Hierarchical Dirichlet process

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 $\mathcal{𝒦}$ functions belong to this family:

Definition: a continuous function $\alpha :[0,a)\to [0,\mathrm{\infty })$ is said to belong to class $\mathcal{𝒦}$ if:

• it is strictly increasing;
• it is s.t. $\alpha (0)=0$.

Definition: a continuous function $\alpha :[0,a)\to [0,\mathrm{\infty })$ is said to belong to class ${\mathcal{𝒦}}_{\mathrm{\infty }}$ if:

• it belongs to class $\mathcal{𝒦}$;
• it is s.t. $a=\mathrm{\infty }$;
• it is s.t. $\underset{r\to \mathrm{\infty }}{\lim }\alpha (r)=\mathrm{\infty }$.

See also

• H. K. Khalil, Nonlinear systems, Prentice-Hall 2001. Sec. 4.4 - Def. 4.2.