sadly I wasn't able to find a 'control systems' community, so I'm posting here. I'm working through the book of H. Khalil - Nonlinear Systems and I'm stuck at the following part:
He states the following:
$$\Omega_C = \{x \in \mathbb{R}^n | V(x) \le c \} $$
For $\Omega_C$ to be in the interior of a ball $B_r$, $c$ must satisfy $c < \inf_{||x||\ge r} V(x)$. If $$l = \lim_{r\to \infty} \inf_{||x|| \ge r} V(x) < \infty$$ then $\Omega_c$ will be bounded if $c<l$.
Can anyone explain this to me in simpler words. Khalil also gives the following example:
$$l = \lim_{r\to\infty} \min_{||x||=r} \left( \frac{x_{1}^2}{1+x_{1}^2} + x_2^2 \right) = \lim_{|x_1|\to\infty} \frac{x_1^2}{1+x_1^2} = 1 $$
where I also don't understand how he evaluates this.
Another big question for me is the claim, that $V(x)$ needs to be radially unbounded, hence,
$$V(x) \to \infty ~ \text{as} ~ ||x|| \to \infty $$
that the equilibrium point is globally asymptotically stable.
I'm confused regarding this last statement because at the top I want that $V(x)$ fits into a 'ball' to be compact. But if $V(x)\to\infty$ it's not compact anymore.
I would be very grateful for your explanations and help.
