# Uniform weak attractivity and criteria for practical global asymptotic stability

## TL;DR

This paper introduces a new concept of uniform weak attractivity and establishes its equivalence to practical uniform global asymptotic stability (pUGAS) across various complex systems, including PDEs and delay systems.

## Contribution

It defines uniform weak attractivity, proves its equivalence to pUGAS, and links the existence of non-coercive Lyapunov functions to pUGAS in broad classes of systems.

## Key findings

- Uniform weak attractivity is equivalent to pUGAS.
- Existence of a non-coercive Lyapunov function implies pUGAS.
- Uniform weak attractivity is stronger than weak attractivity in infinite-dimensional systems.

## Abstract

A subset $A$ of the state space is called uniformly globally weakly attractive if for any neighborhood $S$ of $A$ and any bounded subset $B$ there is a uniform finite time $\tau$ so that any trajectory starting in $B$ intersects $S$ within the time not larger than $\tau$. We show that practical uniform global asymptotic stability (pUGAS) is equivalent to the existence of a bounded uniformly globally weakly attractive set. This result is valid for a wide class of distributed parameter systems, including time-delay systems, switched systems, many classes of PDEs and evolution differential equations in Banach spaces. We apply our results to show that existence of a non-coercive Lyapunov function ensures pUGAS for this class of systems. For ordinary differential equations with uniformly bounded disturbances, the concept of uniform weak attractivity is equivalent to the well-known notion of weak attractivity. It is however essentially stronger than weak attractivity for infinite-dimensional systems, even for linear ones.