Characterizations of integral input-to-state stability for systems with multiple invariant sets

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