  We consider killed Markov decision processes for countable models on a finite
time-interval. Existence of a uniform $\varepsilon$-optimal policy is proven.
We show the correctness of the fundamental equation. The optimal control
problem is reduced to a similar problem for the derived model. We receive an
optimality equation and a method for the construction of simple optimal
policies. The sufficiency of simple policies for countable models is proven. We
show the correctness of the Markovian property. Additionally, a dynamic
programming principle is considered.
