Demand occasionally depends positively on the amount of displayed stock, especially in the case of novelty or impulse purchase items. Such inventory-level dependence of demand rate has been incorporated into some continuous review inventory control models, but only ones with demand rate which does not vary with time, no shortages, infinite production rate and per-unit-time cost minimization. We propose an optimal control model which relaxes all these assumptions. That model generalizes a known optimal control model by adding inventory-level dependence of demand to the state dynamics. The framework is one of discounted profit maximization. We apply the maximum principle and obtain three possible singular regimes. The special case of time-invarying demand is solved explicitly. The uncapacitated case is analyzed in detail for a "wave-mode" time dependence of demand, and a numerical example is given.