  The present paper considers the constrained optimal control problem with
total undiscounted criteria for a continuous-time Markov decision process
(CTMDP) in Borel state and action spaces. Under the standard compactness and
continuity conditions, we show the existence of an optimal stationary policy
out of the class of general nonstationary ones. In the process, we justify the
reduction of the CTMDP model to a discrete-time Markov decision process (DTMDP)
model based on the studies of the undiscounted occupancy and occupation
measures. We allow that the controlled process is not necessarily absorbing,
and the transition rates are not necessarily separated from zero, and can be
arbitrarily unbounded; these features count for the main technical difficulties
in studying undiscounted CTMDP models.
