The sharp Belgian chocolate stabilization threshold
Call a nonzero real polynomial stable if all its roots have strictly negative real part; nonzero constants are stable. Let $\mathcal{D}$ be the set of $\delta \in (0,1)$ for which there are stable polynomials $x$ and $y$ with $\deg y \le \deg x$ such that $$(s^2 - 2\delta s + 1)\, x(s) + (s^2 - 1)\, y(s)$$ is stable. The degrees of $x$ and $y$ are not bounded. This is Blondel's Belgian chocolate problem: for which $\delta$ can the plant $(s^2 - 1)/(s^2 - 2\delta s + 1)$ be stabilized by a proper controller that is itself stable and minimum-phase. Charles and Boston showed that if $\delta \in \mathcal{D}$ then every smaller positive $\delta$ is in $\mathcal{D}$. Patel, Deodhare and Viswanath showed that $0.9 \in \mathcal{D}$ (2002), and Charles and Boston that every $\delta \le 0.9808348$ is in $\mathcal{D}$ (2018). Bergweiler and Eremenko proved $\sup \mathcal{D} < 0.999579$ (2013).
Problem. Determine the exact value of $\delta_* = \sup \mathcal{D}$.
Source: Blondel (1994, p. 149f), the Belgian chocolate problem; in the polynomial form of Charles and Boston (2018), (1.1).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- CV
- Keywords
- AIM Open Applied Problems, Belgian chocolate problem, simultaneous stabilization, stable polynomials
References
@book{blondel1994,
author = {Blondel, Vincent},
title = {Simultaneous Stabilization of Linear Systems},
series = {Lecture Notes in Control and Information Sciences},
volume = {191},
publisher = {Springer},
year = {1994},
doi = {10.1007/3-540-19862-8},
}
@article{charles2018,
author = {Charles, Zachary and Boston, Nigel},
title = {Exploiting algebraic structure in global optimization and the {B}elgian chocolate problem},
journal = {J. Global Optim.},
volume = {72},
number = {2},
year = {2018},
pages = {241--254},
doi = {10.1007/s10898-018-0659-5},
eprint = {1708.08114},
archivePrefix = {arXiv},
}
@article{patel2002,
author = {Patel, Vijay V. and Deodhare, Girish and Viswanath, T.},
title = {Some applications of randomized algorithms for control system design},
journal = {Automatica},
volume = {38},
number = {12},
year = {2002},
pages = {2085--2092},
doi = {10.1016/S0005-1098(02)00112-7},
}
@article{bergweiler2013,
author = {Bergweiler, Walter and Eremenko, Alexandre},
title = {Goldberg's constants},
journal = {J. Anal. Math.},
volume = {119},
number = {1},
year = {2013},
pages = {365--402},
doi = {10.1007/s11854-013-0012-3},
eprint = {1111.2296},
archivePrefix = {arXiv},
}
@misc{aim-belgian-chocolate-threshold,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The sharp Belgian chocolate stabilization threshold}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/308-belgian-chocolate-threshold.md},
note = {Entry 308 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00500, "The sharp Belgian chocolate stabilization threshold", clopen.solutions.