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.
The study of collective behavior of autonomous agents has recently attracted great interest in various scientific applicative areas, such as biology, sociology, robotics, economics (see [2, 3, 5, 7, 11, 15, 28, 29, 31, 32, 39, 40, 42]). The main motivation is to model and explain the possible emergence of self-organization or global pattern formation in a large group of agents having mutual interactions, where individual agents may interact either globally or even only at the local scale.
The well known Cucker-Smale model has been proposed and studied in [19, 20] as a paradigmatic model for flocking, namely for modelling the evolution of dynamics where autonomous agents reach a consensus based on limited environmental information. Consider agents and let be their phase-space coordinates. One can think of by standing for the position of the agent and for its velocity, but for instance in social sciences these variables may stand for other notions such as opinions. The general finite-dimensional Cucker-Smale model is the following:
where the parameter is a nonnegative coupling strength and the communication are of the form
The function is called the potential. Here and throughout, the notation stands for the Euclidean norm in . Along any solution of (1.1), we define the (position and velocity) variances
We say that a solution of (1.1) converges to consensus (or flocking) if
The potential initially considered by Cucker and Smale in [19, 20] is the function with . They proved that there is unconditional convergence to flocking whenever . If , there is convergence to flocking under appropriate assumptions on the values of the initial variances on positions and speeds (see ). Their analysis relies on a Lyapunov approach with quadratic functionals, which we will refer to in the sequel as an analysis. This approach allows to treat symmetric communication rates. An extension of the flocking result to the case of nonsymmetric communication rates has been proposed by Motsch and Tadmor , with a different approach that we will refer to in the sequel as an analysis, which we will describe further.
From the mathematical point of view, there have been a number of generalizations and of results on convergence to consensus for variants of Cucker-Smale models, involving more general potentials (friction, attraction-repulsion), cone-vision constraints, leadership (see [13, 17, 26, 35, 37, 44, 46]), clustering emergence (see [30, 36]), social networks (see ), pedestrian crowds (see [16, 33]), stochastic or noisy models (see [18, 24]), kinetic models in infinite dimension (see [1, 4, 8, 12, 21, 27, 43]), and the control of such models (see [6, 9, 10, 41, 45]).
Cucker-Smale with time-varying delays.
In the present paper, we introduce time-delays in the Cucker-Smale model and we perform an asymptotic analysis of the resulting model. Time-delays reflect the fact that, for a given individual agent, information from other agents does not always propagate at infinite speed and is received after a certain time-delay, of reflect the fact that the agent needs time to elaborate its reaction to incoming stimuli.
We assume throughout that the delay is time-varying. This models the fact that the amplitude of the delay may exhibit some seasonal effects or that it depends on the age of the agents for instance. Our model is the following:
with initial conditions
where are given functions and are suitable communication rates. In the symmetric case, we have
The time-delay function is assumed to be bounded: we assume that there exists such that
We assume moreover that the function is almost everywhere differentiable, and that there exists such that
The potential function in (1.4) is assumed to be continuous and bounded. Without loss of generality (if necessary, do a time reparametrization), we assume that
Note that, in the model (1.3) above, not only the delay is time-varying but also, more importantly, there is no delay in in the equation for velocity . This assumption in our model is realistic because one expects that every agent receives information coming from the other agents with a certain delay while its own velocity is known exactly at every time , but this makes the analysis considerably more complex, as we explain below.
State of the art.
Simpler delay Cucker-Smale models have been considered in several contributions, with a constant delay
Firstly, a time-delayed model has been introduced and studied in , where the equation for velocities (which actually also involves noise terms in that paper) is
with a constant delay . Considering instead of in the equation for is less natural because one can suppose that only the information on the velocities of the other agents is known with a delay . However, this assumption in the model makes the analysis much easier because it allows to keep one of the most important features of the standard Cucker-Smale system (1.1), namely the fact that the mean velocity remains constant, i.e., , as in the undelayed Cucker-Smale model. This fact significantly simplifies the arguments in the asymptotic analysis. In contrast, the mean velocity is not constant for our model (1.3), which makes the problem much more difficult to address.
Secondly, in  the authors consider as equation for the velocities
where and the coupling coefficients are such that , . Compared with (1.3), the sum is running over all indices, including , and thus (1.8) involves, with respect to (1.3), the additional term at the right-hand side. But on the one part, this term has no physical meaning. On the other part, the authors of  claim to study (1.3) but their claim is actually erroneous and their result (unconditional flocking for all delays) actually only applies to (1.8) (cf [34, Eq. (7)]). Note that (1.8) can be rewritten as
with a negative coefficient, independent of the time , for the undelayed velocity of the agent. This allows to obtain a strong stability result: unconditional flocking for all time delays.
