Adriano Festa

A decomposition technique for pursuit evasion games with many pursuers

Memory storage constraints impose ultimate limits on the complexity of differential games for which optimal strategies can be computed via direct solution of the associated Hamilton-Jacobi-Isaacs equations. It is of interest therefore to explore whether, for certain specially structured differential games of interest, it is possible to decompose the original problem into a family of simpler differential games. In this paper we exhibit a class of single evader-multiple pursuers games for which a reduction in complexity of this nature is possible. The target set is expressed as a union of smaller, sub-target sets. The individual differential games are obtained by substituting a sub-target set in place of the original target and are simpler because of geometric features of the dynamics and constraints. We give conditions under which the value function of the original problem can be characterized as the lower envelope of the value functions for the simpler problems and show how optimal strategies can be constructed from those for the simpler problems. The methodology is illustrated by several examples.

Kinetic description of collision avoidance in pedestrian crowds by sidestepping

In this paper we study a kinetic model for pedestrians, who are assumed to adapt their motion towards a desired direction while avoiding collisions with others by stepping aside. These minimal microscopic interaction rules lead to complex emergent macroscopic phenomena, such as velocity alignment in unidirectional flows and lane or stripe formation in bidirectional flows. We start by discussing collision avoidance mechanisms at the microscopic scale, then we study the corresponding Boltzmann-type kinetic description and its hydrodynamic mean-field approximation in the grazing collision limit. In the spatially homogeneous case we prove directional alignment under specific conditions on the sidestepping rules for both the collisional and the mean-field model. In the spatially inhomogeneous case we illustrate, by means of various numerical experiments, the rich dynamics that the proposed model is able to reproduce.

A Semi-Lagrangian Scheme for a Modified Version of the Hughes’ Model for Pedestrian Flow

In this paper, we present a semi-Lagrangian scheme for a regularized version of the Hughes’ model for pedestrian flow. Hughes originally proposed a coupled nonlinear PDE system describing the evolution of a large pedestrian group trying to exit a domain as fast as possible. The original model corresponds to a system of a conservation law for the pedestrian density and an eikonal equation to determine the weighted distance to the exit. We consider this model in the presence of small diffusion and discuss the numerical analysis of the proposed semi-Lagrangian scheme. Furthermore, we illustrate the effect of small diffusion on the exit time with various numerical experiments.

Collision avoidance in pedestrian dynamics

There is a lot of experimental evidence that collision avoidance is among the main driving principles in pedestrian dynamics: individuals actively anticipate the future to predict a possible collision time and adjust their velocity accordingly, see [1]. At the same time they show no intention to change their direction when walking close to each other in the same direction. In this paper we propose an optimal control model based upon a simple two-player pursuer evader game. We shall use Bellmans approach to study the embedded multi-player game for collision avoidance and discuss related theoretical as well as numerical aspects.

Decomposition of Differential Games with Multiple Targets

This paper provides a decomposition technique for the purpose of simplifying the solution of certain zero-sum differential games. The games considered terminate when the state reaches a target, which can be expressed as the union of a collection of target subsets considered as ‘multiple targets’; the decomposition consists in replacing the original target by each of the target subsets. The value of the original game is then obtained as the lower envelope of the values of the collection of games, resulting from the decomposition, which can be much easier to solve than the original game. Criteria are given for the validity of the decomposition. The paper includes examples, illustrating the application of the technique to pursuit/evasion games and to flow control.

A Brief Survey on Semi-Lagrangian Schemes for Image Processing

In this survey we present some semi-Lagrangian schemes for the approximation of weak solutions of first and second order differential problems related to image processing and computer vision. The general framework is given by the theory of viscosity solutions and, in some cases, of calculus of variations. The schemes proposed here have interesting stability properties for evolutive problems since they allow for large time steps, can deal with degenerate problems and are more accurate if compared to standard finite difference/element methods of the same order. Several examples on classical problems will illustrate these properties.