The control of a finite dam with penalty cost function: Wiener process input

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The long-run average cost per unit time of operating a finite dam controlled by a Pτ,λM policy (Attia, 1987) is determined when the cumulative input process is a Wiener process with drift. A penalty cost which accrues continuously at the rate g(Z(t)), where g is a bounded measurable function of the content, is also introduced. We first obtain the resolvent operator Rα of a Wiener process with a reflecting boundary at 0 and the expansion of the associated kernel Kα as a power series in α. Then we use these results to determine the long-run average cost per unit time.