{
  "id": "math/0701458",
  "version": "v3",
  "published": "2007-01-16T23:01:59.000Z",
  "updated": "2008-05-20T06:20:40.000Z",
  "title": "Optimal control of a large dam, taking into account the water costs",
  "authors": [
    "Vyacheslav M. Abramov"
  ],
  "comment": "18 pages, 1 table. Revision is submitted",
  "categories": [
    "math.PR",
    "math.CA"
  ],
  "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 $$J=p_1J_1+p_2J_2+\\sum_{i=L^{lower}+1}^{L^{upper}}q_ic_i.$$",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-05-20T06:20:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K30",
      "40E05",
      "90B05",
      "60K25"
    ],
    "keywords": [
      "water cost",
      "large dam",
      "optimal control",
      "upper level",
      "output stream"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1458A"
    }
  }
}