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 $𝒦ℒ$ functions belong to this family:

Definition: A continuous function $\beta :[0,a)\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is said to belong to class $𝒦ℒ$ if:

• for each fixed $s$, the function $\beta (r,s)$ belongs to class kappa;
• for each fixed $r$, the function $\beta (r,s)$ is decreasing with respect to $s$ and is s.t. $\beta (r,s)\to 0$ for $s\to \mathrm{\infty }$.

See also

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