The sharp local maneuvering time for a shallow-water tank
A tank containing a shallow fluid is moved horizontally, its acceleration $a(t)$ being the control. For $0 < x < 1$ consider the normalized Saint-Venant tank system $$H_t + (Hv)_x = 0, \qquad v_t + (H + v^2/2)_x = -a(t), \qquad v(t,0) = v(t,1) = 0, \qquad s' = a, \quad D' = s,$$ for the fluid depth $H$, the fluid velocity $v$ relative to the tank, and the tank's velocity $s$ and displacement $D$. The control $a$ is continuous, and $H, v$ are $C^1$. The equilibrium is $(H, v, s, D) = (1, 0, 0, 0)$. Let $\mathcal{Y}$ consist of the $(H, v, s, D) \in C^1([0,1])^2 \times \mathbb{R}^2$ with $H > 0$, $\int_0^1 H = 1$, $v(0) = v(1) = 0$ and $H_x(0) = H_x(1)$, with the norm $\|H - 1\|_{C^1} + \|v\|_{C^1} + |s| + |D|$. In this normalization waves of the linearized system cross the tank in time $1$. Coron proved local controllability around the equilibrium for a large enough time, by the return method. Coron, Koenig and Nguyen proved that local controllability fails for every time less than $2$, even between states in which the tank is at rest.
Problem. Is the following local controllability property valid for every $T > 2$? For every $\varepsilon > 0$ there exists $\eta > 0$ such that any two states in $\mathcal{Y}$ within $\eta$ of the equilibrium can be joined in time $T$ by a classical trajectory satisfying $$\sup_{0 \le t \le T} \bigl( \|H(t) - 1\|_{C^1} + \|v(t)\|_{C^1} + |s(t)| + |D(t)| + |a(t)| \bigr) < \varepsilon.$$ The endpoint $T = 2$ is not included.
Source: Coron (2004), Problem 7.1, in Blondel and Megretski (eds.), Unsolved Problems in Mathematical Systems and Control Theory, for the time 2 in this normalization; stated in this 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, local controllability, Saint-Venant equations, water tank
References
@incollection{coron2004,
author = {Coron, Jean-Michel},
title = {Problem 7.1: Time for local controllability of a 1-{D} tank containing a fluid modeled by the shallow water equations},
booktitle = {Unsolved Problems in Mathematical Systems and Control Theory},
editor = {Blondel, Vincent D. and Megretski, Alexandre},
publisher = {Princeton University Press},
address = {Princeton, NJ},
year = {2004},
pages = {247--250},
doi = {10.1515/9781400826155.247},
}
@article{coron2002,
author = {Coron, Jean-Michel},
title = {Local controllability of a 1-{D} tank containing a fluid modeled by the shallow water equations},
journal = {ESAIM Control Optim. Calc. Var.},
volume = {8},
year = {2002},
pages = {513--554},
doi = {10.1051/cocv:2002050},
}
@article{coron2024,
author = {Coron, Jean-Michel and Koenig, Armand and Nguyen, Hoai-Minh},
title = {Lack of local controllability for a water-tank system when the time is not large enough},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {41},
number = {6},
year = {2024},
pages = {1327--1365},
doi = {10.4171/AIHPC/123},
}
@misc{aim-shallow-water-tank-sharp-control-time,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The sharp local maneuvering time for a shallow-water tank}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/387-shallow-water-tank-sharp-control-time.md},
note = {Entry 387 at commit a153f46, 2026-10-13},
}
Resolution claims
We consider a one-dimensional tank governed by the normalized Saint--Venant equations, with the tank acceleration as its only control. We show that local exact controllability around the flat equilibrium, with uniformly small classical trajectories in $C^1$, fails at time $T=21/10$. The terminal states form an explicit family tending to equilibrium, and the tank starts and ends with zero velocity and …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/obstruction.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00769, "The sharp local maneuvering time for a shallow-water tank", clopen.solutions.