Class kappa 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 ${\mathcal {K}}$ functions belong to this family:

Definition: a continuous function $\alpha :[0,a)\rightarrow [0,\infty )$ is said to belong to class ${\mathcal {K}}$ if:

it is strictly increasing;
it is s.t. $\alpha (0)=0$ .

Definition: a continuous function $\alpha :[0,a)\rightarrow [0,\infty )$ is said to belong to class ${\mathcal {K}}_{\infty }$ if:

it belongs to class ${\mathcal {K}}$ ;
it is s.t. $a=\infty$ ;
it is s.t. $\lim _{r\rightarrow \infty }\alpha (r)=\infty$ .

See also