Coercive Quadratic ISS Lyapunov Functions for Analytic Systems

We investigate the relationship between input-to-state stability (ISS) of linear infinite-dimensional systems and existence of coercive ISS Lyapunov functions. 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 underlying semigroup is analytic, and the input operator is bounded. However, if in addition the semigroup is similar to a contraction semigroup on a Hilbert space, then a quadratic ISS Lyapunov function always exists. Next we consider analytic and similar to contraction semi-groups in Hilbert spaces with unbounded input operator
$B$
. If
$B$
is slightly stronger than 2-admissible, we construct explicitly a coercive
$L^{2}$
-ISS Lyapunov function. If the generator of a semigroup is additionally self-adjoint, this Lyapunov function is precisely a square norm in the state space.