Consensus for the Hegselmann-Krause model with time variable time delays

In this paper, we analyze a Hegselmann-Krause opinion formation model with time-variable time delay and prove that, if the influence function is always positive, then there is exponential convergence to consensus without requiring any smallness assumptions on the time delay function. The analysis is then extended to a model with distributed time delay.


Introduction
Multiagent systems attracted, in recent years, the attention of many researchers in several scientific disciplines, such as biology [5,13], economics [24], robotics [4,21], control theory [1,3,33,28,27], social sciences [2,31,8].In particular, we mention the celebrated Hegselmann-Krause opinion formation model [20] (see [6,7] for the related PDE model) and its secondorder version, i.e. the Cucker-Smale model [13] introduced to describe flocking phenomena.A typical feature is the emergence of a collective behavior, namely, under quite general assumptions, solutions to such systems converge to consensus or, in the case of the CS-model, to flocking.It is also natural to include in such models time delays, taking into account the times necessary for each agent to receive information from other agents or reaction times.
In this paper, we deal with an opinion formation model with time-variable time delay.Multiagent systems with time delays have already been studied by some authors.Flocking results for the CS-model with delay have been proved [22,9,10,12,30,19] in different settings, under a smallness assumption on the time delay size.We mention also [14] for the analysis of a thermomechanical CS-model with delay.
Concerning the Hegselmann-Krause model for opinion formation, convergence to consensus results have been proved in presence of small time delays in [11,25,17].More recently, in [18], a consensus result is proved, in the case of a constant time delay, without requiring any upper bound on the time delay.We mention also [23] for a consensus result without any smallness assumptions on the time delay size but in the particular case of constant interaction coefficients.Finally, a flocking result has been recently obtained by [32] for a CS-model with constant time delay without any restrictions on the time delay size, applying a step by step procedure.We mention also [29] for a flocking result without smallness assumption on the time delay related to a CS-model with leadership.Here, we extend the argument of [32] to the Hegselmann-Krause opinion formation model in the case of a time variable time delay.We then improve previous convergence to consensus results by removing the smallness assumption on the time variable time delay.We are also able to consider a more general influence function, without monotonicity assumptions.
Consider a finite set of N ∈ N particles, with N ≥ 2. Let x i (t) ∈ IR d be the opinion of the i-th particle at time t.We shall denote with |•| and •, • the usual norm and scalar product on IR d , respectively.The interactions between the elements of the system are described by the following Hegselmann-Krause type model with variable time delays with weights a ij of the form where ψ : IR d × IR d → IR is a positive function, and the time delay for some positive constant τ .The initial conditions are assumed to be continuous functions.
The influence function ψ is assumed to be continuous.Moreover, we assume that it is bounded and we denote K := ψ ∞ .
For existence results for the above model, we refer e.g. to [15,16].Here, we will concentrate on the asymptotic behavior of solutions.For each t ≥ −τ , we define the diameter d(•) as Definition 1.1.We say that a solution {x i } i=1,...,N to system (1.1) converges to consensus if We will prove the following convergence to consensus result.
for a suitable positive constant γ independent of N.
We will also consider the continuity type equation obtained as mean-field limit of the particle model when the number N of the agents tends to infinity.Indeed, since the constants appearing in the exponential consensus estimate for the discrete model are independent of the number of the agents, we can extend the consensus result to the related PDE model.
Moreover, we extend the results obtained for the Hegselmann-Krause model with a pointwise time delay to a model with distributed time delay, namely each agent is influenced by other agents' opinions in a certain time interval (cf.[12,25]).In particular, we consider the system where the time delays for some positive constant τ .
As before, the initial conditions are assumed to be continuous functions.Moreover, the influence function ψ is assumed to be continuous and bounded, and let us denote K := ψ ∞ .Also in this case, we obtain an exponential consensus estimate without any restrictions on the time delays sizes.This extends and improves the analysis in [25] where a consensus estimate has been obtained, in the case τ 1 (t) ≡ 0, ∀t ≥ 0, subject to a smallness assumption on the time delay size.Moreover, here, as for the pointwise time delay case, we do not require any monotonicity properties on the influence function ψ that is assumed only continuous and bounded.
Even in the distributed case, since the constants in the consensus estimate are independent of the number of the agents, one can extend the consensus theorem to the related PDE model.
The rest of the paper is organized as follows.In Sect. 2 we give some preliminary results based on continuity arguments and Gronwall's inequality.In Sec. 3 we prove our consensus result for the particle model with pointwise time variable time delay, while in Sect. 4 we formulate its extension to the related continuity type equation.Finally, in Sect. 5 we analyze the HK-model with distributed time delay (1.6).

