“…The existence of a deterministic stationary optimal policy is proved under a different and general set of conditions as compared to the previous literature; the controlled process can be explosive, the transition rates can be arbitrarily unbounded and are weakly continuous, the multifunction defining the admissible action spaces can be neither compact-valued nor upper semi-continuous, and the cost rate is not necessarily inf-compact.Firstly, all the aforementioned works on CTMDPs [13,14,15,16,19,27,32,34] assume the underlying process to be non-explosive; and most of them achieve this by assuming the existence of a Lyapunov function bounding the growth of the transition rates. In the present article we remove this condition, and allow the transition rates to be essentially arbitrarily unbounded, and the controlled process to be possibly explosive.The development of the theory covering such CTMDPs was once regarded quite challenging in the survey [15];for the discounted criteria it has been done in e.g., [7], see also [31].Secondly, we assume the weak continuity on the underlying signed kernel defining the transition rates, while all the previous literature on average CTMDPs in Borel spaces is based on the strong continuity condition, except for [20], which establishes the existence of a randomized stationary optimal policy for the constrained CTMDPs.It is relevant to point out that recently the developments of the theory of average DTMDPs (discrete-time Markov decision processes) and SMDPs (semi-Markov decision processes) with weakly continuous (also called Feller) transition probabilities have received much attention from the research community [5,6,8,24,25,26].In a nutshell, as compared to the strongly continuous case, the proofs with weakly continuous transition rates are more technical, and the construction of the solution to the optimality inequality would involve the notion of the generalized lower limit and the generalized Fatou's lemma. Moreover, based on a neat generalization of the Berge theorem [9], which is partially summarized in Lemma 5.1 below, and as in [8] for the average DTMDP, we allow the multifunction defining the admissible action spaces to be neither compact-valued nor upper semi-continuous.If the state space is countable, then the concepts of weak and strong continuity coincide.…”