This paper studies a class of discrete-time discounted semi-Markov control model on
Borel spaces. We assume possibly unbounded costs and a non-stationary exponential
form in the discount factor which depends of on a rate, called the discount rate.
Given an initial discount rate the evolution in next steps depends on both the previous
discount rate and the sojourn time of the system at the current state. The new results
provided here are the existence and the approximation of optimal policies for this
class of discounted Markov control model with non-stationary rates and the horizon
is finite or infinite. Under regularity condition on sojourn time distributions and
measurable selector conditions, we show the validity of the dynamic programming algorithm
for the finite horizon case. By the convergence in finite steps to the value functions,
we guarantee the existence of non-stationary optimal policies for the infinite horizon
case and we approximate them using non-stationary epsilon-optimal policies. We illustrated
our results a discounted semi-Markov linear-quadratic model, when the evolution of
the discount rate follows an appropriate type of stochastic differential equation.