Further Equivalences and Semiglobal Versions of Integral Input to State Stability

This paper continuesthe study of the integral input-to-state stability ( iiss) property.It is shown that the iiss property is equivalent to one whicharises from the consideration of mixed norms on states and inputs,as well as to the superposition of a ``bounded energy boundedstate' requirement and the global asymptotic stability of theunforced system. A semiglobal version of iiss is shown to implythe global version, though a counterexample shows that the analogousfact fails for input to state stability ( iss). The results inthis note complete the basic theoretical picture regarding iissand iss.