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arXiv · 1411.0880

Optimal double control problem for a PDE model of goodwill dynamics

Dominika Bogusz·Mariusz Górajski
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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.

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Dominika Bogusz, Mariusz Górajski. 2014-11-04. Optimal double control problem for a PDE model of goodwill dynamics. https://arxiv.org/abs/1411.0880

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