Minimization of communication expenditure for seasonal products

We consider a firm that sells seasonal goods. The firm seeks to reach a fixed level of goodwill at the end of the selling period, with the minimum total expenditure in promotional activities. We consider the linear optimal control problem faced by the firm which can only control the communication expenditure rate; communication is performed by means of advertising and sales promotion. Goodwill and sales levels are considered as state variables and word-of-mouth effect and saturation aversion are taken into account. The optimal control problem is addressed by means of the classical Pontryagin Maximum Principle and the solution can be easily found solving, in some cases numerically, a system of two non linear equations. Moreover, a parametric analysis is performed to understand how the total expenditure in communication should be divided between advertising and sales promotion