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