A non-uniform Datko-Pazy theorem for bounded operator semigroups

TL;DR

This paper extends the Datko-Pazy theorem to a non-uniform setting, showing that integrability conditions on specific orbits imply polynomial stability of bounded operator semigroups, with applications to Lyapunov and observability conditions.

Contribution

It introduces a non-uniform version of the Datko-Pazy theorem, linking orbit integrability in fractional domains to polynomial stability, and provides new proofs under weaker assumptions.

Findings

Integrability on fractional domain orbits implies polynomial stability.

Polynomial stability established under non-uniform Lyapunov conditions.

New proof of polynomial stability from non-uniform observability.

Abstract

We present a non-uniform analogue of the classical Datko-Pazy theorem. Our main result shows that an integrability condition imposed on orbits originating in a fractional domain of the generator (as opposed to all orbits) implies polynomial stability of a bounded $C_0$-semigroup. As an application of this result we establish polynomial stability of a semigroup under a certain non-uniform Lyapunov-type condition. We moreover give a new proof, under slightly weaker assumptions, of a recent result deducing polynomial stability from a certain non-uniform observability condition.