Note that the jump map $G$, at points $\tau \in {\mathcal{X}}$, is set valued by definition of $g_i$ in . From these points there exist solutions to ${{{\mathcal}H}}_N$ that jump out of ${\mathcal{X}}$. In fact, consider the case $\tau \in {\mathcal{X}}$. We have that $\tau_{i} = \tau_r$ for some $i,r \in I$. Then, after the jump it follows that $g_{i}(\tau) \in \{0, (1+{\varepsilon}){\bar{\tau}}\}$ and $g_{r}(\tau) \in \{0, (1+{\varepsilon}){\bar{\tau}}\},$ and there exist $g_{i}$ and $g_{r}$ such that $g_{i} = g_{r}$ or $g_{i} \neq g_{r}$. Since for every point in ${\mathcal{X}}$ there exists a solution that converges to ${\mathcal{A}}$ and also a solution that stays in ${\mathcal{X}}$, ${\mathcal{X}}$ is weakly forward invariant.[^9]

Characterization of Time of Convergence {#sec:timetoconverge}
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In this section, we characterize the time to converge to a neighborhood of ${\mathcal{A}}$. The proposed (upper bound) of the time to converge depends on the initial distance to the set ${\widetilde}{\mathcal{A}}$ and the parameters of the hybrid system $(\varepsilon,{\bar{\tau}})$.

\[thm:timetoconverge\] For every $N \in {\mathbb{N}}$, $N > 1$, and every $c_1, c_2$ such that $\overline{c} > c_2 > c_1 > 0$ with $\overline{c} = \max_{x \in {\mathcal{X}}} |x|_{\widetilde{\mathcal{A}}}$, every maximal solution to ${{{\mathcal}H}}_N$ with initial condition $\tau(0,0) \in (P_{N} \setminus {\mathcal{X}}) \cap 