On an Abstraction of Lyapunov and Lagrange Stability

TL;DR

This paper introduces a set-theoretic framework unifying Lyapunov and Lagrange stability for abstract systems, revealing a duality and enabling generalized stability theorems applicable to various control properties.

Contribution

It provides a novel set-theoretic abstraction of stability notions, establishing their duality and extending classical theorems to a broader class of systems and properties.

Findings

Unveiled a duality between Lyapunov and Lagrange stability.

Generalized stability theorems for interconnected systems.

Linked Lagrange stability to safety and positivity properties.

Abstract

This paper studies a set-theoretic generalization of Lyapunov and Lagrange stability for abstract systems described by set-valued maps. Lyapunov stability is characterized as the property of inversely mapping filters to filters, Lagrange stability as that of mapping ideals to ideals. These abstract definitions unveil a deep duality between the two stability notions, enable a definition of global stability for abstract systems, and yield an agile generalization of the stability theorems for basic series, parallel, and feedback interconnections, including a small-gain theorem. Moreover, it is shown that Lagrange stability is abstractly identical to other properties of interest in control theory, such as safety and positivity, whose preservation under interconnections can be thus studied owing to the developed stability results.