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 $s\rightarrow \infty$ .

See also