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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.

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The theorem is formulated and proved in:

J. P. LaSalle, Some extensions of Liapunov's second method, IRE Trans. Circuit Theory 7 (1960), 520-527.

See Theorem 1. Available (depending on subscription) at https://ieeexplore.ieee.org/document/1086720

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  • $\begingroup$ 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. $\endgroup$ – Harry Dudley Jan 25 at 0:16
  • $\begingroup$ @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 $\mathbb R^n$). $\endgroup$ – John B Jan 25 at 1:14
  • $\begingroup$ Thank you for the response John. However, I would like to know why it is the same. $\endgroup$ – Harry Dudley Jan 26 at 5:24
  • $\begingroup$ @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). $\endgroup$ – John B Jan 26 at 6:43

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