  This paper continues the study of the integral input-to-state stability
(IISS) property. It is shown that the IISS property is equivalent to one which
arises from the consideration of mixed norms on states and inputs, as well as
to the superposition of a ``bounded energy bounded state'' requirement and the
global asymptotic stability of the unforced system. A semiglobal version of
IISS is shown to imply the global version, though a counterexample shows that
the analogous fact fails for input to state stability (ISS). The results in
this note complete the basic theoretical picture regarding IISS and ISS.
