Advertising a social event in a heterogeneous market

We study the problem of advertising a social event in a segmented market using different media. The result of such advertising process is the evolution of a goodwill vector. We describe the problem as an optimal control one, assuming that the demand is a concave function of goodwill. We prove the existence of an optimal solution and characterize it using the Pontryagin Maximum Principle. This leads to the analysis of a family of linear programming problems. We provide a more explicit and detailed description of optimal solutions for the case of goodwill linear demand.