  This paper concentrates on the minimal hitting probability of continuous-time
controlled Markov systems (CTCMSs) with countable state and finite admissible
action spaces. The existence of an optimal policy is first proved. In
particular, for a special and important case of controlled branching processes
(CBPs), it is proved that the minimal hitting probability is the unique
solution to an improved optimal system of equations. Furthermore, a novel and
precise improved-policy iteration algorithm of an optimal policy and the
minimal hitting probability (minimal extinction probability) is presented for
CBPs.
