On the analytical structure of some kinds of target functionals associated with the control problems of semi-Markov processes

The present author investigates the analytical structure of three kinds of functionals from a controllable semi-Markov process with a finite set of states. It is proved that all these mathematical objects can be represented in the form of a fractional-linear integral functional defined on a finite set of probability measures that determine the control strategy of the corresponding semi-Markov process. For each of these functionals, explicit representations for the integrand functions of the numerator and denominator through the initial probabilistic characteristics of the controlled semi-Markov process are obtained. This result allows one to reduce the problem of optimal control of a semi-Markov process with a particular target functional to the problem of investigation on the global extremum of a given function of a finite number of variables.