given initial condition. These correspond to different routings of tokens in places with several output transitions. However, each of these trajectories satisfies the conservation law: $$\label{eq:flow_conservation_eq}
  \dot{m}(t) + \dot{w}(t) = C f(t) \, ,$$ where $C$ is the token flow matrix of the Petri net. Recall that a *P-invariant* of the Petri net refers to a nonnegative solution $y$ of the system $y{^\mathsf{T}}C = 0$. In the discrete setting, a P-invariant corresponds to a weighting of places that is constant for any reachable marking, meaning that the quantity $y {^\mathsf{T}}M$ is preserved under any firing of transition. An analogous statement holds in the continuous setting:

Given a P-invariant $y$ of the Petri net, the quantity $y{^\mathsf{T}}(m(t) + w(t))_{p \in {\mathcal{P}}}$ is independent of $t$.

In particular, if the entries of $y$ are all positive, then the Petri net is bounded, [*i.e.*]{}, each function $t \mapsto M_p(t)$ is bounded.

The proof consists in multiplying both sides of  by the invariant $y$. It follows that the derivative of $y{^\mathsf{T}}(m(t) + w(t))_{p \in {\mathcal{P}}}$ is zero. If the entries of $y$ are all positive, then $\bigl(y{^\mathsf{T}}(m(0) + w(0))_{p \in {\mathcal{P}}}\bigr)/y_p$ is an upper bound to the marking of any place $p$.

The following 