Input-to-state stability in integral norms for linear infinite-dimensional systems

TL;DR

This paper investigates integral input-to-state stability for infinite-dimensional linear systems in $L^p$ spaces, establishing new characterizations and Lyapunov methods, especially for analytic semigroups and diffusion equations.

Contribution

It develops the admissibility theory for unbounded input operators and characterizes $L^p$-$L^q$-ISS using maximal regularity, extending stability analysis beyond exponential stability.

Findings

Characterization of $L^p$-$L^q$-ISS for analytic semigroups

Lyapunov functions for integral ISS in infinite-dimensional systems

Examples include diagonal systems and diffusion equations

Abstract

We study integral-to-integral input-to-state stability for infinite-dimensional linear systems with inputs and trajectories in $L^p$-spaces. We start by developing the corresponding admissibility theory for linear systems with unbounded input operators. While input-to-state stability is typically characterised by exponential stability and finite-time admissibility, we show that this equivalence does not extend directly to integral norms. For analytic semigroups, we establish a precise characterisation using maximal regularity theory. Additionally, we provide direct Lyapunov theorems and construct Lyapunov functions for $L^p$-$L^q$-ISS and demonstrate the results with examples, including diagonal systems and diffusion equations.