As stated in the title, I hope someone could hint a good reference on the fact that the region of attraction of a stable equilibrium point can be estimated by the level set of Lyapunov function.
From what I read from the book Nonlinear Systems - Hassan K. Khalil, on section 8.2, it is stated that the conditions (1) and (2) are not sufficient to guarantee the region $D$ is an estimation of region of attraction, since the trajectories might run out of the region, and then the $\dot V(x)\leq 0$ is not guaranteed anymore. Thus other than the two conditions, we need to moreover impose that the region $D$ is compact, and positive invariant.
Thus overall there are three conditions to insure that $D$ can be an estimation of region of attraction.
(1) there exists a function $V(x)$ that is positive definite in $D$
(2) $\dot V(x)<0$ or $\leq 0$ in $D$
(3) $D$ is compact, positive invariant.
Then it is stated in the book that
The simplest such estimate is the set $\Omega_c=\{x\in\mathbb{R}^n|V(x)\leq c\}$ when $\Omega_c$ is bounded and contained in $D$.
I am not sure that, whether this sentence means that the sublevel set of $V(x)$ automatically satisfies the condition (3), i.e. sublevel set of $V(x)$ is positive invariant? If this is true, how to prove it?
The reason that I have this confusion is also that it seems in the literature, when people trying to estimate ROA by sublevel set of Lyapunov function, the positive invariance is not examined (I am not 100% sure, it might be I missed something).
Thus with this question, I hope someone could give a good reference regarding this method. Thanks in advance!
I found the following references, but not in book: