The optimal control problem for the Vidale-Wolfe advertising model revisited

The optimal control problem for the Vidale-Wolfe advertising model with unbounded controls is reformulated rigorously. It is shown that even though the optimal control is a measure of bounded variation, there exist physically realizable approximations to it. These smoother controls converge to the optimal control together with the induced trajectories and costs. Also, closed-form expressions are given for the optimal state trajectories and the induced costs.