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PreprintCC BY-NC-ND 4.0arXiv

Maximal Multiplicity Method for Optimal Damping Problem

  • Maxim Arnold
  • Oleg Gendelman
  • Vadim Zharnitsky
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Abstract

We study the problem of designing optimal damping for a system of n coupled linear oscillators. Using the Weyl--Horn theorem, we construct an explicit damping matrix that realizes the optimal asymptotic decay rate of the system. We prove that this decay rate is sharp and is determined by the geometric mean of the natural frequencies. The resulting damping strategy is shown to outperform the classical Rayleigh (proportional) damping approach. We conclude by characterizing the parameter regimes in which the optimal damping matrix necessarily ceases to be positive definite and discuss the implications of this phenomenon.

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Cite

BibTeXRIS
BibTeX
@misc{arnold2026maximal,
  title = {{Maximal Multiplicity Method for Optimal Damping Problem}},
  author = {Maxim Arnold and Oleg Gendelman and Vadim Zharnitsky},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25736},
  eprint = {2609.25736},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25736},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Maximal Multiplicity Method for Optimal Damping Problem
AU  - Maxim Arnold
AU  - Oleg Gendelman
AU  - Vadim Zharnitsky
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25736
UR  - https://arxiv.org/abs/2609.25736
AB  - We study the problem of designing optimal damping for a system of n coupled linear oscillators. Using the Weyl--Horn theorem, we construct an explicit damping matrix that realizes the optimal asymptotic decay rate of the system. We prove that this decay rate is sharp and is determined by the geometric mean of the natural frequencies. The resulting damping strategy is shown to outperform the classical Rayleigh (proportional) damping approach. We conclude by characterizing the parameter regimes in which the optimal damping matrix necessarily ceases to be positive definite and discuss the implications of this phenomenon.
ER  -