27 research outputs found
A discrete contact model for crowd motion
The aim of this paper is to develop a crowd motion model designed to handle
highly packed situations. The model we propose rests on two principles: We
first define a spontaneous velocity which corresponds to the velocity each
individual would like to have in the absence of other people; The actual
velocity is then computed as the projection of the spontaneous velocity onto
the set of admissible velocities (i.e. velocities which do not violate the
non-overlapping constraint). We describe here the underlying mathematical
framework, and we explain how recent results by J.F. Edmond and L. Thibault on
the sweeping process by uniformly prox-regular sets can be adapted to handle
this situation in terms of well-posedness. We propose a numerical scheme for
this contact dynamics model, based on a prediction-correction algorithm.
Numerical illustrations are finally presented and discussed.Comment: 22 page
Optimal Control of Sweeping Processes in Robotics and Traffic Flow Models
The paper is mostly devoted to applications of a novel optimal control theory for perturbed sweeping/Moreau processes to two practical dynamical models. The first model addresses mobile robot dynamics with obstacles, and the second one concerns control
and optimization of traffic flows. Describing these models as controlled sweeping processes with pointwise/hard control and state constraints and applying new necessary optimality conditions for such systems allow us to develop efficient procedures to solve
naturally formulated optimal control problems for the models under consideration and completely calculate optimal solutions in particular situations
Handling congestion in crowd motion modeling
We address here the issue of congestion in the modeling of crowd motion, in
the non-smooth framework: contacts between people are not anticipated and
avoided, they actually occur, and they are explicitly taken into account in the
model. We limit our approach to very basic principles in terms of behavior, to
focus on the particular problems raised by the non-smooth character of the
models. We consider that individuals tend to move according to a desired, or
spontanous, velocity. We account for congestion by assuming that the evolution
realizes at each time an instantaneous balance between individual tendencies
and global constraints (overlapping is forbidden): the actual velocity is
defined as the closest to the desired velocity among all admissible ones, in a
least square sense. We develop those principles in the microscopic and
macroscopic settings, and we present how the framework of Wasserstein distance
between measures allows to recover the sweeping process nature of the problem
on the macroscopic level, which makes it possible to obtain existence results
in spite of the non-smooth character of the evolution process. Micro and macro
approaches are compared, and we investigate the similarities together with deep
differences of those two levels of description
Sweeping process by prox-regular sets in Riemannian Hilbert manifolds
In this paper, we deal with sweeping processes on (possibly
infinite-dimensional) Riemannian Hilbert manifolds. We extend the useful
notions (proximal normal cone, prox-regularity) already defined in the setting
of a Hilbert space to the framework of such manifolds. Especially we introduce
the concept of local prox-regularity of a closed subset in accordance with the
geometrical features of the ambient manifold and we check that this regularity
implies a property of hypomonotonicity for the proximal normal cone. Moreover
we show that the metric projection onto a locally prox-regular set is
single-valued in its neighborhood. Then under some assumptions, we prove the
well-posedness of perturbed sweeping processes by locally prox-regular sets.Comment: 27 page