Abstract: The system of controlled time-inhomogeneous Markov chains (MC) is considered. The principal problem is the ``curse of dimension'' which appears here as the necessity of solving the system of ordinary differential equations of high dimension. Moreover, even the development of the system itself is a serious issue since these equations are linked and the standard parallelization approach developed in existing software packages are not very effective. Meanwhile, one can observe that the minimization procedure needed for the right hand side of this system may be easily parallelized since for each equation the minimization procedure may be realized independently. As an example we consider the management of linked dams under seasonal random inflows/outflows and customers' demands. The current state of each dam is the state of continuous-time Markov chain corresponding to the water level. So the state of the dams system is represented in tensor form. The connection of Markov chains is due to the controlled flow between dams. The aim of the control is to keep balance under the natural perturbation and at the same time to satisfy the customers' demands. The general approach to the solution is based on the dynamic programming method which leads to the solution of Bellman type equation in tensor form. This equation may be reduced to the system of ordinary differential equations. Here we suggested the automatic procedure of this system generation and an approach to the minimization which may be realized for each state independently.

Abstract: В статье рассмотрена задача Монжа-Канторовича в одномерном случае с выпуклой ценовой функцией. Найден явный вид ее решения. Найден явный вид барицентра конечного набора мер. Для пространства мер с компактным носителем и метрикой Монжа-Канторовича определен барицентр распределения и найдена явная формула для него. Также доказан аналог ЗБЧ для пространства мер с компактным носителем.