Optimal double control problem for a PDE model of goodwill dynamics
Abstract
We propose a new optimal model of product goodwill in a segmented market where the state variable is described by a partial differential equation of the Lotka--Sharp--McKendrick type. In order to maximize the sum of discounted profits over a finite time horizon, we control the advertising efforts which influence the state equation and the boundary condition. Moreover, we introduce the mathematical representation of consumer recommendations in a segmented market. Based on the semigroup approach, we prove the existence and uniqueness of optimal controls. Using a maximum principle, we construct a numerical algorithm to find the optimal solution. Finally, we examine several simulations on the optimal goodwill model and discover two types of advertising strategies.
Keywords
Cite
@article{arxiv.1411.0880,
title = {Optimal double control problem for a PDE model of goodwill dynamics},
author = {Dominika Bogusz and Mariusz Górajski},
journal= {arXiv preprint arXiv:1411.0880},
year = {2014}
}