Problem
I am looking for a reference for a converse Lyapunov function theorem that proves, given an asymptotically stable set there exists a Lyapunov function for it.
I have seen a lot of proofs that given a given an asymptotically stable fixed point there exists a Lyapunov function for it. I am looking for something similar but for some asymptotically stable subset of $\mathbb{R}^n$. I am not assuming that the set is well behaved(i.e. the interior of the set may be chaotic). The definition of asymptotically stable I am using is the following: The system $$\dot{x}=f(x),x(0)=x_0$$ has an asymptotically stable fixed point $\bar{x}$, if for every $\epsilon>0$ there exists a $\delta>0$ such that,if $||x(0)-\bar{x}||<\delta$ the for every $t\geq 0$ we have $||x(0)-\bar{x}|| <\epsilon$ and $\lim_{t\to\infty}||x(0)-\bar{x}||=0$. For an asymptotically stable set my definition is similar except that $x^*\in D$ and the point closest(by the euclidean metric) to $x(t)$. $D$ is a compact a asymptotically stable set. If $x(t)\in D$ then $x^*=x(t)$.
Notes
- If any clarification is needed tell me and I will update my question.
- If there is some way I can reinterpret other converse Lyapunov theorem functions into the type described in my problem please let me know.