  The supervisory control of probabilistic discrete event systems (PDESs) is
investigated under the assumptions that the supervisory controller (supervisor)
is probabilistic and has a partial observation. The probabilistic P-supervisor
is defined, which specifies a probability distribution on the control patterns
for each observation. The notions of the probabilistic controllability and
observability are proposed and demonstrated to be a necessary and sufficient
conditions for the existence of the probabilistic P-supervisors. Moreover, the
polynomial verification algorithms for the probabilistic controllability and
observability are put forward. In addition, the infimal probabilistic
controllable and observable superlanguage is introduced and computed as the
solution of the optimal control problem of PDESs. Several examples are
presented to illustrate the results obtained.
