  The objective of this work is to study continuous-time Markov decision
processes on a general Borel state space with both impulsive and continuous
controls for the infinite-time horizon discounted cost. The continuous-time
controlled process is shown to be non explosive under appropriate hypotheses.
The so-called Bellman equation associated to this control problem is studied.
Sufficient conditions ensuring the existence and the uniqueness of a bounded
measurable solution to this optimality equation are provided. Moreover, it is
shown that the value function of the optimization problem under consideration
satisfies this optimality equation. Sufficient conditions are also presented to
ensure on one hand the existence of an optimal control strategy and on the
other hand the existence of an $\varepsilon$-optimal control strategy. The
decomposition of the state space in two disjoint subsets is exhibited where
roughly speaking, one should apply a gradual action or an impulsive action
correspondingly to get an optimal or $\varepsilon$-optimal strategy. An
interesting consequence of our previous results is as follows: the set of
strategies that allow interventions at time $t=0$ and only immediately after
natural jumps is a sufficient set for the control problem under consideration.
