Management science applications of optimal control theory

Optimal control theory was first invented in the late 1950's by the Russian mathematician Pontryagin and some of his colleagues to solve the problem of controlling a rocket in flight. The state equation of that problem was the differential equation of motion of a free body together with control variables, which represent the actions of vernier rockets mounted on the rocket. The problem is to find the optimal way of controlling the rocket by adjusting these control variables in order to reach a given destination. They stated the basic idea of using the so-called maximum principle to obtain necessary conditions that such an optimal solution must obey. When their book was translated into English this theory was quickly taken up by economists who used it to formulate and solve a number of economic problems. It was also of interest to operations researchers, who also solved a number of different problems using it. The topics to be discussed in this talk will be chosen from: the cash balance problem of a company; the optimal financing of a startup firm; a production-inventory smoothing problem; the Vidale-Wolfe advertising mode; and a model for optimal preventive maintenance of a machine. In these examples the ideas of singular control and state space constraints will be briefly illustrated and discussed.