# A Characterization of Semiglobal, Practical, Asymptotic Stability for Gain-Parametrized Systems

## TL;DR

This paper introduces a new stability concept called semiglobal, practical, asymptotic stability (SPAS) for gain-parametrized nonlinear discrete-time systems, providing a Lyapunov-based characterization without requiring an attractor.

## Contribution

It defines SPAS for a broad class of systems and offers a Lyapunov criterion that does not depend on the existence of a stable attractor.

## Key findings

- SPAS is characterized via Lyapunov functions.
- The theorem applies to systems without stable attractors.
- Provides a new stability analysis framework for gain-parametrized systems.

## Abstract

We consider a general class of nonlinear, constrained, discrete-time systems whose dynamics are parametrized by a set of gains. We define the semiglobal, practical, asymptotic stability (SPAS) of compact sets for this class of systems, and we provide a Lyapunov characterization of such sets. A set A that is SPAS with respect to a given system need not be an attractor for that system. Relative to existing characterizations of similar qualitative behaviors, our SPAS theorem does not require the existence of an asymptotically stable attractor associated to a nominal counterpart of the given dynamics.