Solution of the unconditional extremum problem for a linear-fractional integral functional on a set of probability measures

Resumo

A new, generalized and strengthened, form of an assertion about an extremum of a linear-fractional integral functional given on a set of probability measures is presented. It is shown that the solution of the extremal problem for such a functional is completely determined by the extremal properties of the so-called test function, which is the ratio of the integrands of the numerator and the denominator. On the basis of this assertion, a theorem on an optimal strategy for controlling a semi-Markov process with a finite set of states is proved. In particular, it is established that if the test function of the objective functional of a control problem attains a global extremum, then an optimal control strategy exists, is deterministic, and is determined by the point of global extremum. The corresponding assertions are also obtained for the case where the test function does not attain the global extremum.