Sharp vanishing-viscosity control time for downstream transport
Fix $L, M > 0$. For $\varepsilon > 0$ and $T > 0$ consider $$y_t + M y_x - \varepsilon y_{xx} = 0 \ \text{ in } (0,T) \times (0,L), \qquad y(t,0) = h(t), \quad y(t,L) = 0,$$ with $y(0) = y_0 \in L^2(0,L)$ and a boundary control $h \in L^2(0,T)$ at the inflow endpoint, the solution taken in the sense of transposition, and let $$C(T,\varepsilon) = \sup_{\|y_0\|_{L^2(0,L)} \le 1} \inf\bigl\{\|h\|_{L^2(0,T)} : y(T) = 0\bigr\}.$$ For every $\varepsilon > 0$ and $T > 0$ the system is null-controllable, so $C(T,\varepsilon)$ is finite. The transport equation $y_t + M y_x = 0$ obtained for $\varepsilon = 0$ is null-controllable from $x = 0$ if and only if $T \ge L/M$. Coron and Guerrero proved that $C(T,\varepsilon)$ grows exponentially in $1/\varepsilon$ as $\varepsilon \to 0$ when $T < L/M$, and tends to zero when $T > 4.3\,L/M$. Glass, Lissy, and Dardé and Ervedoza lowered the constant $4.3$ to $4.2$, $2\sqrt{3}$ and $4\sqrt{K_0} \approx 3.34$ respectively. For negative speed $M < 0$, Lissy proved that the cost does not stay bounded as $\varepsilon \to 0$ for times $T < 2\sqrt{2}\,L/|M|$, beyond the threshold $2L/|M|$ that Coron and Guerrero's lower bound suggested.
Problem. Is it true that for every $T > L/M$, $$\lim_{\varepsilon \downarrow 0} C(T,\varepsilon) = 0?$$
Source: Coron and Guerrero (2005), as a conjecture on the optimal time; see Lissy (2014), Conjecture 3, and Laurent and Léautaud (2023), §1. Stated for positive speed by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, null controllability, transport-diffusion equations, vanishing viscosity limits
References
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pages = {237--257},
doi = {10.3233/ASY-2005-707},
}
@article{lissy2014,
author = {Lissy, Pierre},
title = {An application of a conjecture due to {E}rvedoza and {Z}uazua concerning the observability of the heat equation in small time to a conjecture due to {C}oron and {G}uerrero concerning the uniform controllability of a convection--diffusion equation in the vanishing viscosity limit},
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volume = {69},
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doi = {10.1016/j.sysconle.2014.04.011},
eprint = {1310.4355},
archivePrefix = {arXiv},
}
@article{laurent2023,
author = {Laurent, Camille and L{\'e}autaud, Matthieu},
title = {On uniform controllability of 1{D} transport equations in the vanishing viscosity limit},
journal = {C. R. Math. Acad. Sci. Paris},
volume = {361},
year = {2023},
pages = {265--312},
doi = {10.5802/crmath.405},
eprint = {2203.03266},
archivePrefix = {arXiv},
}
@article{glass2010,
author = {Glass, Olivier},
title = {A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit},
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}
@article{lissy2012,
author = {Lissy, Pierre},
title = {A link between the cost of fast controls for the 1-{D} heat equation and the uniform controllability of a 1-{D} transport-diffusion equation},
journal = {C. R. Math. Acad. Sci. Paris},
volume = {350},
number = {11--12},
year = {2012},
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}
@article{darde2019,
author = {Dard{\'e}, J{\'e}r{\'e}mi and Ervedoza, Sylvain},
title = {On the cost of observability in small times for the one-dimensional heat equation},
journal = {Anal. PDE},
volume = {12},
number = {6},
year = {2019},
pages = {1455--1488},
doi = {10.2140/apde.2019.12.1455},
}
@article{lissy2015,
author = {Lissy, Pierre},
title = {Explicit lower bounds for the cost of fast controls for some 1-{D} parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-{D} transport--diffusion equation},
journal = {J. Differential Equations},
volume = {259},
number = {10},
year = {2015},
pages = {5331--5352},
doi = {10.1016/j.jde.2015.06.031},
}
@misc{aim-positive-transport-sharp-uniform-control-time,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp vanishing-viscosity control time for downstream transport}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/412-positive-transport-sharp-uniform-control-time.md},
note = {Entry 412 at commit a153f46, 2026-10-13},
}
Resolution claims
We study the uniform null-controllability, in the vanishing viscosity limit, of the transport--diffusion equation $u_t + M u_x = \varepsilon u_{xx}$ on $[0,L]$ with a Dirichlet boundary control at $x=0$. Our results concern the uniform null-controllability time $T_{\mathrm{unif}}$, defined as the infimum of the times for which the null-controllability cost remains bounded as $\varepsilon \to 0$. We determine this time …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: arXiv:2609.35355v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00496, "Sharp vanishing-viscosity control time for downstream transport", clopen.solutions.