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In this paper, a new method for solving the problem of optimal control ofsemi-Markov processes with finitely many states is considered. A new form ofthe assertion on an extremum of a liner-fractional integral functional given ona set of probability measures is formulated and proved. This form underlies thetheorem of optimal control strategy for semi-Markov processes. It is provedthat the solution of the optimal control problem for a semi-Markov process withfinitely many states is completely determined by the extremum properties of theso-called test function of the liner-fractional integral functional which isthe control quality index. At the same time, an explicit analyticrepresentation was obtained for this test function in terms of the initialprobability characteristics of the semi-Markov model.

Solution of the unconditional extremum problem for a liner-fractional  integral functional on a set of probability measures and its application in  the theory of optimal control of semi-Markov processes