A problem in jointly optimal production and advertising decisions

In this paper, a production control model proposed by Holt, Modigliani, Muth and Simon and a marketing model suggested by Nerlove and Arrow, are combined to produce an optimal control problem with two state variables, two control variables and inequality constraints on these variables. The optimal production rate and advertising expenditures over time are determined using the maximum principle. Due to the complex nature of the necessary conditions, the numerical technique known as the initial-value shooting method, is used to generate optimal trajectories. A numerical example is presented to illustrate the solution.