On Evaluating of Final States in the Inventory Processes

This paper deals with a problem of controlling a power system which consists of a hydroelectric power station of limited capacity and a heating plant of unlimited capacity. Given the demand and random independent flows it is required to find the policy which would minimize the total fuel expenses.We introduce the penalty function evaluating the state of the system at the end of the regulation period (a year) which enable us to use the dynamic programming methods. If we suppose the process to be year-periodic it is reasonable to expect that the equality (4) holds, where $f_{n+1}(\Omega)$ is our penalty function.We prove here that the condition (4) uniquely determines the penalty function. It always takes place for $s<1$ in (2), but when $s=1$ we make an assumption which indeed signifies that there exist a moment when the probability of a flood (i.e. of a flow which is large in some sense) is positive. If the last assumption holds, then any stationary policy generates a stationary absolute (periodic) distribution on a state set, and it becomes possible to discuss the mean expenses under the stationary policy. The minimum of these expenses is reached for the policy which is determined by (2) and (4).We prove also the continuity of the solution of (4) as a function of $s$.