Coercive quadratic converse ISS Lyapunov theorems for linear analytic systems

We derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. We show that input-to-state stability of a linear system does not imply existence of a coercive quadratic ISS Lyapunov function, even if the input operator is bounded. If, however, the semigroup is similar to a contraction semigroup on a Hilbert space, then a quadratic ISS Lyapunov function always exists for any input operator that is bounded, or more generally, $p$-admissible with $p<2$. The constructions are semi-explicit and, in the case of self-adjoint generators, coincide with the canonical Lyapunov function being the norm squared. Finally, we construct a family of non-coercive ISS Lyapunov functions for analytic ISS systems under weaker assumptions on $B$.