
journal = {Math. Models Methods Appl. Sci.},
volume = {27},
number = {9},
year = {2017},
pages = {1587--1644},
doi = {10.1142/S0218202517500270},
}
@article{loheac2021,
author = {Loh{\'e}ac, J{\'e}r{\^o}me and Tr{\'e}lat, Emmanuel and Zuazua, Enrique},
title = {Nonnegative control of finite-dimensional linear systems},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {38},
number = {2},
year = {2021},
pages = {301--346},
doi = {10.1016/j.anihpc.2020.07.004},
}
@misc{aim-unique-minimum-time-positive-heat-control,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of nonnegative heat controls at the minimum time}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/138-unique-minimum-time-positive-heat-control.md},
note = {Entry 138 at commit a153f46, 2026-10-13},
}
Resolution claims
We consider the heat equation on an interval with two nonnegative Dirichlet boundary controls. For every pair of distinct positive constant states, we prove that the nonnegative finite Radon measure pair attaining the minimum time is unique, under the terminal sine-moment formulation. The two boundary measures are equal. This answers the continuum uniqueness question posed by Lohéac, Trélat and Zuazua. …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00776, "Uniqueness of nonnegative heat controls at the minimum time", clopen.solutions.