Non-coercive Lyapunov functions for input-to-state stability of infinite-dimensional systems

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 (ISS). This property in turn is equivalent to ISS, if the system has some sort of regularity. 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.