Optimal control of advertising and production of a seasonal product with nondifferentiable demand

We consider an application of optimal control theory to a marketing problem, in which a firm seeks the maximum profit by producing and selling a seasonal product. We assume that the firm can advertise its product in order to improve the firm goodwill and affect the product demand positively. In particular the demand is zero when the goodwill is less than a fixed threshold. The original optimal control problem is shown to be equivalent to a nonlinear programming problem, which has an optimal solution, possibly not unique.