Advertising for the introduction of an age-sensitive product

An age-dependent market segmentation is often observed for real life products. We introduce a simple age-structured model for the advertising process of a firm and the consequent goodwill evolution. The model formal structure is characterized by a first order linear partial differential equation. We formulate the advertising problem for a new product introduction as a distributed parameter optimal control problem and solve it using the suitable maximum principle conditions. Then, we discuss and solve the same problem under the assumption that a finite set of boundedly age-selective advertising channels are available, instead of one infinitely age-selective channel.