Extending a theorem of Datko for evolutionary family

TL;DR

This paper extends Datko's theorem by characterizing exponential stability of evolutionary families through the norm behavior of their pointwise trajectories in a Banach function space.

Contribution

It generalizes Datko's result to a broader class of evolutionary families using Banach function space analysis.

Findings

Exponential stability characterized by pointwise trajectory norms.

Extension of Datko's theorem to evolutionary families.

Use of Banach function space framework.

Abstract

In this note, we extend a Datko's result in the paper \cite[1972]{Dat}. In particular, the exponential stability of an evolutionary family is characterized by its pointwise trajectories in which the norm mapping of each pointwise trajectory lies in a Banach function space.