can easily recover the solution over the whole Petri net. This applies to the original dynamics of the system, on any time interval over which the infimum is reached by a constant policy, as stated in the following proposition.

\[thm:fc\_prio\_unicity\] Suppose that there is a policy $\pi^*$ which reaches the infimum in  for all time $t$ in the interval $[t_i, t_f]$. Then the dynamics of the Petri net with free choice and priority routing reduces to a linear system, which admits a unique solution, given the initial conditions $(m(t_i), w(t_i))$.

If $\pi^*$ reaches the infimum for any $t \in [t_i, t_f]$, then the continuous dynamics of the Petri net reads:

\[eq:const\_pi\_dyn\_system\] $$\begin{aligned}
  {C^-_{\pi^*}}f(t) &= S_{\pi^*} \left(\dfrac{m(t)}{\tau} \right) \label{eq:const_pi_dyn_f} \,, \\
  \dot{m}(t) &= C^+ f(t) - \dfrac{m(t)}{\tau} \label{eq:const_pi_dyn_dot_m} \,, \\
  \dot{w}(t) &= \dfrac{m(t)}{\tau} - C^- f(t) \label{eq:const_pi_dyn_dot_w} \,, \\
  S_{\pi^*} w (t) &= 0  \,, \label{eq:const_pi_dyn_w}
  \end{aligned}$$

which is a linear system.

We multiply  by $S_{\pi^*}$, and replace the term $S_{\pi^*} (m(t)/\tau)$ by its expression given in . This leads to: $$S_{\pi^*} \dot{m}(t) = S_{\pi^*}C^+ f(t) - {C^-_{\pi^*}}f(t) \,.$$

Let $D = \text{diag}(\tau)$ be the ${\mathcal{P}}\times {\mathcal{P}}$ diagonal matrix such that $D_{pp} = \tau_p$, then $D_{\pi} \mathrel{:=} S_{\pi} D {S_{\pi}}{^\mathsf{T}}$ is 