Decomposition of Differential Games

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; the decomposition consists of 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, where the decomposition arises from considering the interaction of individual pursuer/evader pairs.Comment: 10 pages, 2 figure

A decomposition technique for pursuit evasion games with many pursuers

Here we present a decomposition technique for a class of differential games. The technique consists in a decomposition of the target set which produces, for geometrical reasons, a decomposition in the dimensionality of the problem. Using some elements of Hamilton-Jacobi equations theory, we find a relation between the regularity of the solution and the possibility to decompose the problem. We use this technique to solve a pursuit evasion game with multiple agents

Different transport regimes in a spatially-extended recirculating background

Passive scalar transport in a spatially-extended background of roll convection is considered in the time-periodic regime. The latter arises due to the even oscillatory instability of the cell lateral boundary, here accounted for by sinusoidal oscillations of frequency $\omega$. By varying the latter parameter, the strength of anticorrelated regions of the velocity field can be controled and the conditions under which either an enhancement or a reduction of transport takes place can be created. Such two ubiquitous regimes are triggered by a small-scale(random) velocity field superimposed to the recirculating background. The crucial point is played by the dependence of Lagrangian trajectories on the statistical properties of the small-scale velocity field, e.g. its correlation time or its energy.Comment: 9 pages Latex; 5 figure

A discrete Hughes' model for pedestrian flow on graphs

In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law while the minimization principle is described by a graph eikonal equation. We show that the model is well posed and we implement some numerical examples to demonstrate the validity of the proposed model