Further equivalences and semiglobal versions of integral input to state stability

TL;DR

This paper advances the theoretical understanding of integral input-to-state stability (IISS) by establishing new equivalences, introducing semiglobal versions, and clarifying their relationships with global stability and input-to-state stability (ISS).

Contribution

It introduces new equivalences for IISS, explores semiglobal versions, and clarifies their relationship with ISS, completing the theoretical framework.

Findings

IISS is equivalent to mixed norm conditions and superposition of bounded energy and stability.

Semiglobal IISS implies global IISS, but this does not hold for ISS.

The paper completes the theoretical understanding of IISS and ISS.

Abstract

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.