Thirdly, in the recent paper , the authors analyze a Cucker-Smale model with delay and normalized communication weights given by
where the influence function is assumed to be bounded, nonincreasing, Lipschitz continuous on with Thus, in practice, due to the assumption , their model can be written as
to which the same considerations than for the model (1.9) apply. Moreover, the particular form of the communication weights allows to apply some convexity arguments in order to obtain the flocking result for sufficiently small delays. Then, the result strictly relies on the specific form of the interaction between the agents. Note also that the influence function in the definition (1.10) of has as arguments with the state of the agent at the time and the states of the other agents at time This fact does not seem to have physical meaning, but it allows to easily derive the mean-field limit of the problem at hand by obtaining a nice and tractable kinetic equation. In contrast, putting the time-delay also in the state of agent is more suitable to describe the physical model but it makes unclear (at least to us) the passage to mean-field limit (see Section 5).
Framework and structure of the present paper.
In Section 2, we consider the model (1.3) with symmetric interaction weights given by (1.4). In this symmetric case, we perform a analysis, designing appropriate quadratic Lyapunov functionals adapted to the time-delay framework. The main result, Theorem 2.1, establishes convergence to consensus for small enough time-delays.
As in , a structural assumption is required on the matrix of communication rates. We define the Laplacian matrix by
with . The matrix is symmetric, diagonally dominant with nonnegative diagonal entries, has nonnegative eigenvalues, and its smallest eigenvalue is zero. Considering the matrix along a trajectory solution of (1.3), we denote by its smallest positive eigenvalue, also called Fiedler number. The structural assumption that we make throughout is the following:
This is guaranteed for instance if the communication rates are uniformly bounded away from zero, i.e., if there exists such that for all and (but in that case of course there is unconditional convergence to consensus for the undelayed model).
where the communication rates are arbitrary. They may of course be symmetric as above, e.g.
or nonsymmetric, for instance
for a suitable bounded function To analyze such models, we perform a analysis as in , by considering, instead of Euclidean norms, the time-evolution of the diameters in position and velocity phase space. The main result, Theorem 3.1, establishes convergence to consensus under appropriate assumptions.
In Section 5, we provide a conclusion and further comments.
2 Consensus for symmetric potentials: analysis
2.1 The main result
Following , we set
The set is the eigenspace of associated with the zero eigenvalue. Its orthogonal in is
Given any , we denote the mean by , and we define by
and we have Moreover,
Under the structural assumption , setting
if , then every solution of system satisfies
2.2 Proof of Theorem 2.1
We start with the following lemma.
We consider an arbitrary solution of (1.3). Setting
for every .
The term is due to the presence of the time delay. Indeed, we have two quantities at the right-hand side of the inequality (2.7): the “classical” term (coming from the undelayed model), and the term caused by the delay effect.
Given any solution of (1.3), we have
for every .
Using (1.3), we compute
where we have used that and
Therefore, thanks to (2.2), we infer that
The second term at the right-hand side of the above equality is bounded by
where is defined by
and is estimated by
Therefore, we get
where we have used the Young inequality333This inequality states that, given any positive real numbers , and , we have . for some arbitrary . Choosing where is the constant in the structural assumption (1.11), we infer that
The second inequality of (2.11) yields a first restriction on the size of the delay, namely, that . Let us now choose the constant in the definition (2.10) of so that both conditions in (2.11) are satisfied:
This is possible only if
which is equivalent to
with defined by (2.3). We conclude that, if then we can choose such that
To conclude, it suffices to write that
3 Consensus for nonsymmetric potentials: analysis
3.1 The main result
In this section, we consider nonsymmetric potentials, and we perform a analysis as in . We consider the Cucker-Smale system
with initial conditions, for
where are given functions and quantifies the pairwise influence of agent on the alignment of agent. By rescaling if necessary (or by time reparametrization), we assume that
This includes for instance the case considered in previous section, that is
with satisfying for every but we can consider a nonsymmetric interaction, for instance like in (1.13),
for a suitable bounded function
As said before, an analogous delay model has been also investigated in  for constant and under a restrictive assumption on the potential interaction. Indeed, the authors there consider the problem