I am copying in verbatim a standard reference on Lasalle's theorem (which is Hassan Khalil's textbook, page 128)
Theorem:
Let $\Omega \subset D$ be a compact set that is positively invariant with respect to an ODE $\dot x = f(x)$. Let $V: D \to \mathbb{R}$ be a continuously differentiable function such that $\dot V(x) \leq 0$ in $\Omega$. Let $E$ be the set of all points in $\Omega$ where $\dot V(x) = 0$. Let $M$ be the largest invariant set in $E$. Then every solution trajectory starting in $\Omega$ approaches $M$ as $t \to \infty$.
Here are my two questions about this theorem:
How would you know if there exists such a positively invariant set $\Omega$ in the first place? And how would you construct this set $\Omega$?
How would you know there exists an invariant set $M$ in $E$ in the first place? And how would you construct this set $M$?
Thank you.
For added context, this question involves an extremely well studied/known theorem in dynamical system theory. The text in reference is cited 40,000 times, hence not some obscure result. However, very little is said about the existence (or construction) of certain sets used in this theorem and most examples do not verify the existence of such sets, or does so in a case-by-case basis with no universal method of constructing these sets.
Question 1: As explained by Khalil just after the proof of the theorem, you typically take $\Omega$ to be a sublevel set $\{ x : V(x) \le c \}$.
Question 2: The set $L^+$ in the proof is a nonempty invariant set contained in $E$. And finding $M$ is often quite easy. Just look at the set $E$ where $\dot V=0$ and check how the vector field is pointing at those points. If it's pointing out of $E$ at some point, then that point is not in $M$. For a point to belong to $M$, the whole trajectory through that point must lie in $E$. Put differently, $M$ is the union of all trajectories that are completely contained in $E$.