226 research outputs found
Singular Cucker-Smale Dynamics
The existing state of the art for singular models of flocking is overviewed,
starting from microscopic model of Cucker and Smale with singular communication
weight, through its mesoscopic mean-filed limit, up to the corresponding
macroscopic regime. For the microscopic Cucker-Smale (CS) model, the
collision-avoidance phenomenon is discussed, also in the presence of bonding
forces and the decentralized control. For the kinetic mean-field model, the
existence of global-in-time measure-valued solutions, with a special emphasis
on a weak atomic uniqueness of solutions is sketched. Ultimately, for the
macroscopic singular model, the summary of the existence results for the
Euler-type alignment system is provided, including existence of strong
solutions on one-dimensional torus, and the extension of this result to higher
dimensions upon restriction on the smallness of initial data. Additionally, the
pressureless Navier-Stokes-type system corresponding to particular choice of
alignment kernel is presented, and compared - analytically and numerically - to
the porous medium equation
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
Binary interaction algorithms for the simulation of flocking and swarming dynamics
Microscopic models of flocking and swarming takes in account large numbers of
interacting individ- uals. Numerical resolution of large flocks implies huge
computational costs. Typically for interacting individuals we have a cost
of . We tackle the problem numerically by considering approximated
binary interaction dynamics described by kinetic equations and simulating such
equations by suitable stochastic methods. This approach permits to compute
approximate solutions as functions of a small scaling parameter
at a reduced complexity of O(N) operations. Several numerical results show the
efficiency of the algorithms proposed
Particle based gPC methods for mean-field models of swarming with uncertainty
In this work we focus on the construction of numerical schemes for the
approximation of stochastic mean--field equations which preserve the
nonnegativity of the solution. The method here developed makes use of a
mean-field Monte Carlo method in the physical variables combined with a
generalized Polynomial Chaos (gPC) expansion in the random space. In contrast
to a direct application of stochastic-Galerkin methods, which are highly
accurate but lead to the loss of positivity, the proposed schemes are capable
to achieve high accuracy in the random space without loosing nonnegativity of
the solution. Several applications of the schemes to mean-field models of
collective behavior are reported.Comment: Communications in Computational Physics, to appea
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