Input-to-state stability, integral input-to-state stability, and unbounded level sets

We provide partial Lyapunov characterizations for a recently proposed generalization of input-to-state and integral input-to-state stability (ISS and iISS, respectively). This generalization relies on the notion of stability with respect to two measures originally introduced by Movchan [1960]. We show that the two classical Lyapunov characterizations of ISS-type properties, i.e., decrease conditions in an implication or dissipative form, correspond to ISS and iISS, respectively. We also demonstrate via an example that, for the generalization considered here, ISS does not necessarily imply iISS