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 α : [ 0 , a ) → [ 0 , ∞ ) is said to belong to class K if:

Definition: a continuous function α : [ 0 , a ) → [ 0 , ∞ ) is said to belong to class K ∞ if:

A nondecreasing positive definite function β satisfying all conditions of class K ( K ∞ ) other than being strictly increasing can be upper and lower bounded by class K ( K ∞ ) functions as follows:

Thus, to proceed with the appropriate analysis, it suffices to bound the function of interest with continuous nonincreasing positive definite functions.