Cucker-Smale model with time delay

Abstract

<p style='text-indent:20px;'>We study the flocking model for continuous time introduced by Cucker and Smale adding a positive time delay <inline-formula><tex-math id="M1">\begin{document}$ \tau $\end{document}</tex-math></inline-formula>. The goal of this article is to prove that the same unconditional flocking result for the non-delayed case is valid in the delayed case. A novelty is that we do not need to impose any restriction on the size of <inline-formula><tex-math id="M2">\begin{document}$ \tau $\end{document}</tex-math></inline-formula>. Furthermore, when the unconditional flocking occurs, velocities converge exponentially fast to a common one.</p>

Asymptotic flocking for the Cucker-Smale model with time variable time delays

“…A flocking result for the Cucker-Smale model with leadership and time delay without upper bounds is obtained by [30]. Here, we extend the argument of [33] to a Cucker-Smale flocking model with varying time delay. We succed in improving previous flocking results by removing upper bounds on the time delay function.…”

“…We succed in improving previous flocking results by removing upper bounds on the time delay function. Moreover, we are able to weaken the monotonicity hypothesis on the influence function that is assumed in [33].…”

“…We also refer to [24] for a consensus result without any restrictions on the constant time delay but in the particular case of constant interaction coefficients. For the Cucker-Smale model with constant time delay, asymptotic flocking is achieved by [33] without assuming the smallness of the time delay size. A flocking result for the Cucker-Smale model with leadership and time delay without upper bounds is obtained by [30].…”

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

“…From the mathematical point of view, the system (3)-( 5) is a system of functional differential equations with state-dependent delay, which stems from the fact that the delay τ ij (t) in (5) depends on the configuration of the system in a nontrivial (even implicit) way through (3). This poses new analytical challenges: in particular, the standard well-posedness theory for ODE systems (the classical theorems of Peano and Picard-Lindelöf) does not apply to (3)- (5).…”

“…Cucker-Smale-type systems with delay and their flocking behavior have been studied in a series of recent papers. However, to our best knowledge, all the previous works [3,4,5,6,9,12,14,15,19,21,22,23,25] assume the delay to be state-independent, i.e., given a-priori either as a constant, time-dependent function or probability distribution. State-dependent delay induced by finite speed of information propagation was considered in [13] for a Hegselmann-Krause-type model of consensus formation [16].…”