Input-to-state stability and admissibility in integral norms for linear distributed parameter systems⁎

Abstract

We investigate input-to-state stability (ISS) for infinite-dimensional linear systems with inputs and trajectories in Lp-spaces. We begin by developing the corresponding admissibility theory for linear systems with unbounded input operators. We show that ISS in integral norms cannot be in general characterized by exponential stability together with finite-time admissibility, in contrast to classical ISS theory. For analytic semigroups, we derive a precise characterization based on maximal regularity theory. Finally, we establish direct Lyapunov theorems and construct Lyapunov functions for L^p-L^q-ISS systems. We illustrate the results with examples, including diagonal systems and diffusion equations.