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

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

Cristina Pignotti1    Emmanuel Trélat 1
11Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università di L’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila Italy (pignotti@univaq.it).
###### Abstract

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.

## 1 Introduction

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:

 ˙xi(t)=vi(t),˙vi(t)=λNN∑j=1ψij(t)(vj(t)−vi(t)),i=1,…,N, (1.1)

where the parameter is a nonnegative coupling strength and the communication are of the form

 ψij(t)=ψ(|xi(t)−xj(t)|).

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

 X(t)=12N2N∑i,j=1|xi(t)−xj(t)|2

and

 V(t)=12N2N∑i,j=1|vi(t)−vj(t)|2. (1.2)
###### Definition 1.1.

We say that a solution of (1.1) converges to consensus (or flocking) if

 supt>0X(t)<+∞\rm andlimt→+∞V(t)=0.

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 [23]). 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 [36], 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 [4]), 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:

 ˙xi(t)=vi(t),˙vi(t)=λNN∑j=1j≠iψij(t−τ(t))(vj(t−τ(t))−vi(t)),i=1,…,N, (1.3)

with initial conditions

 xi(t)=fi(t),vi(t)=gi(t),t∈[−τ(0),0],

where are given functions and are suitable communication rates. In the symmetric case, we have

 ψij(t)=ψ(|xi(t)−xj(t)|),i,j∈{1,…,N}. (1.4)

The time-delay function is assumed to be bounded: we assume that there exists such that

 0⩽τ(t)⩽¯¯¯τ,∀ t>0. (1.5)

We assume moreover that the function is almost everywhere differentiable, and that there exists such that

 |τ′(t)|⩽c<1,∀ t>0. (1.6)

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

 ψij(t)⩽1,∀t∈[−τ(0),+∞),∀i,j∈{1,…,N}. (1.7)

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 [22], where the equation for velocities (which actually also involves noise terms in that paper) is

 ˙vi(t)=λNN∑j=1ψij(t−τ)(vj(t−τ)−vi(t−τ)),

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 [34] the authors consider as equation for the velocities

 ˙vi(t)=αN∑j=1aij(t−τ)(vj(t−τ)−vi(t)), (1.8)

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 [34] 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

 ˙vi(t)=αN∑j=1aij(t−τ)vj(t−τ)−αvi(t), (1.9)

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 [14], the authors analyze a Cucker-Smale model with delay and normalized communication weights given by

 Φij(x,τ)=⎧⎪ ⎪⎨⎪ ⎪⎩ψ(|xj(t−τ)−xi(t)|)∑k≠iψ(|xk(t−τ)−xi(t)|)\rm if%  j≠i,0\rm if j=i, (1.10)

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

 ˙vi(t)=N∑j=1Φij(x,t−τ)vj(t−τ)−vi(t),

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 [22], a structural assumption is required on the matrix of communication rates. We define the Laplacian matrix by

 Lij=−λNψij,\rm for i≠j,Lii=λN∑j≠iψij,

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:

 ∃γ>0 ∣ μ(t)⩾γ,∀t>0. (1.11)

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).

In Section 3, we consider the model (1.3) with possibly nonsymmetric potentials:

 ˙xi(t)=vi(t),˙vi(t)=λN∑j≠iaij(t−τ(t))(vj(t−τ(t))−vi(t)),i=1,…,N,

where the communication rates are arbitrary. They may of course be symmetric as above, e.g.

 aij(t)=ψ(|xi(t)−xj(t)|), (1.12)

or nonsymmetric, for instance

 aij(t)=Nψ(|xi(t)−xj(t)|)∑Nk=1ψ(|xk(t)−xi(t)|), (1.13)

for a suitable bounded function To analyze such models, we perform a analysis as in [36], 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: L2 analysis

### 2.1 The main result

#### Several notations.

Following [19], we set

 Δ={(v1,v2,…,vN)∈(IRd)N ∣ v1=⋯=vN}={(v,v,…,v) ∣ v∈IRd}.

The set is the eigenspace of associated with the zero eigenvalue. Its orthogonal in is

 Δ⊥={(v1,v2,…,vN)∈(IRd)N ∣ N∑i=1vi=0}.

Given any , we denote the mean by , and we define by

 wi=vi−¯v,i=1,…,N,

so that

 v=(¯v,…,¯v)+w∈Δ+Δ⊥,

and we have Moreover,

 12N2N∑i,j=1|wi−wj|2=1N∥w∥2, (2.1)

and

 ⟨Lv,v⟩=12λNN∑i,j=1ψij|vi−vj|2. (2.2)
###### Theorem 2.1.

Under the structural assumption , setting

 τ0=γ22λ21−c2λ2+γ2, (2.3)

if , then every solution of system satisfies

 V(t)⩽Ce−rt, (2.4)

with

 r=min{γ−4λ2γλ2¯¯¯τ2(1−c)e−¯¯τ−2λ2¯¯¯τ2,1}, (2.5)
 C=V(0)+λ2¯¯¯τγN1(1−c)e−¯¯τ−2λ2¯¯¯τ2∫0−τ(0)es∫0sN∑i=1|˙vi(σ)|2dσds.
###### Remark 2.2.

