Analytical solution of the optimal control task of a semi-Markov process with finite set of states

The theoretical verification of the new method of finding the optimal strategy of control of a semi-Markov process with finite set of states is presented. The paper considers Markov randomized strategies of control, determined by a finite collection of probability measures, corresponding to each state. The quality characteristic is the stationary cost index. This index is a linear-fractional integral functional, depending on collection of probability measures, giving the strategy of control. Explicit analytical forms of integrands of numerator and denominator of this linear-fractional integral functional are known. The basis of consequent results is the new generalized and strengthened form of the theorem about an extremum of a linear-fractional integral functional. It is proved that problems of existence of an optimal control strategy of a semi-Markov process and finding this strategy can be reduced to the task of numerical analysis of global extremum for the given function, depending on finite number of real arguments.