On “uniformity” in definitions of global asymptotic stability for time-varying nonlinear systems

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.