Flow-Invariant Sets with Respect to the Markings of Timed Continuous Petri Nets

The paper investigates the existence of flow-invariant sets with respect to the marking of a timed continuous Petri net (TCPN) with infinite server semantics. Such a set has the property that for any initial marking belonging to the set, the marking at any moment in the evolution of the net also belongs to the set. Thus, the traditional concept of marking invariance used in PN theory, which refers to a set of places, is complemented in the sharper sense of the individual monitoring of each place. We take into consideration several types of bounded flow-invariant sets. The join-free TCPNs are treated separately from TCPNs with joins as allowing the development of supplementary investigation tools. Subsidiary to our results we give a consistent and rigorous mathematical proof for the nonnegativeness of the marking in TCPNs.