Class kappa-ell 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{KL} }[/math] functions belong to this family: Definition: A continuous function [math]\displaystyle{ \beta : [0, a) \times [0, \infty) \rightarrow [0, \infty) }[/math] is said to belong to class [math]\displaystyle{ \mathcal{KL} }[/math] if:

• for each fixed [math]\displaystyle{ s }[/math], the function [math]\displaystyle{ \beta(r,s) }[/math] belongs to class kappa;
• for each fixed [math]\displaystyle{ r }[/math], the function [math]\displaystyle{ \beta(r,s) }[/math] is decreasing with respect to [math]\displaystyle{ s }[/math] and is s.t. [math]\displaystyle{ \beta(r,s) \rightarrow 0 }[/math] for [math]\displaystyle{ s \rightarrow \infty }[/math].

See also

• Class kappa function
• H. K. Khalil, Nonlinear systems, Prentice-Hall 2001. Sec. 4.4, Def. 4.3.