${\mathcal{A}}$ for ${{{\mathcal}H}}_N$ is given by the locally Lipschitz function $$\label{eqn:lyapunov}
V(\tau) = \min\{|\tau|_{{\widetilde}\ell_1},|\tau|_{{\widetilde}\ell_2}, \ldots ,|\tau|_{{\widetilde}\ell_k}, \ldots ,|\tau|_{{\widetilde}\ell_N!}\} \quad \forall \ \tau \in P_N \setminus {\mathcal{X}}$$ where, for some $k$, $|\tau|_{{\widetilde}\ell_k}$ is the distance between the point $\tau$ and the set ${\widetilde}\ell_k$.[^6] The following theorem establishes asymptotic stability of ${\mathcal{A}}$ for ${{{\mathcal}H}}_N$. We show that the change in $V$ during flows is zero and that at jumps we have a strict decrease of $V$; namely, $V(G(\tau)) - V(\tau) = -|{\varepsilon}| V(\tau)$. A key step in the proof is in using [@teel2012hybrid Theorem 8.2] on a restricted version of ${{{\mathcal}H}}_N$.

\[thm:stability\] For every $N \in {\mathbb{N}}, N > 1$, ${\bar{\tau}}> 0,\omega > 0$, and $\varepsilon \in (-1,0)$, the hybrid system ${{{\mathcal}H}}_N$ is such that the compact set ${\mathcal{A}}$ is

Let the set ${\mathcal{X}}_{v}$ define the $v$-inflation of ${\mathcal{X}}$ (defined in Lemma \[lem:defX\]), that is, the open set[^7] ${\mathcal{X}}_{v} := \{\tau \in {\mathbb{R}}^N : |\tau|_{{\mathcal{X}}} < v\}$, where $v \in (0,v^{*})$ and $v^{*} = \min_{x\in{\mathcal{X}}, y\in{\widetilde}{\mathcal{A}}} |x-y|$. Given any $v \in (0,v^{*})$, we now consider a restricted hybrid system ${\widetilde}{{{\mathcal}H}}_{N} = (f,{\widetilde}{C},G,{\widetilde}{D})$, where ${\widetilde}{C} := C \setminus {\mathcal{X}}_{v}$ and ${\widetilde}{D} := D \setminus {\mathcal{X}}_{v}$, which are closed. We establish that ${\widetilde}{\mathcal{A}}$ is 