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In this research, we obtained some mathematical programming approaches to reachability problems for place-transition Petri nets which are one of typical models for various kinds of discrete event dynamical systems.
An algorithm for the executable firing sequence with the shortest length in the submarking reachability problems (i.e. SMR) is given by general discrete-time Pontryagin s mini mum principle included linear programmings (i.e., "PMP+LP" ).
We also obtained algorithms for the executable firing sequence with the prescribed length in the basic marking reachability problems with the known firing count. vector (i.e., MR-FV) via special "PMP+LP" as well as "DP+LP" (i.e., the dynamic programming included linear programmings).
These algorithms divide die original problem into d smaller subproblems and have semi-polynomial time complexity provided that checking critical siphons, which is an important future research problem, is neglected, where d is the length of firing sequence of transitions.
Note that SMR (MF-FV, resp.) is the most, fundamental reachability problem without (with, resp.) the known firing count vector in si-ate equation of Petri nets.
For MR-FV, a new reachability criterion is also given, that is, the reachability for a given Petri net is reduced to one for each strongly-connected and maximal siphon because the firing count vector in MR-FV is in advance specified. Another approach to finding the executable firing sequence in MR-FV by using T-invariants and the extended particular solution of state equation. This approach is expected to be useful for reachability problems of live free-choice Petri nets.