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 [math]\displaystyle{ \mathcal{K} }[/math] functions belong to this family:
Definition: a continuous function [math]\displaystyle{ \alpha : [0, a) \rightarrow [0, \infty) }[/math] is said to belong to class [math]\displaystyle{ \mathcal{K} }[/math] if:
- it is strictly increasing;
- it is s.t. [math]\displaystyle{ \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]\displaystyle{ \alpha : [0, a) \rightarrow [0, \infty) }[/math] is said to belong to class [math]\displaystyle{ \mathcal{K}_{\infty} }[/math] if:
- it belongs to class [math]\displaystyle{ \mathcal{K} }[/math];
- it is s.t. [math]\displaystyle{ a = \infty }[/math];
- it is s.t. [math]\displaystyle{ \lim_{r \rightarrow \infty} \alpha(r) = \infty }[/math].
A nondecreasing positive definite function [math]\displaystyle{ \beta }[/math] satisfying all conditions of class [math]\displaystyle{ \mathcal{K} }[/math] ([math]\displaystyle{ \mathcal{K}_{\infty} }[/math]) other than being strictly increasing can be upper and lower bounded by class [math]\displaystyle{ \mathcal{K} }[/math] ([math]\displaystyle{ \mathcal{K}_{\infty} }[/math]) functions as follows:
- [math]\displaystyle{ \beta(x)\frac{x}{x+1}\lt \beta(x)\lt \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]\displaystyle{ \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.
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