  In this work we present further characterizations of integral input-to-state
stability (iISS) for hybrid systems. In particular, the equivalence between
0-input uniform global asymptotic stability (0-UGAS) plus uniform bounded
energy bounded state (UBEBS) and iISS is examined. In order to show this
equivalence, some necessary conditions for 0-UGAS and UBEBS are provided. In
addition, a non-smooth Lyapunov characterization for hybrid systems is proposed
and proven. With the help of the aforementioned equivalence, the combination of
local iISS and practical iISS, which are defined in this work, is also shown to
be equivalent to iISS under one condition on the local and practical
quantifiers.
