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

Minimal time problem for discrete crowd models with a localized vector field

In this work, we study the minimal time to steer a given crowd to a desired configuration. The control is a vector field, representing a perturbation of the crowd velocity, localized on a fixed control set. We characterize the minimal time for a discrete crowd model, both for exact and approximate controllability. This leads to an algorithm that computes the control and the minimal time. We finally present a numerical simulation

Controllability and optimal control of the transport equation with a localized vector field

We study controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling such system with a control being a Lipschitz vector field on a fixed control set $\omega$. We prove that, for each initial and final configuration, one can steer one to another with such class of controls only if the uncontrolled dynamics allows to cross the control set $\omega$. We also prove a minimal time result for such systems. We show that the minimal time to steer one initial configuration to another is related to the condition of having enough mass in $\omega$ to feed the desired final configuration

A parabolic approach to the control of opinion spreading

We analyze the problem of controlling to consensus a nonlinear system modeling opinion spreading. We derive explicit exponential estimates on the cost of approximately controlling these systems to consensus, as a function of the number of agents N and the control time-horizon T. Our strategy makes use of known results on the controllability of spatially discretized semilinear parabolic equations. Both systems can be linked through time-rescalin

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