Advertising and congestion management for a museum temporary exhibition

We propose a model for a museum institution which offers a special exhibition in a definite time period. Attention is given to the museum demand and the quality of the visitors' experience, in particular in relation to 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 nondifferentiable, and necessary conditions in the form of Clarke's generalized maximum principle are derived. We discuss the existence of an optimal solution and some special classes of control functions which are economically meaningful.