A question about Comparison Principle
For a general system, we have $$ V=x^{2}+y^{2} $$ where $x \in \mathbb{R}$ and $y \in \mathbb{R}$ are two independent states, and $V$ is a Lyapunov function. There are two cases: Case 1 is very common, based which I formulate the question in Case 2 .
Case 1. If $$ \dot{V} \leqslant-K\left(x^{2}+y^{2}\right)+\beta $$ holds for $K>0$ and $\beta>0$, we have $$ \dot{V} \leqslant-K V+\beta $$ then, according to Comparison Principle, $$ \limsup _{t \rightarrow \infty} V=\limsup _{t \rightarrow \infty}\left(x^{2}+y^{2}\right) \leqslant \frac{\beta}{K} $$ Case 2. If $$ \dot{V} \leqslant-K\left(x^{2}+100 y^{2}\right)+\beta $$ holds for $K>0$ and $\beta>0$, will we have $$ \limsup _{t \rightarrow \infty}\left(x^{2}+100 y^{2}\right) \leqslant \frac{\beta}{K} ? $$
One way to think about it is that in case 1, you define a new variable $Y$ such that $$Y' = -K Y + \beta.$$ And you assume $Y(0) = V(0)$ and $Y'\geq V'$. Then, $Y(t) \geq V(t)$ for all $t\ge 0$.
Then, by $Y'$, you have that the limit as $t\rightarrow \infty$ for $Y(t)$ approaches $\beta/K$ (from below or above dependent on the initial $Y(0)$. That's one way to conclude that $\limsup_{t\rightarrow\infty} V \leq \beta/K$.
In case 2, you don't need the whole part because you are only concern with $$Y' = -K(x^2 + 100y^2) + \beta$$ where it's implicitly assumed that $Y = x^2 + 100y^2$. Then your conclusion follows. $V$ just coincidentally happens to be always smaller or equal to $Y$.