Technical communique: On "uniformity" in definitions of global asymptotic stability for time-varying nonlinear systems

Authors:
Andrew R. Teel;Luca Zaccarian
Affiliations:
Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA;Dipartimento di Informatica, Sistemi e Produzione, University of Rome, Tor Vergata, 00133 Rome, Italy
Venue:
Automatica (Journal of IFAC)
Year:
2006

Citing 4
Cited 1

Nonlinear systems analysis (2nd ed.)

Nonlinear systems analysis (2nd ed.)
A Smooth Converse Lyapunov Theorem for Robust Stability

SIAM Journal on Control and Optimization
Comments on integral variants of ISS

Systems & Control Letters
Lyapunov Characterizations of Input to Output Stability

SIAM Journal on Control and Optimization

On partial stability theory of nonlinear dynamic systems

Journal of Computer and Systems Sciences International

Quantified Score

Hi-index	22.14

Visualization

Abstract

We discuss the lack of ''uniformity'' in definitions of uniform global asymptotic stability (UGAS) that have been used in various textbooks, monographs, and papers over the years. Sometimes UGAS is taken to be the combination of uniform local stability (ULS) and uniform global attractivity (UGA). Other times it also encompasses uniform global boundedness (UGB). This paper contains an explicit, smooth scalar example that shows that these definitions do not agree in general, even when the right-hand side is locally Lipschitz in the state uniformly in time (and thus bounded in time). We also discuss various notions of global asymptotic stability with relaxed uniformity (with respect to the initial time) requirements for the behavior of the solutions. In particular, we consider class-KL estimates and Lyapunov characterizations.