Coercive quadratic converse ISS Lyapunov theorems for linear analytic systems

Mathematics > Optimization and Control [Submitted on 27 Mar 2023 (v1), last revised 18 Jun 2026 (this version, v3)] Title:Coercive quadratic converse ISS Lyapunov theorems for linear analytic systems View PDF HTML (experimental)Abstract:We derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. While we show that ISS in general does not imply the existence of a coercive quadratic ISS Lyapunov function, even if the input operator is bounded, we prove that indeed quadratic ISS Lyapunov functions always exist for $p$-admissible input operators with $p<2$, provided the semigroup is similar to a contraction on a Hilbert space. The constructions are semi-explicit and rely on classical results on analytic semigroups and similarity to contractive ones. In the case of self-adjoint generators, they coincide with the canonical Lyapunov function being the norm squared. Submission history From: Felix Schwenninger [view email][v1] Mon, 27 Mar 2023 11:01:30 UTC (47 KB) [v2] Wed, 17 Sep 2025 20:30:47 UTC (37 KB) [v3] Thu, 18 Jun 2026 14:44:25 UTC (272 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.