Optimal control of a large dam, taking into account the water costs

Abstract

Consider a dam model, $L^{upper}$ and $L^{lower}$ are upper and, respectively, lower levels, $L = L^{upper}-L^{lower}$ is large and if the level of water is between these bounds, then the dam is said to be in a normal state. Passage across lower or upper levels leads to damage. Let $J_1=j_1L$ and $J_2=j_2L$ denote the damage costs per time unit of crossing the lower and, correspondingly, upper level where $j_1$ and $j_2$ are given real constants. It is assumed that input stream of water is described by a Poisson process, while the output stream is state dependent. Let $L_t$ denote the level of water in time $t$, and $c_{L_t}$ denote the water cost at level $L_t$ ($L^{lower}<L_t\leq L^{upper}$). Assuming that $p_1=\lim_{t\to\infty}\mathbf{P}\{L_t=L^{lower}\}$, $p_2=\lim_{t\to\infty}\mathbf{P}\{L_t>L^{upper}\}$ and $q_i=\lim_{t\to\infty}\mathbf{P}\{L_t=i\}$ ($L^{lower}<i\leq L^{upper}$) exist, the aim of the paper is to choose the parameters of an output stream (specifically defined in the paper) minimizing the long-run expenses $$J=p_1J_1+p_2J_2+\sum_{i=L^{lower}+1}^{L^{upper}}q_ic_i.$$