Class kappa-ell function

In control theory, it is often required to check if a nonautonomous system is stable or not. To cope with this it is necessary to use some special comparison functions. Class $$\mathcal{KL}$$ functions belong to this family:

Definition: A continuous function $$\beta : [0, a) \times [0, \infty) \rightarrow [0, \infty)$$ is said to belong to class $$\mathcal{KL}$$ if:
 * for each fixed $$s$$, the function $$\beta(r,s)$$ belongs to class kappa;
 * for each fixed $$r$$, the function $$\beta(r,s)$$ is decreasing with respect to $$s$$ and is s.t. $$\beta(r,s) \rightarrow 0$$ for $$s \rightarrow \infty$$.