Does Global Asymptotic Stability imply Global Uniform Asymptotic Stability? What conditions need to be satisifed for both types of stability?
2026-03-27 05:04:02.1774587842
Stability theory and Lyapunov functions
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The answer to the first question is no, in general global asymptotic stability (GAS) does not imply global uniform asymptotic stability (GUAS). If the system is autonomous (only depending on the state) GAS and GUAS are the same.
Conditions for uniform asymptotic stability can be set up by a so-called decrescent Lyapunov function.
E.g. consider the system $\dot x=f(x,t)$ which is non-autonomous since it depends not only on the state $x$ but also on time $t$.
A Lyapunov function $V(t,x)$ is decrescent if $ a\|x\|^2 \leq V(x,t) \leq b \|x\|^2$ and the system is said to be uniformly asymptotically stable if $ \dot V \leq -c \|x\|^2$.
The important thing is that the Lyapunov function does not increase with time as $x$ decreases.