Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions

TL;DR

This paper establishes the equivalence between uniform global asymptotic stability and integral input-to-state stability for bilinear systems in infinite-dimensional spaces, providing proofs and explicit Lyapunov functions.

Contribution

It proves the equivalence of stability notions for infinite-dimensional bilinear systems and offers constructive Lyapunov functions in Hilbert spaces.

Findings

Equivalence between UGAS and iISS for bilinear systems in Banach spaces.

Explicit iISS Lyapunov functions derived for Hilbert space systems.

Two different proofs demonstrating the main result.

Abstract

For bilinear infinite-dimensional dynamical systems, we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions.