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.