  We study an optimal process control problem with multiple assignable causes.
The process is initially in-control but is subject to random transition to one
of multiple out-of-control states due to assignable causes. The objective is to
find an optimal stopping rule under partial observation that maximizes the
total expected reward in infinite horizon. The problem is formulated as a
partially observable Markov decision process (POMDP) with the belief space
consisting of state probability vectors. New observations are obtained at fixed
sampling interval to update the belief vector using Bayes' theorem. Under
standard assumptions, we show that a conditional control limit policy is
optimal and that there exists a convex, non-increasing control limit that
partitions the belief space into two individually connected control regions: a
stopping region and a continuation region. We further derive the analytical
bounds for the control limit. An algorithm is devised based on structural
results, which considerably reduces the computation. We also shed light on the
selection of optimal fixed sampling interval.
