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 rec-ommendations 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. AMS 2000 subject classification: 90B60, 49J20; JEL C61, D42, M37. 1