Optimal double control problem for a PDE model of goodwill dynamics

TL;DR

This paper develops a PDE-based model for optimizing advertising strategies in a segmented market, incorporating consumer recommendations, and provides theoretical and numerical methods to determine optimal controls.

Contribution

It introduces a novel PDE model for goodwill dynamics with boundary controls and consumer recommendations, and develops a maximum principle-based numerical solution approach.

Findings

Two distinct optimal advertising strategies identified

Existence and uniqueness of optimal controls proven

Numerical simulations demonstrate model effectiveness

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.