Class kappa function
# 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. $\alpha (0)=0$.

In fact, this is nothing but the definition of the norm except for the triangular inequality.

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. $\lim _{r\rightarrow \infty }\alpha (r)=\infty$.

A nondecreasing positive definite function $\beta$ satisfying all conditions of class ${\mathcal {K}}$ (${\mathcal {K}}_{\infty }$) other than being strictly increasing can be upper and lower bounded by class ${\mathcal {K}}$ (${\mathcal {K}}_{\infty }$) functions as follows:

    $\beta (x){\frac {x}{x+1}}<\beta (x)<\beta (x)\left({\frac {x}{x+1}}+1\right)=\beta (x){\frac {2x+1}{x+1}},\qquad x\in (0,a).\,$

Thus, to proceed with the appropriate analysis, it suffices to bound the function of interest with continuous nonincreasing positive definite functions. In other words, when a function belongs to the (${\mathcal {K}}_{\infty }$) it means that the function is radially unbounded.

