# Asymptotic Analysis of a Cucker–Smale System with Leadership and Distributed Delay

@article{Pignotti2019AsymptoticAO, title={Asymptotic Analysis of a Cucker–Smale System with Leadership and Distributed Delay}, author={Cristina Pignotti and Irene Reche Vallejo}, journal={Trends in Control Theory and Partial Differential Equations}, year={2019} }

We extend the analysis developed in Pignotti and Reche Vallejo (J Math Anal Appl 464:1313–1332, 2018) [34] in order to prove convergence to consensus results for a Cucker–Smale type model with hierarchical leadership and distributed delay. Flocking estimates are obtained for a general interaction potential with divergent tail. We analyze also the model when the ultimate leader can change its velocity. In this case we give a flocking result under suitable conditions on the leader’s acceleration.

## 13 Citations

Cucker-Smale flocking under rooted leadership and time-varying heterogeneous delays

- Computer Science, MathematicsAppl. Math. Lett.
- 2019

Emergence of time‐asymptotic flocking for a general Cucker–Smale‐type model with distributed time delays

- MathematicsMathematical Methods in the Applied Sciences
- 2020

In this paper, we analyze the asymptotic flocking behavior for a Cucker–Smale‐type model with a disturbed delayed coupling, where delays are information processing and reactions of individuals. By…

Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays

- MathematicsNetworks Heterog. Media
- 2019

It is shown that as the number of individuals N tends to infinity, the N-particle system can be well approximated by a delayed Vlasov alignment equation and its large-time asymptotic behavior.

Asymptotic Flocking Behavior of the General Finite-Dimensional Cucker–Smale Model with Distributed Time Delays

- Mathematics
- 2020

In this paper, we study a Cucker–Smale-type flocking model with distributed time delays, in which individuals interact with each other through general communication weights, and delays are…

Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays.

- MathematicsMathematical biosciences and engineering : MBE
- 2020

Using backward-forward and stability estimates on the quadratic velocity fluctuations, sufficient conditions are derived for asymptotic flocking of the solutions of the Cucker-Smale system and the applicability of the theory to particular delay distributions is demonstrated.

Convergence to consensus for a Hegselmann-Krause-type model with distributed time delay

- Mathematics
- 2020

In this paper we study a Hegselmann-Krause opinion formation model with distributed time delay and positive influence functions. Through a Lyapunov functional approach, we provide a consensus result…

Cucker-Smale model with finite speed of information propagation: well-posedness, flocking and mean-field limit

- Mathematics
- 2021

We study a variant of the Cucker-Smale model where information between agents propagates with a finite speed c > 0. This leads to a system of functional differential equations with state-dependent…

The delayed Cucker-Smale model with short range communication weights

- Computer ScienceKinetic & Related Models
- 2021

A simple sufficient condition of the initial data to the non-flocking behavior of the delayed Cucker-Smale model is established and a flocking result is obtained, which also depends upon theinitial data in the short range communication case.

Bifurcation analysis of Friedkin-Johnsen and Hegselmann-Krause models with a nonlinear interaction potential

- MathematicsMath. Comput. Simul.
- 2021

Flocking in the Cucker-Smale model with self-delay and nonsymmetric interaction weights

- MathematicsJournal of Mathematical Analysis and Applications
- 2022

## References

SHOWING 1-10 OF 41 REFERENCES

Flocking estimates for the Cucker–Smale model with time lag and hierarchical leadership

- MathematicsJournal of Mathematical Analysis and Applications
- 2018

Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays

- Mathematics
- 2017

The Cucker-Smale model in finite dimension is considered, modelling interacting collective dynamics and their possible evolution to consensus by a Lyapunov functional approach, and convergence results to consensus for symmetric as well as nonsymmetric communication weights under some structural conditions are provided.

Cucker-Smale model with normalized communication weights and time delay

- Mathematics
- 2016

We study a Cucker-Smale-type system with time delay in which agents interact with each other through normalized communication weights. We construct a Lyapunov functional for the system and provide…

Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system

- Mathematics
- 2009

Abstract. We study a stochastic Cucker-Smale flocking system in which particles interact with the environment through white noise. We provide the definition of flocking for the stochastic system, and…

Emergent phenomena in an ensemble of Cucker–Smale particles under joint rooted leadership

- Mathematics
- 2014

We present an emergent flocking estimate in a group of interacting Cucker–Smale particles under the joint rooted leadership via the discrete-time Cucker–Smale model. It is well known that the network…

A Cucker-Smale Model with Noise and Delay

- MathematicsSIAM J. Appl. Math.
- 2016

Sufficient conditions for flocking for the generalized Cucker--Smale model are derived by using a suitable Lyapunov functional and a new result regarding the asymptotic behavior of delayed geometric Brownian motion is obtained.

Cucker--Smale Flocking under Hierarchical Leadership

- Computer ScienceSIAM J. Appl. Math.
- 2007

The emergent behavior of Cucker–Smale flocking under hierarchical leadership is studied and the rates of convergence towards asymptotically coherent group patterns in different scenarios are established.

Cucker--Smale Flocking under Rooted Leadership with Fixed and Switching Topologies

- MathematicsSIAM J. Appl. Math.
- 2010

This paper studies the discrete Cucker–Smale flocking under rooted leadership, which means that there exists an overall leader such that any other agent is led, directly or indirectly, by the leader.

A simple proof of the Cucker-Smale flocking dynamics and mean-field limit

- Mathematics
- 2009

We present a simple proof on the formation of flocking to the Cucker-Smale system based on the explicit construction of a Lyapunov functional. Our results also provide a unified condition on the…