Optimal Distributed Dynamic Advertising

We propose a novel approach to modeling advertising dynamics for a firm operating over a distributed market domain based on controlled partial differential equations of the diffusion type. Using our model, we consider a general type of finite-horizon profit maximization problem in a monopoly setting. By reformulating this profit maximization problem as an optimal control problem in infinite dimensions, we derive sufficient conditions for the existence of its optimal solutions under general profit functions, as well as state and control constraints, and provide a general characterization of the optimal solutions. Sharper, feedback-form characterizations of the optimal solutions are obtained for two variants of the general problem. The first author gratefully acknowledges financial support by the NSF, the DAAD, the SFB 611 (Bonn), and the Max-Planck-Institut für Mathematik (Leipzig) through an IPDE fellowship.