A Class of Nonlocal Models for Pedestrian Traffic

Abstract

We present a new class of macroscopic models for pedestrian flows. Each individual is assumed to move towards a fixed target, deviating from the best path according to the instantaneous crowd distribution. The resulting equation is a conservation law with a nonlocal flux. Each equation in this class generates a Lipschitz semigroup of solutions and is stable with respect to the functions and parameters defining it. Moreover, key qualitative properties such as the boundedness of the crowd density are proved. Specific models are presented and their qualitative properties are shown through numerical integrations. In particular, the present model accounts for the possibility of reducing the exit time from a room by carefully positioning obstacles that direct the crowd flow.

Nonlocal multi-scale traffic flow models: analysis beyond vector spaces

“…Convolution operators, for example, are preferred to Nemytskii operators in the quasilinear hyperbolic equations. Nonlocal models of hyperbolic type have already been investigated thoroughly by Colombo, Goatin and collaborators, for example (see [2,16,[22][23][24][25]28,36] …”

“…Most of the publications so far specify weak solutions in L 1 with bounded total variation in space so the well established theory about hyperbolic balance laws can be applied (see, e.g., [2,23,25,30]). Clearly, the latter function space opens the door for the coefficients to a larger class of pointwise dependence on space, but it restricts the admissible set of solution values significantly.…”

Coupling Traffic Flow Networks to Pedestrian Motion

“…Traffic models on networks for this equation can be found among many others in [25,26,9,23,18]. In [27,28,15,10] pedestrian traffic modeling with scalar conservation laws based on the solution of the eikonal equation have been presented and investigated, see again [5] for further developments and historical comments. Additionally, in [5,6], a variety of other macroscopic models are discussed.…”

Nonlocal Systems of Conservation Laws in Several Space Dimensions

“…As usual, t is time, x the space variable, U the vector of the unknown densities, F the matrix valued flow and η is a matrix of smooth convolution kernels. Systems of the type (1.1), with the particular coupling considered below, comprise, for instance, sedimentation models [4] and various crowd dynamics models [8,9,10]. The nonlocal nature of (1.1) is particularly suitable in describing the behavior of crowds, where each member moves according to her/his evaluation of the crowd density and its variations within her/his horizon.…”

“…We refer to the several models recently considered in the literature, e.g. [8,10]. In one space dimension, vehicular traffic offers entirely similar situations, see [5,9].…”