A control theoretical approach to crowd management: Comment on "Human behaviours in evacuation crowd dynamics: From modelling to "big data" toward crisis management" by Nicola Bellomo et al.

The formulation of mathematical models for crowd dynamics is one current challenge in many fields of applied sciences. It involves the modelisation of the complex behavior of a large number of individuals. In particular, the difficulty lays in describing emerging collective behaviors by means of a relatively small number of local interaction rules between individuals in a crowd. Clearly, the individual’s free will involved in decision making processes and in the management of the social interactions cannot be described by a finite number of deterministic rules. On the other hand, in large crowds, this individual indeterminacy can be considered as a local fluctuation averaged to zero by the size of the crowd. While at the microscopic scale, using a system of coupled ODEs, the free will should be included in the mathematical description (e.g. with a stochastic term), the mesoscopic and macroscopic scales, modeled by PDEs, represent a powerful modeling tool that allows to neglect this feature and provide a reliable description. In this sense, the work by Bellomo, Clarke, Gibelli, Townsend, and Vreugdenhil [2] represents a mathematical-epistemological contribution towards the design of a reliable model of human behavior. Besides the exhaustive review work and the cognitive contribution to the modeling problem, the essay [2] also inspires new issues concerning the application to crowd management; for instance, the identification and the analysis of control mechanisms in crowd behaviors. Control problems in crowd dynamics represent a rather new topic, which has already attracted the attention of several scientists. Following the characterization of the modeling scales given by [2], we distinguish between micro, meso, and macro models. On the other hand, concerning