Quadratic ISS Lyapunov functions for linear analytic systems

Titelangaben

Mironchenko, Andrii ; Schwenninger, Felix L.:
Quadratic ISS Lyapunov functions for linear analytic systems.
In: Mathematics of Control, Signals, and Systems. Bd. 38 (8 Juli 2026) . - S. 885-908.
ISSN 1435-568X
DOI der Verlagsversion: https://doi.org/10.1007/s00498-026-00455-x

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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.