Skip to content
DraconiticResearch
Search Sources API
206,690 studies31 sourcesUpdated 46 min ago
PreprintarXiv licencearXiv

Converse Lyapunov theorem for input-to-state stability of infinite-dimensional systems

  • Patrick Bachmann
  • Andrii Mironchenko
Open at sourcePDFDOICite

Abstract

We prove converse Lyapunov theorems for input-to-state stability (ISS) of a broad class of infinite-dimensional systems applicable to evolution equations in Banach spaces, boundary control systems, and delay systems. To this end, we employ a novel method that does not require constructing auxiliary closed-loop systems. We start by showing that ISS of a control system implies uniform global asymptotic stability of the same system with respect to trajectory-dominated inputs. Then, under an additional regularity requirement, we prove that an ISS Lyapunov function can be constructed. Based on our general result, we prove that for linear systems with unbounded ∞-admissible operators, ISS is equivalent to the existence of a Lipschitz continuous coercive ISS Lyapunov function.

Related

  1. On the Stability Analysis of Infinite-Dimensional Port-Hamiltonian Systems Within the Variational Framework
    Moritz Leitner, Lukas Ecker, Markus Schöberl
    arXivPreprintCC BY 4.0Full textOptimisation & controlarXiv
  2. BIBO stability in infinite dimensions: extensions and limitations of finite-dimensional theory
    Sahiba Arora, Alexander A. Wierzba
    arXivPreprintarXiv licenceFull textAnalysis & PDEsOptimisation & controlarXiv
  3. Sampling with Autoregressive Clock Jitter: Infinite-Dimensional Stability Failure and Finite-Dimensional Conditioning
    Daniele Gerosa, Thomas Eriksson
    arXivPreprintCC BY-NC-ND 4.0Full textGeneral mathematicsarXiv
  4. A converse to the Erdős-Fuchs theorem
    Quan-Hui Yang, Lilu Zhao
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  5. A Converse to the Skoda L2 division theorem II
    Zhi Li, Xiankui Meng, Jiafu Ning +1
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{bachmann2026converse,
  title = {{Converse Lyapunov theorem for input-to-state stability of infinite-dimensional systems}},
  author = {Patrick Bachmann and Andrii Mironchenko},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.12601},
  eprint = {2611.12601},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.12601},
}
RIS
TY  - UNPB
TI  - Converse Lyapunov theorem for input-to-state stability of infinite-dimensional systems
AU  - Patrick Bachmann
AU  - Andrii Mironchenko
PY  - 2026
DA  - 2026/11/09
PB  - arXiv
DO  - 10.48550/arxiv.2611.12601
UR  - https://arxiv.org/abs/2611.12601
AB  - We prove converse Lyapunov theorems for input-to-state stability (ISS) of a broad class of infinite-dimensional systems applicable to evolution equations in Banach spaces, boundary control systems, and delay systems. To this end, we employ a novel method that does not require constructing auxiliary closed-loop systems. We start by showing that ISS of a control system implies uniform global asymptotic stability of the same system with respect to trajectory-dominated inputs. Then, under an additional regularity requirement, we prove that an ISS Lyapunov function can be constructed. Based on our general result, we prove that for linear systems with unbounded ∞-admissible operators, ISS is equivalent to the existence of a Lipschitz continuous coercive ISS Lyapunov function.
ER  -