Semi-uniform Input-to-state Stability of Infinite-dimensional Systems

TL;DR

This paper introduces semi-uniform input-to-state stability concepts for infinite-dimensional systems, providing characterizations, sufficient conditions for linear systems, and stability results for bilinear systems under certain assumptions.

Contribution

It defines semi-uniform and polynomial input-to-state stability for infinite-dimensional systems and offers new characterizations and stability conditions for linear and bilinear cases.

Findings

Characterization of semi-uniform input-to-state stability based on attractivity.

Sufficient conditions for polynomial input-to-state stability in linear systems.

Polynomial stability of certain bilinear systems under smoothness assumptions.

Abstract

We introduce the notions of semi-uniform input-to-state stability and its subclass, polynomial input-to-state stability, for infinite-dimensional systems. We establish a characterization of semi-uniform input-to-state stability based on attractivity properties as in the uniform case. Sufficient conditions for linear systems to be polynomially input-to-state stable are provided, which restrict the range of the input operator depending on the rate of polynomial decay of the product of the semigroup and the resolvent of its generator. We also show that a class of bilinear systems are polynomially integral input-to-state stable under a certain smoothness assumption on nonlinear operators.