Sharp exponential cost of fast boundary quantum control
Fix $L > 0$ and consider complex-valued solutions of $$i y_t + y_{xx} = 0 \ \text{ on } (0,T) \times (0,L), \qquad y(t,0) = h(t), \quad y(t,L) = 0,$$ with initial data $y(0) = y_0 \in H^{-1}(0,L)$ and boundary control $h \in L^2(0,T)$, solutions and boundary values being understood by transposition. The system is null controllable in every time $T > 0$. Let $$C_S(T,L) = \sup_{\|y_0\|_{H^{-1}(0,L)} = 1} \inf \bigl\{ \|h\|_{L^2(0,T)} : y(0) = y_0,\ y(T) = 0 \bigr\},$$ the norm of the map sending $y_0$ to its null control of least $L^2(0,T)$ norm. Miller proved that $\liminf_{T \downarrow 0} T \log C_S(T,L) \ge L^2/4$, Tenenbaum and Tucsnak that $\limsup_{T \downarrow 0} T \log C_S(T,L) \le 3L^2/2$, and Lissy improved the upper bound to $\sqrt[4]{27}\, L^2/4$.
Problem. Is it true that $$\lim_{T \downarrow 0} T \log C_S(T,L) = L^2/4?$$
Source: Lissy (2025), §3.2, the conjecture preceding Theorem 3.1, that the lower bound L²/4 of Miller (2004) is optimal.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, control costs, null controllability, Schrödinger equations
References
@misc{lissy2025,
author = {Lissy, Pierre},
title = {Effective multipliers for weights whose log are {H}{\"o}lder continuous. {A}pplication to the cost of fast boundary controls for the {1D} {S}chr{\"o}dinger equation},
year = {2025},
eprint = {2502.04859},
archivePrefix = {arXiv},
}
@article{miller2004,
author = {Miller, Luc},
title = {How violent are fast controls for {S}chr{\"o}dinger and plate vibrations?},
journal = {Arch. Ration. Mech. Anal.},
volume = {172},
number = {3},
year = {2004},
pages = {429--456},
doi = {10.1007/s00205-004-0312-y},
}
@article{tenenbaum2007,
author = {Tenenbaum, G{\'e}rald and Tucsnak, Marius},
title = {New blow-up rates for fast controls of {S}chr{\"o}dinger and heat equations},
journal = {J. Differential Equations},
volume = {243},
number = {1},
year = {2007},
pages = {70--100},
doi = {10.1016/j.jde.2007.06.019},
}
@misc{aim-schrodinger-sharp-fast-control-exponent,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp exponential cost of fast boundary quantum control}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/403-schrodinger-sharp-fast-control-exponent.md},
note = {Entry 403 at commit a153f46, 2026-10-13},
}
Resolution claims
We consider the one-dimensional Schrödinger equation on an interval of length $L$, driven by an $L^2$ Dirichlet control at one endpoint, with initial data measured in $H^{-1}$. We prove that its null-control norm cost satisfies $C_S(T,L)\ge c_L T^{7/2}\exp(L^2/(\pi T))$ for all sufficiently small $T$. Thus $\liminf_{T\downarrow0}T\log C_S(T,L)\ge L^2/\pi$, excluding the proposed coefficient $L^2/4$. The proof uses one sine initial state: …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | A lower bound for the small-time cost of boundary control of the one-dimensional Schrödinger equation v1 recorded from AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00551, "Sharp exponential cost of fast boundary quantum control", clopen.solutions.