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For large classes of infinite-dimensional time-varying control systems, theequivalence between integral input-to-state stability (iISS) and thecombination of global uniform asymptotic stability under zero input (0-GUAS)and uniformly bounded-energy input/bounded state (UBEBS) is established under areasonable assumption of continuity of the trajectories with respect to theinput, at the zero input. By particularizing to specific instances ofinfinite-dimensional systems, such as time-delay, or semilinear over Banachspaces, sufficient conditions are given in terms of the functions defining thedynamics. In addition, it is also shown that for semilinear systems whosenonlinear term satisfies an affine-in-the-state norm bound, it holds that iISSbecomes equivalent to just 0-GUAS, a fact known to hold for bilinear systems.An additional important aspect is that the iISS notion considered is moregeneral than the standard one.

Characterization of integral input-to-state stability for nonlinear time-varying systems of infinite dimension