Solution to the problem of optimal control of a stochastic semi-Markov model of continuous supply of product management under the condition of constantly happening consumption

The solution of the optimal control problem in the semi-Markov model in question is theoretically justified. To achieve this goal, formal analytic transformations of the integral representations obtained by the authors earlier for the basic probabilistic characteristics of the model were carried out. These transformations made it possible to use the theorem on the analytical representation of the stationary value of the management effectiveness of a semi-Markov process in the form of a fractional-linear integral functional. In the sequel, the authors use the general theorem on the extremum of a fractional-linear integral functional, proved by P. V. Shnurkov. This theorem makes it possible to reduce the problem of optimal reserve management to the problem of investigating the global extremum of a given function from a finite number of real nonnegative variables that can be effectively solved in practice using the known numerical methods.