Two-stage optimal control problems and optimality conditions

When an optimal investment decision is studied for a firm whose capital goods are subject to a delivery lag, a control problem with a time-delay argument in the objective function emerges. Such a problem under a set of simple assumptions is shown to be reducible to a two-stage optimal control problem. The significance of this is that familiar tools from standard optimal control theory are applicable to the resulting two-stage problem. Necessary conditions are presented for a general two-stage problem with an adjustable switching time. Some specific results are also obtained for several special cases.