Weak input-to-state stability: characterizations and counterexamples

TL;DR

This paper characterizes weak input-to-state stability in dynamical systems, compares it with other stability concepts, and provides counterexamples to clarify its properties and open questions.

Contribution

It offers new characterizations of weak input-to-state stability and addresses open questions with counterexamples, expanding understanding of stability concepts.

Findings

Characterizations of weak input-to-state stability established.

Counterexamples clarify relations between stability concepts.

Answers to open questions about weak input-to-state stability.

Abstract

We establish characterizations of weak input-to-state stability for abstract dynamical systems with inputs, which are similar to characterizations of uniform and of strong input-to-state stability established in a recent paper by A. Mironchenko and F. Wirth. We also answer, by means of suitable counterexamples, two open questions concerning weak input-to-state stability (and its relation to other common stability concepts) raised in the aforementioned paper.