I came accross this link about the necessity of the Lyapunov function being radially unbounded. My understanding is that this condition is unnecessary if the time derivative along solution trajectories is bounded away from zero everywhere. That is, if I have the system $x'=F(x)$ with $F(x^*)=0$, and I find a function $V(x)$ that satisfies $V(x)>0$ for all $x\neq x^*$, $V(x^*)=0$, and in addition $$\dot{V}(x)<\epsilon<0,$$ then the system is assymptotically globaly stable. More specifically, if these conditions are only satisfied for a subset of the state space then this subset is contained in the basin of attraction
Is this true? Does anybody know where I can look for alternative statements of Lyapunov's theorem?
Thanks so much!
