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} 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.