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: In fact, this is nothing but the definition of the norm except for the triangular inequality.
 * 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. $$\lim_{r \rightarrow \infty} \alpha(r) = \infty $$.

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.