The threshold (given by (2.3)) on the time-delay depends on the parameter and on the lower bound in (1.11) for the Fiedler number.

### 2.2 Proof of Theorem 2.1

We start with the following lemma.

###### Lemma 2.3.

We consider an arbitrary solution of (1.3). Setting

 Rτ(t)=1N∫tt−τ(t)N∑i=1|˙vi(s)|2ds, (2.6)

we have

 1NN∑i=1|˙vi(t)|2⩽4λ2N∥w(t)∥2+2λ2¯¯¯τRτ(t), (2.7)

for every .

###### Proof.

Using (1.3), we compute

 ˙vi(t)=λN∑j≠iψij(t−τ(t))(wj(t)−wi(t))+λN∑j≠iψij(t−τ(t))(vj(t−τ(t))−vj(t))=λN∑j≠iψij(t−τ(t))(wj(t)−wi(t))−λN∑j≠iψij(t−τ(t))∫tt−τ(t)˙vj(s)ds.

Now, using (1.7), we get that

 |˙vi(t)|⩽λN∑j≠i|wj(t)−wi(t)|+λN∑j≠i∫tt−τ(t)∣∣˙vj(s)∣∣ds.

Then,

 |˙vi(t)|2⩽2λ2N2(N∑j=1|wi(t)−wj(t)|)2+2λ2N2(N∑j=1∫tt−τ(t)∣∣˙vj(s)∣∣ds)2⩽2λ2NN∑j=1|wj(t)−wi(t)|2+2λ2NN∑j=1(∫tt−τ(t)∣∣˙vj(s)∣∣ds)2.

Using (2.1), the Cauchy-Schwarz inequality and (1.5), we infer that

 N∑i=1|˙vi(t)|2⩽2λ2NN∑i,j=1|wi(t)−wj(t)|2+2λ2τ(t)∫tt−τ(t)N∑i=1|˙vi(s)|2ds⩽4λ2∥w(t)∥2+2λ2¯¯¯τ∫tt−τ(t)N∑i=1|˙vi(s)|2ds,

which gives (2.7).

###### Remark 2.4.

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.

###### Lemma 2.5.

Given any solution of (1.3), we have

 ddt(1N∥w(t)∥2)⩽−γN∥w(t)∥2+λ2¯¯¯τγRτ(t), (2.8)

for every .

###### Proof.

Using (1.3), we compute

 ˙wi(t)=˙vi(t)−1NN∑k=1˙vk(t)=λN∑j≠iψij(t−τ(t))(vj(t−τ(t))−vi(t))−λN2N∑k=1∑j≠kψkj(t−τ(t))(vj(t−τ(t))−vk(t))=λN∑j≠iψij(t−τ(t))(vj(t)−vi(t))+λN∑j≠iψij(t−τ(t))(vj(t−τ(t))−vj(t))−λN2N∑k=1∑j≠kψkj(t−τ(t))(vj(t)−vk(t))−λN2N∑k=1∑j≠kψkj(t−τ(t))(vj(t−τ(t))−vj(t))=λN∑j≠iψij(t−τ(t))(wj(t)−wi(t))+λN∑j≠iψij(t−τ(t))(vj(t−τ(t))−vj(t))−λN2N∑k=1∑j≠kψkj(t−τ(t))(wj(t)−wk(t))−λN2N∑k=1∑j≠kψkj(t−τ(t))(vj(t−τ(t))−vj(t)).

Then,

 N∑i=1wi(t)˙wi(t)=−12λNN∑i,j=1ψij(t−τ(t))|wi−wj|2+λNN∑i=1∑j≠iψij(t−τ(t))(vj(t−τ(t))−vj(t))wi,

where we have used that and

 ∑j≠iψij(t−τ(t))(wj(t)−wi(t))wi(t) = ∑j≠iψij(t−τ(t))(wj(t)−wi(t))(wi(t)−wj(t))+∑j≠iψij(t−τ(t))(wj(t)−wi(t))wj(t) = −12∑j≠iψij(t−τ(t))|wi(t)−wj(t)|2.

Therefore, thanks to (2.2), we infer that

 ddt(12∥w(t)∥2)=−⟨L(t−τ(t))w(t),w(t)⟩+λN∑i∑j≠iψij(t−τ(t))(vj(t−τ(t))−vj(t))wi(t)=−12λNN∑i,j=1ψij(t−τ(t))|wi(t)−wj(t)|2+λN∑i∑j≠iψij(t−τ(t))(vj(t−τ(t))−vj(t))wi(t).

