Optimal open-loop control of online advertisement

Online advertisement through social media is studied in this paper. We describe the combined effect of two mechanisms: the acceleration service offered by online media platforms and the effect of content sharing which occurs in parallel over multiple online channels. In particular, the latter operates by reinforcing the effect of acceleration and may ultimately determine the viral diffusion of the advertisement. We describe the concurrence of these two phenomena by means of a generalized Bass model, which is inspired by the dynamics which is observed in real data. Also, the source of the advertisement pays the platform owner in order to accelerate its diffusion. We hence focus on the optimal control problem that solves the trade-off between the acceleration cost and the diffusion of the advertisement. We derive the structure of the optimal control in open loop form and we discuss its structure in connection with the virality footprint of the advertisement.