17 research outputs found
Local well-posedness of the generalized Cucker-Smale model
In this paper, we study the local well-posedness of two types of generalized
Cucker-Smale (in short C-S) flocking models. We consider two different
communication weights, singular and regular ones, with nonlinear coupling
velocities for . For the singular
communication weight, we choose with and in dimension . For the regular case, we
select belonging to (L_{loc}^\infty \cap
\mbox{Lip}_{loc})(\mathbb{R}^d) and . We also
remark the various dynamics of C-S particle system for these communication
weights when
Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays
We consider the celebrated Cucker-Smale model in finite dimension, modelling
interacting collective dynamics and their possible evolution to consensus. The
objective of this paper is to study the effect of time delays in the general
model. By a Lyapunov functional approach, we provide convergence results to
consensus for symmetric as well as nonsymmetric communication weights under
some structural conditions
Mean-Field Sparse Optimal Control
We introduce the rigorous limit process connecting finite dimensional sparse
optimal control problems with ODE constraints, modeling parsimonious
interventions on the dynamics of a moving population divided into leaders and
followers, to an infinite dimensional optimal control problem with a constraint
given by a system of ODE for the leaders coupled with a PDE of Vlasov-type,
governing the dynamics of the probability distribution of the followers. In the
classical mean-field theory one studies the behavior of a large number of small
individuals freely interacting with each other, by simplifying the effect of
all the other individuals on any given individual by a single averaged effect.
In this paper we address instead the situation where the leaders are actually
influenced also by an external policy maker, and we propagate its effect for
the number of followers going to infinity. The technical derivation of the
sparse mean-field optimal control is realized by the simultaneous development
of the mean-field limit of the equations governing the followers dynamics
together with the -limit of the finite dimensional sparse optimal
control problems.Comment: arXiv admin note: text overlap with arXiv:1306.591
Stochastic Mean-Field Limit: Non-Lipschitz Forces \& Swarming
We consider general stochastic systems of interacting particles with noise
which are relevant as models for the collective behavior of animals, and
rigorously prove that in the mean-field limit the system is close to the
solution of a kinetic PDE. Our aim is to include models widely studied in the
literature such as the Cucker-Smale model, adding noise to the behavior of
individuals. The difficulty, as compared to the classical case of globally
Lipschitz potentials, is that in several models the interaction potential
between particles is only locally Lipschitz, the local Lipschitz constant
growing to infinity with the size of the region considered. With this in mind,
we present an extension of the classical theory for globally Lipschitz
interactions, which works for only locally Lipschitz ones
Control to flocking of the kinetic Cucker-Smale model
The well-known Cucker-Smale model is a macroscopic system reflecting
flocking, i.e. the alignment of velocities in a group of autonomous agents
having mutual interactions. In the present paper, we consider the mean-field
limit of that model, called the kinetic Cucker-Smale model, which is a
transport partial differential equation involving nonlocal terms. It is known
that flocking is reached asymptotically whenever the initial conditions of the
group of agents are in a favorable configuration. For other initial
configurations, it is natural to investigate whether flocking can be enforced
by means of an appropriate external force, applied to an adequate time-varying
subdomain.
In this paper we prove that we can drive to flocking any group of agents
governed by the kinetic Cucker-Smale model, by means of a sparse centralized
control strategy, and this, for any initial configuration of the crowd. Here,
"sparse control" means that the action at each time is limited over an
arbitrary proportion of the crowd, or, as a variant, of the space of
configurations; "centralized" means that the strategy is computed by an
external agent knowing the configuration of all agents. We stress that we do
not only design a control function (in a sampled feedback form), but also a
time-varying control domain on which the action is applied. The sparsity
constraint reflects the fact that one cannot act on the whole crowd at every
instant of time.
Our approach is based on geometric considerations on the velocity field of
the kinetic Cucker-Smale PDE, and in particular on the analysis of the particle
flow generated by this vector field. The control domain and the control
functions are designed to satisfy appropriate constraints, and such that, for
any initial configuration, the velocity part of the support of the measure
solution asymptotically shrinks to a singleton, which means flocking