the definition of the set $\ell_k$.

The solution to $\Gamma\tau_s = b$ is the result of a single case of $\tau \in D \setminus {\mathcal{X}}$. As indicated above, to get a full definition of the set ${\mathcal{A}}$, the $N!$ sets $\ell_k$ should be computed. For arbitrary $N$, the set ${\mathcal{A}}$ is given as a collection of sets $\ell_{k}$ given by $${\mathcal{A}}= \bigcup_{k =1}^{N!}\ell_{k} \label{eqn:AN},$$ where, for each $k \in \{1,2,\dots,N!\}$, $\ell_k := \{\tau  :\tau =  {\widetilde}\tau^k + {\mathbf 1}s \in P_N, s \in {\mathbb{R}}\}.$

Lyapunov Stability {#sec:lyapunov}
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Lyapunov theory for hybrid systems is employed to show that the set of points ${\mathcal{A}}$ is asymptotically stable. Our candidate Lyapunov-like function, which is defined below and uses the distance function, is built by observing that there exist points where the distance to ${\mathcal{A}}$ may increase during flows. This is due to the sets $\ell_{k}$ being a subset $P_{N}$. To avoid this issue, we define $${\widetilde}{\mathcal{A}}= \bigcup_{k=1}^{N!} {\widetilde}\ell_k \supset {\mathcal{A}}$$ where ${\widetilde}\ell_k$ is the extension of $\ell_k$ given by $$\begin{aligned}
{\widetilde}\ell_k = \left\{\tau \in {\mathbb{R}}^{N} : \tau = {\widetilde}\tau^k + {\mathbf 1}s, s \in {\mathbb{R}}\right\}. \label{eqn:wtellk}\end{aligned}$$ Then, with this extended version of ${\mathcal{A}}$, the proposed candidate Lyapunov-like function for asymptotic stability of 