the ${\mathcal{Q}}\times {\mathcal{Q}}$ diagonal matrix such that $(D_{\pi})_{qq} = \tau_{p_{\pi}(q)}$. Equation  then writes $S_{\pi^*} m(t) = D_{\pi^*} {C^-_{\pi^*}}f(t)$. This leads to: $$\dot{f}(t) = ({C^-_{\pi^*}})^{-1} D_{\pi^*}^{-1} \left( S_{\pi^*} C^+  -  {C^-_{\pi^*}}\right) f(t) \, ,$$ which is an ordinary differential system. Moreover, the $f(t_i)$ can be obtained from the $m(t_i)$ by , so that this system admits a unique solution $f$ for all $t \in [t_i, t_f]$.

Given this solution $f$, one can successively solve the differential system in ${m}$ given by  and the differential system in ${w}$ given by , whose initial conditions are known, so that the whole dynamics admits a unique trajectory.

Stationary solutions {#sec:4}
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In this section, we prove that the stationary solutions of the continuous and discrete dynamics of a timed Petri net with free-choice and priority routing are the same. To do so, we first recall in Section \[sec:stationary\_discrete\] the formulation of the discrete dynamics and the associated stationary solutions given in [@allamigeon2015performance].

Stationary solutions of the discrete dynamics {#sec:stationary_discrete}
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The discrete dynamics of Petri nets with free choice and priority is expressed in terms of *counter variables* associated with transitions and places. Given a transition $q$, the counter variable $z_q: {\mathbb{R}_{\geq 0}}\to {\mathbb{N}}$ 