In this paper we propose a classification of crowd models in built environments based on the assumed pedestrian ability to foresee the movements of other walkers. At the same time, we introduce a new family of macroscopic models, which make it possible to tune the degree of predictiveness of the individuals. By means of these models we describe both the natural behavior of pedestrians, i.e., their expected behavior according to their real limited predictive ability, and a target behavior, i.e., a particularly efficient behavior one would like them to assume (for, e.g., logistic or safety reasons). Then we tackle a challenging shape optimization problem, which consists in controlling the environment in such a way that the natural behavior is as close as possible to the target one, thereby inducing pedestrians to behave more rationally than what they would naturally do. We present numerical tests which elucidate the role of rational/predictive abilities and show some promising results about the shape optimization problem.

We study in what sense one can determine the flux functions k = k(x) and f = f(u), k piecewise constant, in the scalar hyperbolic conservation law u(t) + (k(x)f (u))(x) = 0 by observing the solution u(t, center dot) of the Cauchy problem with suitable piecewise constant initial data u vertical bar(t=0) = u(o).

We consider stochastic differential games with N players, linear-Gaussian dynamics in arbitrary state-space dimension, and long-time-average cost with quadratic running cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of N Hamilton-Jacobi-Bellman (HJB) and N Kolmogorov-Fokker-Planck (KFP) partial differential equations. We give necessary and sufficient conditions for the existence and uniqueness of quadratic-Gaussian solutions in terms of the solvability of suitable algebraic Riccati and Sylvester equations. Under a symmetry condition on the running costs and for nearly identical players, we study the large population limit, N tending to infinity, and find a unique quadratic-Gaussian solution of the pair of mean-field game HJB-KFP equations. Examples of explicit solutions are given, in particular for consensus problems.

We consider stochastic differential games with N nearly identical players, linear-Gaussian dynamics, and infinite horizon discounted quadratic cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of N Hamilton-Jacobi-Bellman and N Kolmogorov-Fokker-Planck partial differential equations, proving that for small discount factors quadratic-Gaussian solutions exist and are unique. Then, we prove the convergence of such solutions to the unique quadratic-Gaussian solution of the pair of Mean Field equations. We also discuss some singular limits, such as vanishing discount, vanishing noise, and cheap control.