Definition (Class $\mathcal{K}$)

A Lipschitz Continuous function $\alpha:[0,a)\to[0,\infty)$, for $a>0$ is said to belong to class $\mathcal{K}$ if:

1. It is strictly increasing and;
2. $\alpha(0)=0$.

Definition (Extended Class $\mathcal{K}$)

A Lipschitz Continuous function $\alpha:[-b,a)\to[-\infty,\infty)$ is said to belong to extended class $\mathcal{K}$ if:

3. It is strictly increasing and;
4. $\alpha(0)=0$.