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.

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