Coercive quadratic converse ISS Lyapunov theorems for linear analytic systems

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.