Understanding Lyapunov functions and notation I'm trying to wrap my mind around Lyapunov functions, but I'm having an hard time and I need some help. The reference I'm using is Khalil's "Nonlinear systems" book.
Let's consider the autonomous system
\begin{equation}
\dot{x}=f(x)
\end{equation}
where $f: D \to \mathbb{R}^n$ is a locally Lipschitz map from a domain $D \subset \mathbb{R}^n$ to $\mathbb{R}^n$, and let $V : D \to \mathbb{R}^n$ be a continuously differentiable function defined on $D$.
The time derivative of $V$ along the trajectories of the system, denoted by $\dot{V}$, is given by
$$
\dot{V}=\sum_{i=1}^n \frac{\partial V}{\partial x_i}\dot{x}_i=\sum_{i=1}^n \frac{\partial V}{\partial x_i}f_i=\nabla{V} \cdot f(x).
$$
If $\phi(x;t)$ is the solution of the system that starts at $x$ at the time $t_0=0$, then
$$
\dot{V}=\frac{d}{dt}V(\phi(x;t))\Bigr|_{t=0}
$$
but why do we evaluate $\frac{d}{dt}V(\phi(x;t))$ at $t=0$?
From my understanding $\dot{V}(x)=\dot{V}(x(t))=\frac{d}{dt}{V}(x(t))$, and hence it's time dependant, and we are omitting $t$ in the equation just for the sake of a lighter notation, but in the second and equivalent equation by evaluating in $t=0$ aren't we removing the time dependence? It would make more sense to me to just leave $\frac{d}{dt}V(\phi(x;t))$; what am I missing?
Hopefully everything is worded well enough to be understandable; the question may be a triviality that shows the lack of understanding that I have on the subject, but I'd really appreciate if someone can help me on this.
 A: It's a good question! And the short answer is that you don't really have to.
The first thing to spot is that the systems considered when using Lyapunov functions are autonomous—that is, they have no explicit time-dependence. So given any function $x(t)$ that serves as a solution to the autonomous system, any statement about the behavior of $V$ or $\dot{V}$ at $t = 0$ can always be translated into a statement about $V$ or $\dot{V}$ for specific values of $x$ (and by extension, $\dot{x}$).
The statements we can make about Lyapunov functions, including the ones Khalil makes, are always things like if $\dot{V}(x) <$ or $\leq 0 $ for all $x \neq 0$. Since the system is autonomous, we can simply use any $x_1$ in $\mathbb{R}/\{0\}$ as an initial condition, because there has to be a solution $x(t)$ to the system such that $x(t=0) = x_1$ for all $x_1 \in \mathbb{R}/\{0\}$. So it actually doesn't matter if you evaluate it at $t = 0$; you still need to check whether or not the Lyapunov function is negative or non-positive for all $x \neq 0$.
