On a structure of an accumulation function constructed on trajectories of controlled semi-Markov process with finite state space

We consider a controlled semi-Markov process with finite state space whose embedded Markov chain has several closed classes of communicating states. We prove the theorem which states that the accumulation function depends in a linear fractional manner on probability measures defining a Markov homogeneous randomizing strategy. This result determines a structure of distributions providing extremal values of objective functionals.