Finding legal firing sequences in submarking reachability problems of Petri nets by discrete-time Pontryagin's minimum principle

The relaxed problems for SMR/spl sup/MR have been formulated by PMP with a free terminal time; SMR using Theorem Al and MR using Corollary Al; note that MR-FV has been solved by PMP-LP with a fixed terminal time. Secondly, we have proven that each problem for SMR/spl sup/MR can have nonnegative integer solution by using PMP-LP of which time complexity is semi-polynomial time in live TCC and SCC nets. But, checking critical siphons at each time is needed in the upper classes. We have also assured the above execution by computer simulations. These approaches can give useful approximation algorithms for various important reachability problems in practical applications.