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## Solution of the Unconditional Extremal Problemfor a Linear-Fractional Integral FunctionalDepending on the Parameter

Working papers by Cornell University. Series math "arxiv.org". 2019. No. arXiv:1906.05824v1. P. 1-14.
P. V. Shnurkov, K. A. Adamova.

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