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

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

• it is strictly increasing;
• it is s.t. ${\displaystyle \alpha (0)=0}$.

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

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

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

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

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

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 (${\displaystyle {\mathcal {K}}_{\infty }}$) it means that the function is radially unbounded.

• Class kappa-ell function
• H. K. Khalil, Nonlinear systems, Prentice-Hall 2001. Sec. 4.4 - Def. 4.2.