Further equivalences and semiglobal versions of integral input to state stability

This paper continues the study of the integral input-to-state stability (II SS) 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 an d the global asymptotic stability of the unforced system. A semiglobal vers ion of IISS is shown to imply the global version, though a counterexample s hows 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.