A universal example for quantitative semi‐uniform stability

Authors

Sahiba Arora, Felix L. Schwenninger, Ingrid Vukusic, Marcus Waurick

Abstract

              We characterise quantitative semi‐uniform stability for ‐semigroups arising from port‐Hamiltonian systems, complementing recent works on exponential and strong stability. With the result, we present a simple universal example class of port‐Hamiltonian ‐semigroups exhibiting arbitrary decay rates slower than . The latter is based on results from the theory of Diophantine approximation as the decay rates will be strongly related to approximation properties of irrational numbers by rationals given through cut‐offs of continued fraction expansions.

Citation

• Journal: Journal of the London Mathematical Society
• Year: 2026
• Volume: 113
• Issue: 2
• Pages:
• Publisher: Wiley
• DOI: 10.1112/jlms.70472

BibTeX

@article{Arora_2026,
  title={{A universal example for quantitative semi‐uniform stability}},
  volume={113},
  ISSN={1469-7750},
  DOI={10.1112/jlms.70472},
  number={2},
  journal={Journal of the London Mathematical Society},
  publisher={Wiley},
  author={Arora, Sahiba and Schwenninger, Felix L. and Vukusic, Ingrid and Waurick, Marcus},
  year={2026}
}

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References

• Ammari K, Tucsnak M (2001) Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: COCV 6:361–386. https://doi.org/10.1051/cocv:200111 – 10.1051/cocv:2001114
• Arendt W, Batty CJK, Hieber M, Neubrander F (2011) Vector-valued Laplace Transforms and Cauchy Problems. Springer Base – 10.1007/978-3-0348-0087-7
• Baker A (1984) A Concise Introduction to the Theory of Number – 10.1017/cbo9781139171601
• Bastin G., Prog. Nonlinear Differ. Equ. Appl (2016)
• Batty CJK, Chill R, Tomilov Y (2016) Fine scales of decay of operator semigroups. J Eur Math Soc 18(4):853–929. https://doi.org/10.4171/jems/60 – 10.4171/jems/605
• {“status”:”error” – 10.1007/s00028‐008‐0424‐1
• {“status”:”error” – 10.1007/s00208‐009‐0439‐0
• Bugeaud Y (2012) Distribution Modulo One and Diophantine Approximatio – 10.1017/cbo9781139017732
• Chill R, Paunonen L, Seifert D, Stahn R, Tomilov Y (2023) Nonuniform stability of damped contraction semigroups. Analysis & PDE 16(5):1089–1132. https://doi.org/10.2140/apde.2023.16.108 – 10.2140/apde.2023.16.1089
• Chill R, Seifert D, Tomilov Y (2020) Semi-uniform stability of operator semigroups and energy decay of damped waves. Phil Trans R Soc A 378(2185):20190614. https://doi.org/10.1098/rsta.2019.061 – 10.1098/rsta.2019.0614
• Jacob B, Zwart H (2018) An operator theoretic approach to infinite‐dimensional control systems. GAMM-Mitteilungen 41(4). https://doi.org/10.1002/gamm.20180001 – 10.1002/gamm.201800010
• Jacob B., Oper. Theory: Adv. Appl (2012)
• Jarník V., Zur metrischen Theorie der diophantischen Approximationen. Pr. Mat.‐Fiz. (1929)
• Khintchine A (1924) Einige S�tze �ber Kettenbr�che, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math Ann 92(1–2):115–125. https://doi.org/10.1007/bf0144843 – 10.1007/bf01448437
• Khintchine A. Y., Continued fractions (1963)
• Kuipers L, Meulenbeld B (1952) Some properties of continued fractions. Acta Math 87(0):1–12. https://doi.org/10.1007/bf0239227 – 10.1007/bf02392279
• Legendre A-M (1798) Essai sur la théorie des nombre – 10.5962/bhl.title.18546
• Rozendaal J, Seifert D, Stahn R (2019) Optimal rates of decay for operator semigroups on Hilbert spaces. Advances in Mathematics 346:359–388. https://doi.org/10.1016/j.aim.2019.02.00 – 10.1016/j.aim.2019.02.007
• Rzepnicki Ł, Schnaubelt R (2018) Polynomial stability for a system of coupled strings. Bull London Math Soc 50(6):1117–1136. https://doi.org/10.1112/blms.1221 – 10.1112/blms.12212
• Trostorff S., Characterisation for exponential stability of port‐Hamiltonian systems. Israel J. Math. (2023)
• van der Schaft A (2007) Port-Hamiltonian systems: an introductory survey. Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006 1339–136 – 10.4171/022-3/65
• Wacker P (2023) Please, Not Another Note About Generalized Inverse – 10.2139/ssrn.4549719
• Waurick M, Zwart H (2024) Asymptotic Stability of Port-Hamiltonian Systems. Trends in Mathematics 91–12 – 10.1007/978-3-031-64991-2_4
• {“status”:”error” – 10.1016/j.jde.2004
• Zwart H, Le Gorrec Y, Maschke B, Villegas J (2009) Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: COCV 16(4):1077–1093. https://doi.org/10.1051/cocv/200903 – 10.1051/cocv/2009036