Killed Markov Decision Processes on Finite Time Interval for Countable Models

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We consider killed Markov decision processes for countable models on a finite time-interval. Existence of a uniform ε-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.