# Lyapunov functions for determining the stability of invariant sets?

Suppose we have a dynamical system $$\dot{x}=f(x)$$ with an equillibrium $$x_0$$. It is known that $$x_0$$ is Lyapunov stable in this sense if there exists $$V:\mathbb{R}^n\rightarrow\mathbb{R}$$ such that $$V(x_0)=0$$ and $$(\nabla V\cdot f)(x)\leq 0$$ for $$x\neq x_0$$.

Now suppose $$A\subset\mathbb{R}^n$$ is a bounded invariant subset of our dynamical system, every trajectory initialized in $$A$$ stays in $$A$$. What I'm wondering is whether or not we can say that $$A$$ is Lyapunov stable in this sense (yes that article is using discrete time but I'm pretty sure the definitions carry over easily) if there exists a function $$V$$ with $$V(a)=0$$ for all $$a\in A$$ and $$(\nabla V\cdot f)(x)\leq 0$$ for all $$x\not\in A$$ (as before).

One problem with this that I imediately foresee is the geometrical nature of $$A$$; if $$A$$ is a fractal then the condition $$V(a)=0$$ might make it impossible for $$\nabla V$$ to exists. Is it possible to say anything if we put restrictions on $$A$$? Is there any literature related to this sort of generalization of Lyapunov functions?

• Given your requirements $V(x)=0$ would satisfy, also in the fractal case. This of course would not show Lyapunov stability, but does show that $V(x)$ doesn't have to be positive for all $x \notin A$ to show Lyapunov stability. If instead you want to show that all $x \notin A$ converge to $A$ you probably need LaSalle or $(\nabla V\cdot f)(x)< 0\ \forall\,x\notin A$ and $V(x)>0\ \forall\,x\notin A$, which probably would put restrictions on $A$. – fibonatic May 12 at 1:51