A time-dependent switching mean-field game on networks motivated by optimal visiting problems

Motivated by an optimal visiting problem, we study a switching mean-field game on network, where both decisional and time-variable is at disposal of the agents for what concerns, respectively, instant to decide perform switch. Every switch between nodes network represents from $0$ $1$ one component string $p = (p_1,\ldots, p_n)$ which, in interpretation, gives information visited targets, being targets labeled $i=1,\ldots, n$. The goal reach final $(1, \ldots, 1)$ time $T$, minimizing cost also depending congestion nodes. We prove existence suitable definition approximated $\varepsilon$-mean-field equilibrium then address passage limit when $\varepsilon$ goes 0.