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

TL;DR

This paper establishes the equivalence between integral input-to-state stability (iISS) and a combination of stability properties for a broad class of infinite-dimensional time-varying control systems, including specific cases like time-delay and semilinear systems.

Contribution

It provides new theoretical conditions linking iISS with other stability notions for infinite-dimensional systems, extending known results to more general and nonlinear cases.

Findings

iISS is equivalent to 0-GUAS plus UBEBS under certain conditions

For semilinear systems with affine-in-the-state bounds, iISS reduces to 0-GUAS

The results apply to systems like time-delay and semilinear Banach space systems

Abstract

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