& \mbox{if } \tau_1 < {\bar{\tau}}, \tau_2 = {\bar{\tau}}\end{array} \right.
\qquad \qquad  g_2 (\tau) = \left\{ \begin{array}{ll}  0 & \mbox{if } \tau_{2} = {\bar{\tau}}, \tau_1 < {\bar{\tau}}\\ \{0 , \tau_{2}(1+\varepsilon) \} & \mbox{if } \tau_{2} = {\bar{\tau}}, \tau_1 = {\bar{\tau}}\\
(1+\varepsilon)\tau_{2} & \mbox{if } \tau_2 < {\bar{\tau}}, \tau_1 = {\bar{\tau}}\end{array} \right. .$$

Basic Properties of ${{{\mathcal}H}}_{N}$ {#sec:BasicCond}
-----------------------------------------

\[sec:ModelForAnalysis\]

### Hybrid Basic Conditions

To apply analysis tools for hybrid systems in [@teel2012hybrid], which will be summarized in Section \[sec:analysis\], the data of the hybrid system ${{{\mathcal}H}}_{N}$ must meet certain mild conditions. These conditions, referred to as the [*hybrid basic conditions*]{}, are as follows:

1.  $C$ and $D$ are closed sets in ${\mathbb{R}}^N$.

2.  $f: {\mathbb{R}}^N\to{\mathbb{R}}^N$ is continuous on $C$.

3.  $G :{\mathbb{R}}^N{\rightrightarrows}{\mathbb{R}}^N$ is an outer semicontinuous[^4] set-valued mapping, locally bounded on $D$, and such that $G(x)$ is nonempty for each $x \in D$.

\[lem:BasicConds\] ${{{\mathcal}H}}_{N}$ satisfies the hybrid basic conditions.

Note that satisfying the hybrid basic conditions [[ implies that ${{{\mathcal}H}}_N$ is well-posed [@teel2012hybrid Theorem 6.30], which automatically gives robustness to vanishing state disturbances; see [@teel2012hybrid; @Goebel.ea.09.CSM]. Section \[sec:robustness\] considers different types of perturbations that ${{{\mathcal}H}}_N$ can withstand. ]{}]{}

### Solutions to ${{{\mathcal}H}}_N$

\[lem:solutions2HS\] From every point in $C \cup 