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Optimal double control problem for a PDE model of goodwill dynamics. (English) Zbl 1376.49003

Summary: We propose a new optimal control model of product goodwill in a segmented market where the state variable behavior 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 marketing communication activities which influence the state equation and the boundary condition. Moreover, we introduce the mathematical representation of heterogeneous electronic word of mouth. Based on the semigroup approach, we prove the existence and uniqueness of optimal controls. Using a maximum principle, we describe a numerical algorithm to find the optimal solution. Finally, we examine several examples on the optimal goodwill model and discover two types of marketing strategies.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
49M05 Numerical methods based on necessary conditions
90B60 Marketing, advertising
91B60 Trade models
91B15 Welfare economics
91B24 Microeconomic theory (price theory and economic markets)

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