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.
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)
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