# Looking for a proof of LaSalle's invariance principle for a dynamical system on a manifold.

I found the following version of LaSalle's theorem and it appears to be stayed differently from the original.

Consider the smooth dynamical system on an $$n-$$manifold $$M$$ given by $$\dot{x} = X(x)$$ and let $$\Omega$$ be a compact set in $$M$$ that is (positively) invariant under the flow of $$X$$. Let $$V: \Omega \to \mathbb{R}$$, $$V \geq 0$$, be a $$C^1$$ function such that $$\dot{V}(x) = \frac{\partial V}{\partial x} \cdot X \leq 0$$ in $$\Omega$$. Let $$S$$ be the largest invariant set in $$\Omega$$ where $$\dot{V}(x) = 0$$. Then every solution with initial point in $$\Omega$$ tends asymptotically to $$S$$ as $$t \to \infty$$. In particular, if $$S$$ is an isolated equilibrium, it is asymptotically stable.

I have seen LaSalle's proof, but I was wondering if anyone can cite a proof of this theorem - specifically regarding a dynamical system on an $$n-$$manifold.

• I am familiar with this paper. My question is about how the proof changes when we move from a dynamical system on $\mathbb{R}^n$ to a dynamical system on a manifold. – Harry Dudley Jan 25 at 0:16
• @HarryDudley Indeed: LaSalle does not say explicitly that we are on $\mathbb R^n$. In any event the proof is the same (you can check as I mentioned the proof of Theorem 1, short and simple without any need to be on $\mathbb R^n$). – John B Jan 25 at 1:14
• @HarryDudley You said that you are familiar with the paper: did you notice any need for $\mathbb R^n$ in the argument? Did LaSalle make this requirement? I did check before replying (and I have already replied to what you asked, if you want to ask something different please ask a new question). – John B Jan 26 at 6:43