Generalized submarking reachability problems under unknown firing count vectors and their inverse problems of Petri nets

Petri nets are useful in modeling concurrent/parallel systems. But, their applications in practice have been slow due to lack of computational tools and techniques capable of dealing with large scale nets. In this paper, first, optimal-control-base analysis aspect of the sub-marking reachability problems (SMR) included the well-known reachability problems (MR) and MR with a firing count vector (MR-FV) is discussed and a semi-polynomial time algorithm for SMR⊃MR is shown by applying the discrete-time Pontryagin´s minimum principle (PMP) which includes the linear programming (LP) for each sub-problem optimization, where the checking procedure for critical siphons at each time, however, is neglected in the above time-complexity evaluation. Secondly, it is shown that the reachability problems and their inverse problems with no intermediate marking constraints, including the generalized sub-marking problems (GSMR) and the generalized sub-marking reachability on a minimum initial marking (MIS), are classified into five problems. Thirdly, it is briefly discussed that other problems with an unknown firing count vector can be reduced to SMR⊃MR