\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left. - \sum_{i = 2}^{N} (1+{\varepsilon})\tau_{i}\right) \\
& = \frac{{\varepsilon}}{({\varepsilon}+1)^{N} - 1}{\bar{\tau}}- \frac{({\varepsilon}+1)(({\varepsilon}+1)^{N} - 1) - N{\varepsilon}}{{\varepsilon}N(({\varepsilon}+1)^{N} - 1)}{\bar{\tau}}\\ 
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \frac{1}{N}\sum_{i = 2}^{N} (1+{\varepsilon})\tau_{i} \\
 &= (1+{\varepsilon})\left(-\frac{(({\varepsilon}+1)^{N} - 1) - N{\varepsilon}}{{\varepsilon}N(({\varepsilon}+1)^{N} - 1)}{\bar{\tau}}+ \frac{1}{N}\sum_{i = 2}^{N} \tau_{i}\right).\end{aligned}$$ For each element $m > 1$, it follows that

Combining the expressions for each of the elements inside the norm of $V(G(\tau))$, it follows that $V(G(\tau)) = (1+{\varepsilon}) V(\tau)$.

Then, the change during jumps is given by $V(G(\tau)) - V(\tau) = {\varepsilon}V(\tau)$ where ${\varepsilon}\in (-1,0)$. With the property of $V$ during flows established above, the change of $V$ along solutions is bounded during flows and jumps by the nonpositive functions $u_{{\widetilde}{C}}$ and $u_{{\widetilde}{D}}$, respectively, defined as follows: $u_{{\widetilde}{C}} (z)= 0$ for each $z \in {\widetilde}{C}$ and $u_{{\widetilde}{C}} (z)= -\infty$ otherwise; $u_{{\widetilde}{D}}(z) = {\varepsilon}V(z)$ for each $z \in {\widetilde}{D}$ and $u_{{\widetilde}{D}}(z) = -\infty$ otherwise. Using Lemma \[lem:BasicConds\], the fact that ${\widetilde}{C}$ and ${\widetilde}{D}$ are closed, and the fact that every maximal solution to ${\widetilde}{{{\mathcal}H}}$ is bounded and complete, by [@teel2012hybrid Theorem 8.2], every maximal solution to ${\widetilde}{{{\mathcal}H}}_N$ approaches the largest weakly invariant subset of $L_{V}(r') \cap 