The second term at the right-hand side of the above equality is bounded by

 ∣∣λN∑i∑j≠iψij(t−τ(t))(vj(t−τ(t))−vj(t))wi(t)∣∣⩽λN∥w(t)∥∥U(t)∥,

where is defined by

 Ui(t)=∑j≠iψij(t−τ(t))(vj(t−τ(t))−vj(t)),i=1,…,N,

and is estimated by

 ∥U(t)∥⩽N∑i=1|Ui(t)|⩽N∑i=1∑j≠iψij(t−τ(t))∫tt−τ(t)∣∣˙vj(s)∣∣ds⩽∑ijψij(t−τ(t))∫tt−τ(t)∣∣˙vj(s)∣∣ds.

Therefore, we get

 ddt(12∥w(t)∥2)⩽−12λNN∑i,j=1ψij(t−τ(t))|wi(t)−wj(t)|2+λN∑ijψij(t−τ(t))∫tt−τ(t)∣∣˙vj(s)∣∣ds∥w(t)∥⩽−12λNN∑i,j=1ψij(t−τ(t))|wi(t)−wj(t)|2+λN∑j=1∫tt−τ(t)∣∣˙vj(s)∣∣ds∥w(t)∥⩽−12λNN∑i,j=1ψij(t−τ(t))|wi(t)−wj(t)|2+λδ2∥w∥2+λ2δ(N∑j=1∫tt−τ(t)∣∣˙vj(s)∣∣ds)2,

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

 ddt(12∥w(t)∥2)⩽−⟨L(t−τ(t))w(t),w(t)⟩+γ2∥w(t)∥2+λ22γ(N∑j=1∫tt−τ(t)∣∣˙vj(s)∣∣ds)2⩽−γ2∥w(t)∥2+λ2τ(t)2γN∑j=1∫tt−τ(t)∣∣˙vj(s)∣∣2ds, (2.9)

which, using (1.5) and the definition (2.6) of gives (2.8).

We are now in a position to prove Theorem 2.1. Let be a positive constant to be chosen later. We consider the Lyapunov functional along solutions of (1.3), defined by

 L(t)=12N∥w(t)∥2+βN∫tt−τ(t)e−(t−s)∫tsN∑i=1|˙vi(σ)|2dσds, (2.10)

Using (2.9) and Lemma 2.3, we have

 ˙L(t)⩽−γ2N∥w(t)∥2+λ2¯¯¯τ2γRτ(t)+βτ(t)NN∑i=1|˙vi(t)|2−βN(1−τ′(t))e−τ(t)∫tt−τ(t)N∑i=1∣∣˙vj(s)∣∣2ds−βN∫tt−τ(t)e−(t−s)∫tsN∑i=1|˙vi(σ)|2dσds⩽−1N(γ2−4λ2β¯¯¯τ)∥w(t)∥2−(β(1−c)e−¯¯τ−λ2¯¯¯τ2γ−2βλ2¯¯¯τ2)Rτ(t)−βN∫tt−τ(t)e−(t−s)∫tsN∑i=1|˙vi(σ)|2dσds,

where we have used (1.5)–(1.6). Convergence to consensus will then be ensured if

 γ2−4βλ2¯¯¯τ>0,β(1−c)e−¯¯τ−λ2¯¯¯τ2γ−2βλ2¯¯¯τ2⩾0. (2.11)

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:

 λ2¯¯¯τ2γ1(1−c)e−¯¯τ−2λ2¯¯¯τ2⩽β<γ8λ2¯¯¯τ.

This is possible only if

 λ2¯¯¯τ2(1−c)e−¯¯τ−2λ2¯¯¯τ2<γ24λ2,

which is equivalent to

 ¯¯¯τ2e¯¯τ<τ0,

with defined by (2.3). We conclude that, if then we can choose such that

 dLdt(t)⩽−rL(t), (2.12)

for a suitable positive constant . In particular, in order to obtain the best decay rate with our procedure, we fix obtaining (2.12) with as in (2.5).

To conclude, it suffices to write that

 1N∥w(t)∥2⩽2L(t)⩽2L(0)e−rt.

Then (2.4) follows from the latter inequality, (1.2) and (2.1) with as in the statement.

## 3 Consensus for nonsymmetric potentials: L∞ analysis

### 3.1 The main result

In this section, we consider nonsymmetric potentials, and we perform a analysis as in [36]. We consider the Cucker-Smale system

 ˙xi(t)=vi(t),˙vi(t)=λN∑j≠iaij(t−τ(t))(vj(t−τ(t))−vi(t)),i=1,…,N, (3.1)

with initial conditions, for

 xi(t)=fi(t),vi(t)=gi(t),t∈[−τ(0),0],

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

 1N∑j≠iaij<1. (3.2)

This includes for instance the case considered in previous section, that is

 aij(t)=ψ(|xi(t)−xj(t)|),

with satisfying for every but we can consider a nonsymmetric interaction, for instance like in (1.13),

 aij(t)=Nψ(|xi(t)−xj(t)|)∑Nk=1ψ(|xk(t)−xi(t)|),

for a suitable bounded function

As said before, an analogous delay model has been also investigated in [14] for constant and under a restrictive assumption on the potential interaction. Indeed, the authors there consider the problem