Advertising and congestion management policies for a museum temporary exhibition

In the economic literature there has recently been a strong interest on cultural organizations, especially museums. Nevertheless, only a few mathematical models have been developed in this field and most of them are econometric models. The aim of this work is to propose a special optimal control model in order to analyze the behaviour of a museum institution in terms of advertising and congestion management policies. We focus on a museum which offers a special exhibition in a definite time period. The model takes into account some of the basic features of the museum institutions, which distinguish them from the traditional economic firms. Attention is given to the cultural demand and the quality of the visitors experience, in particular in relation with the occurrence of congestion situations. The laws governing the behaviour of the system through time are defined by three distinct dynamical systems, depending on the position of the visitor attendance rate with respect to two critical levels: the congestion threshold which is a fixed parameter of the system and the extreme congestion threshold which depends on the congestion management policy. Because of the regime switching the optimal control problem is non-differentiable and the standard Pontryagin maximum principle necessary conditions are not suitable for characterizing the optimal solutions. The Clarke's generalized maximum principle necessary conditions for an optimal solution are derived, which provide some useful information to define special classes of control functions. We discuss the existence of an optimal solution which is economically meaningful and some special classes of control functions.