View source for Class kappa function

In [[Control theory|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 <math>\mathcal{K}</math> functions belong to this family:

'''Definition''': a continuous function <math>\alpha : [0, a) \rightarrow [0, \infty)</math> is said to belong to class <math>\mathcal{K}</math> if:
* it is strictly increasing;
* it is s.t. <math>\alpha(0) = 0</math>.
In fact, this is nothing but the definition of the norm except for the triangular inequality.


'''Definition''': a continuous function <math>\alpha : [0, a) \rightarrow [0, \infty)</math> is said to belong to class <math>\mathcal{K}_{\infty}</math> if:
* it belongs to class <math>\mathcal{K}</math>;
* it is s.t. <math>a = \infty</math>;
* it is s.t. <math>\lim_{r \rightarrow \infty} \alpha(r) = \infty </math>.

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

[[Category:Control theory]]

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