# Solution of the Unconditional Extremal Problem for a Linear-Fractional Integral Functional Depending on the Parameter

## TL;DR

This paper investigates an extremal problem for a fractional linear integral functional depending on a parameter, providing new theoretical results and potential applications in stochastic control.

## Contribution

It extends previous work by analyzing a fractional linear integral functional with parameter-dependent integrands on probability distributions.

## Key findings

- Proved three key extremum theorems for the functional.
- Established that solutions are determined by the extremal properties of a test function.
- Connected the results to applications in stochastic system control.

## Abstract

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.