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.
Highlights
Stability and robustness are fundamental for control systems and typically they have been addressed within two different concepts
After a short discussion of infinite-dimensional linear systems, we prove that bilinear systems with bounded input operators which are uniformly globally asymptotically stable for a zero input are necessarily integral input-to-state stability (iISS)
First we address this question for systems whose state space is an arbitrary Banach space
Summary
Stability and robustness are fundamental for control systems and typically they have been addressed within two different concepts. In order to prove equivalence between iISS and uniform global asymptotic stability for systems over Banach spaces, this paper develops a different method Another difficulty is a need to use various density arguments, since the direct check of the properties of Lyapunov functions on the whole state space is often not possible. After a short discussion of infinite-dimensional linear systems, we prove that bilinear systems with bounded input operators which are uniformly globally asymptotically stable for a zero input are necessarily iISS. First we address this question for systems whose state space is an arbitrary Banach space.
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