A Note on the Optimal Control of Stocks Accumulating with a Delay

We study a generic optimal control problem with a stock that accumulates with a constant delay. We show that the optimal system dynamics reduces to a system of ordinary differential equations, implying monotonic optimal paths, if the objective function is additively separable in the stock and the control. This is, however, not true for general objective functions which may exhibit non-monotonic and oscillatory optimal paths. The reason is that the impact of the stock on the objective depends on the current level of the control, while the control influences the dynamcis of the stock with a delay