A large dam model is the object of study of this paper. The parameters Llower and Lupper define its lower and upper levels, L = Lupper - Llower is large, and if the current level of water is between these bounds, the dam is assumed to be in a normal state. Passage across one or other of the levels leads to damage. Let J1 and J2 denote the damage costs of crossing the lower and, respectively, the upper levels. It is assumed that the input stream of water is described by a Poisson process, while the output stream is state dependent. Let Lt denote the dam level at time t, and let p1 = limt→∞P{Lt = Llower} and p2 = limt→∞P{Lt > Lupper} exist. The long-run average cost, J = p1J1 + p2J2, is a performance measure. The aim of the paper is to choose the parameter controlling the output stream so as to minimize J.

We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior of this problem in numerical examples.