  We consider a class of optimal advertising problems under uncertainty for the
introduction of a new product into the market, on the line of the seminal
papers of Vidale and Wolfe, 1957, and Nerlove and Arrow, 1962. The main
features of our model are that, on one side, we assume a carryover effect (i.e.
the advertisement spending affects the goodwill with some delay); on the other
side we introduce, in the state equation and in the objective, some mean field
terms which take into account the presence of other agents. We take the point
of view of a planner who optimizes the average profit of all agents, hence we
fall into the family of the so-called "Mean Field Control" problems. The
simultaneous presence of the carryover effect makes the problem infinite
dimensional hence belonging to a family of problems which are very difficult in
general and whose study started only very recently, see Cosso et Al, 2023. Here
we consider, as a first step, a simple version of the problem providing the
solutions in a simple case through a suitable auxiliary problem.
