  We consider an abstract class of infinite-dimensional dynamical systems with
inputs. For this class, the significance of noncoercive Lyapunov functions is
analyzed. It is shown that the existence of such Lyapunov functions implies
norm-to-integral input-to-state stability. This property in turn is equivalent
to input-to-state stability, if the system satisfies certain mild regularity
assumptions. For a particular class of linear systems with unbounded admissible
input operators, explicit constructions of noncoercive Lyapunov functions are
provided. The theory is applied to a heat equation with Dirichlet boundary
conditions.
