an asymptotically stable set for ${\widetilde}{{{\mathcal}H}}_N$.

Note that the continuous function $V$, given by , is defined as the minimum distance from $\tau$ to ${\widetilde}{\mathcal{A}}$, where ${\widetilde}{\mathcal{A}}$ is the union of $N!$ sets ${\widetilde}\ell_{k}$ in . To determine the change of $V$ during flows[^8], we consider the relationship between the flow map and the sets ${\widetilde}\ell_{k}$. The inner product between a vector pointing in the direction of the set ${\widetilde}\ell_{k}$ and the flow map on ${\widetilde}{C}$ satisfies $${\mathbf 1}^{\top}f(\tau) = {\mathbf 1}^{\top}(\omega{\mathbf 1}) = \omega N=|{\mathbf 1}||\omega {\mathbf 1}| = |{\mathbf 1}||f(\tau)|\cos \theta$$, which is only true if $\theta$ is zero. Therefore, the direction of the flow map and of the vector defining ${\widetilde}\ell_{k}$ are parallel, implying that the distance to the set ${\widetilde}{\mathcal{A}}$ is constant during flows.

The change in $V$ during jumps is given by $V(G(\tau)) - V(\tau)$ for $\tau \in {\widetilde}D \setminus {\widetilde}{\mathcal{A}}$. Due to the fact that we can rearrange the components of $\tau \in P_N \setminus {\mathcal{X}}$, without loss of generality, we consider a single jump condition, namely, we consider $\tau$ such that ${\bar{\tau}}= \tau_{1} > \tau_{2} > \ldots > \tau_{N-1} > \tau_{N}$. Using the formulation in Section \[sec:AN\] and Lemma \[eqn:tauSsolution\], the elements of 