Exact uniform control time near a viscous Burgers shock
Fix $L > 0$. For viscosity $\varepsilon > 0$ let $U^\varepsilon(x) = -\tanh\bigl(x/(2\varepsilon)\bigr)$, the stationary viscous Burgers shock, and consider the Burgers equation linearized at it, with a boundary control $h$ at the left endpoint: $$\partial_t u - \varepsilon \partial_x^2 u + \partial_x(U^\varepsilon u) = 0 \quad (-L < x < L), \qquad u(t,-L) = h(t), \quad u(t,L) = 0, \quad u(0) = u_0.$$ Let $$C(T,\varepsilon) = \sup_{\|u_0\|_{L^2(-L,L)} = 1} \inf\bigl\{\|h\|_{L^2(0,T)} : u(T) = 0\bigr\}, \qquad T_{\mathrm{unif}} = \inf\Bigl\{T > 0 : \limsup_{\varepsilon \downarrow 0} C(T,\varepsilon) < \infty\Bigr\}.$$ For each $\varepsilon > 0$ and $T > 0$ the system is null controllable. In the formal limit $\varepsilon = 0$ the equation becomes transport towards the shock at $x = 0$, which is null controllable exactly when $T > L$. Laheurte proved $$(4\sqrt{2} - 2)L \le T_{\mathrm{unif}} \le 4\sqrt{3}\,L.$$
Problem. Determine the exact value of the dimensionless constant $T_{\mathrm{unif}}/L$.
Source: Stated by AIM — Open Applied Problems, after Laheurte (2026), Theorem 1.1 and Remark 1.2, which give (4√2 − 2)L ≤ T_unif ≤ 4√3 L.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Burgers equation, null controllability, vanishing viscosity
References
@article{laheurte2026,
author = {Laheurte, Vincent},
title = {Cost of controllability of the {B}urgers equation linearized at a steady shock in the vanishing viscosity limit},
journal = {SIAM J. Math. Anal.},
volume = {58},
number = {3},
year = {2026},
pages = {2702--2737},
doi = {10.1137/25M1723311},
eprint = {2411.12267},
archivePrefix = {arXiv},
}
@misc{aim-burgers-shock-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 = {{Exact uniform control time near a viscous Burgers shock}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/067-burgers-shock-uniform-control-time.md},
note = {Entry 067 at commit a153f46, 2026-10-13},
}
Resolution claims
We determine the minimal time for boundary null control of the Burgers equation linearized at the stationary shock $-\tanh(x/(2\eps))$ on $(-L,L)$, with a control at the left endpoint. For arbitrary $L^2$ initial data, the worst-case $L^2$ control cost remains bounded as the viscosity tends to zero at every time $T>4\sqrt2L$ and diverges at every time $0<T<4\sqrt2L$. Thus the exact uniform …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00742, "Exact uniform control time near a viscous Burgers shock", clopen.solutions.