Solution of the unconditional extremal problem for a linear-fractional integral functional dependent on the parameter

The paper is devoted to the study of the unconditional extremal problem for a fractional linear integral functional defined on a set of probability distributions. In contrast to results proved earlier, the integrands of the integral expressions in the numerator and the denominator in the problem under consideration depend on a real optimization parameter vector. Thus, the optimization problem is studied on the Cartesian product of a set of probability distributions and a set of admissible values of a real parameter vector. Three statements on the extremum of a fractional linear integral functional are proved. It is established that, in all the variants, the solution of the *original problem is completely determined by the extremal properties of the test function of the linear-fractional integral functional; this function is the ratio of the integrands of the numerator and the denominator. Possible applications of the results obtained to problems of optimal control of stochastic systems are described.