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 ${𝒦}$ functions belong to this family:

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

• it belongs to class ${𝒦}$;
• it is s.t. $a=\mathrm{\infty }$;
• it is s.t. $\underset{r\to \mathrm{\infty }}{\lim }\alpha (r)=\mathrm{\infty }$.

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

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

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

See also

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