40 research outputs found
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
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
The Hughes model for pedestrian dynamics and congestion modelling
In this paper we present a numerical study of some variations of the Hughes
model for pedestrian flow under different types of congestion effects. The
general model consists of a coupled non-linear PDE system involving an eikonal
equation and a first order conservation law, and it intends to approximate the
flow of a large pedestrian group aiming to reach a target as fast as possible,
while taking into account the congestion of the crowd.
We propose an efficient semi-Lagrangian scheme (SL) to approximate the
solution of the PDE system and we investigate the macroscopic effects of
different penalization functions modelling the congestion phenomena.Comment: 6 page
A PDE approach to centroidal tessellations of domains
We introduce a class of systems of Hamilton-Jacobi equations that
characterize critical points of functionals associated to centroidal
tessellations of domains, i.e. tessellations where generators and centroids
coincide,
such as centroidal Voronoi tessellations and centroidal power diagrams. An
appropriate version of the Lloyd algorithm, combined with a Fast Marching
method on unstructured grids for the Hamilton-Jacobi equation, allows computing
the solution of the system. We propose various numerical examples to illustrate
the features of the technique
Error estimates for the Euler discretization of an optimal control problem with first-order state constraints
International audienceWe study the error introduced in the solution of an optimal control problem with first order state constraints, for which the trajectories are approximated with a classical Euler scheme. We obtain order one approximation results in the L ∞ norm (as opposed to the order 2/3 obtained in the literature). We assume either a strong second order optimality condition, or a weaker one in the case where the state constraint is scalar, satisfies some hypotheses for junction points, and the time step is constant. Our technique is based on some homotopy path of discrete optimal control problems that we study using perturbation analysis of nonlinear programming problems