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