Integral ISS for systems with multiple invariant sets

We generalize the classical integral Input-to-State Stability (iISS) theory for systems evolving on manifolds and admitting multiple disjoint invariant sets, so as to embed a much broader variety of dynamical behaviors of interest. Building upon a recent extension of the Input-to-State (ISS) theory for this kind of systems, we provide here equivalent characterizations of the iISS concept in terms of dissipation inequalities as well as connections with the Strong iISS notion. Finally, we discuss some examples within the domain of mechanical systems.