Does Lyapunov theorem extend to dynamical system that have "equilibrium sets"? Suppose that I have some dynamical system
$$\dot x = f(x)$$
$f$ is locally Lipschitz, etc. I know that $f(x) = 0$ whenever $x \in \Gamma$, where $\Gamma$ is some closed (and possibly bounded) set in $\mathbb{R}^n$, could be a small disc, a line, a plane. In other words, not just an equilibrium.
How can dynamical system theory deal with this case?
Normally the approach is through Lyapunov theorem, but it is usually with respect to a single point.
Is it possible to extend the Lyapunov theorem by defining a Lyapunov function $V$ such that it is $V(\Gamma) = 0$ and $V(x) > 0$ elsewhere and proceed as usual by showing $\dot V(\Gamma) = 0$ and $\dot V(x) < 0$ elsewhere?
Is this possible? Are there any catch for using the above? I have went through several books but could not find this extension.
 A: You can use https://en.wikipedia.org/wiki/LaSalle%27s_invariance_principle, though I don't like the statement off of Wikipedia. The following comes from Nonlinear Systems, Third Edition by Khalil.
Consider the autonomous system
\begin{align}
\dot{x} = f(x) \qquad (1)
\end{align}
where $f: D\to \mathbb{R}^{n}$ is a locally Lipschitz map from a domain $D\subset \mathbb{R}^{n}$ into $\mathbb{R}^{n}$.
Theorem 4.4 in Nonlinear Systems, Third Edition by Khalil:
Let $\Omega \subset D$ be a compact set that is positively invariant with respect to $(1)$. Let $V: D \to \mathbb{R}$ be a continuously differentiable function such that $\dot{V} \leq 0$ in $\Omega$. Let $E$ be the set of all points in $\Omega$ where $\dot{V}=0$. Let $M$ be the largest invariant set in $E$. Then every solution starting in $\Omega$ approaches $M$ as $t\to\infty$.
Note that this theorem is more general than the extension you mention, $V$ doesn't need to be positive definite. As long as you can show that $\Omega$ is a compact set and positively invariant. If $V$ is positive definite then $\Omega$ can be obtained as the sublevel set of $V$, i.e., $\Omega := \{ x\ |\ V(x) \leq c\}$ for some $c$ and showing that $\dot{V}\leq 0$ for all elements in $\Omega$. If you can find a Lyapunov function such that your set $\Gamma = M$ then you can use LaSalles to show that you converge to the set $\Gamma$.
