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

Question

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.

Answer 1

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