Preliminaries
Let {x i } i=1,...,N be solution to (1.1) under the initial conditions (1.4).In this section we present some auxiliary lemmas.We assume that the hypotheses of Theorem 1.1 are satisfied.
The following arguments generalize and extend the ones developed in [32] in the case of a Cucker-Smale model with constant time delay.
Proof.First of all, we note that the inequalities in (2.1) are satisfied for every t For all ǫ > 0, let us define By continuity, we have that K ǫ = ∅.Thus, denoted with We claim that S ǫ = +∞.Indeed, suppose by contradiction that S ǫ < +∞.Note that by definition of S ǫ it turns out that max i=1,...,N and lim For all i = 1, . . ., N and t ∈ (T, S ǫ ), we compute Combining this last fact with (2.4), we can write Then, from Gronwall's inequality we get for all t ∈ (T, S ǫ ).We have so proved that, ∀i = 1, . . ., N, Thus, we get max i=1,...,N Letting t → S ǫ− in (2.5), from (2.3) we have that which is a contraddiction.Thus, S ǫ = +∞, which means that max i=1,...,N From the arbitrariness of ǫ we can conclude that max i=1,...,N which proves the second inequality in (2.1).Now, to prove the other inequality, let v ∈ IR d and define Then, for all i = 1, . . ., N and t > T , by applying the second inequality in (2.1) to the vector Thus, also the first inequality in (2.1) is fullfilled.
We now introduce some notation.
Definition 2.1.We define and in general, ∀n ∈ N, Note that inequality (1.5) can be written as Let us denote with N 0 := N ∪ {0}.
Lemma 2.2.For each n ∈ N 0 and i, j = 1, . . ., N , we get It turns out that v is a unit vector and, by using (2.1) with T = nτ and the Cauchy-Schwarz inequality, we can write which proves (2.6).
Remark 2.3.Let us note that from (2.6), in particular, it follows that With an analogous argument, one can find a bound on |x i (t)|, uniform with respect to t and i = 1, . . ., N. Indeed, we have the following lemma.
Lemma 2.4.For every i = 1, . . ., N, we have that where which is a unit vector for which we can write Then, by applying (2.1) for T = 0 and by using the Cauchy-Schwarz inequality we get which proves (2.9).
Lemma 2.6.For all i, j = 1, . . ., N , unit vector v ∈ IR d and n ∈ N 0 we have that for all t ≥ t 0 ≥ nτ .Moreover, for all n ∈ N 0 , we get (2.12) Proof.Fix n ∈ N 0 and v ∈ IR d such that |v| = 1.We set Then, it is easy to see that M n − m n ≤ D n .Now, for all i = 1, . . ., N and t ≥ t 0 ≥ nτ we have that Note that, being t ≥ nτ , x i (t), v ≤ M n from (2.1).Therefore, we have that M n − x i (t), v ≥ 0 and we can write Thus, from the Gronwall's inequality it comes that On the other hand, for all i = 1, . . ., N and t ≥ t 0 ≥ nτ it holds that Note that from (2.1) x i (t), v ≥ m n since t ≥ nτ .Thus, m n − x i (t), v ≤ 0 and, by recalling that ψ is bounded, we get Hence, by using the Gronwall's inequality it turns out that Therefore, for all i, j = 1, . . ., N and t ≥ t 0 ≥ nτ , by using (2.13) and (2.14) and by recalling that M n − m n ≤ D n , we finally get i.e. (2.11) holds true.Now we prove (2.12).Given n ∈ N 0 , let i, j = 1, . . ., N and s, t ∈ [nτ , nτ + τ ] be such that So we can assume |x i (s) − x j (t)| > 0. Let us define the unit vector Hence, we can write Now, by using (2.13) with t 0 = nτ , we have that Thus, since s ≤ nτ + τ and x i (nτ ), v − M n ≤ 0 from (2.1), we get Similarly, by taking into account of (2.1) and (2.14), we have that Therefore, combining (2.15) and (2.16), we can write Then, by recalling that M n − m n ≤ D n and by using the Cauchy-Schwarz inequality, we can conclude that Lemma 2.7.There exists a constant C ∈ (0, 1), independent of N ∈ N, such that for all n ≥ 2.
Proof.Trivially, if d(nτ ) = 0, then of course inequality (2.17) holds for any constant C ∈ (0, 1).So, suppose d(nτ ) > 0. Let i, j = 1, . . ., N be such that d(nτ Then, v is a unit vector for which we can write Let us define . Now, we distinguish two different situations.Case I. Assume that there exists t 0 ∈ [nτ − 2τ , nτ ] such that Then from (2.11) with nτ ≥ t 0 ≥ nτ − 2τ we have that Case II.Suppose that Then, for every t ∈ [nτ − τ , nτ ] we have that (2.19) Therefore, using (2.9), we get and Combining this last fact with (2.19) it comes that Therefore, since from (2.18) Hence, from Gronwall's inequality it comes that In particular, for t = nτ it comes that Then, by recalling that Finally, by using (2.6) and (2.8) we have that that (2.20) Now, we set Then, taking into account (2.20), we can conclude that C is the constant for which inequality (2.17) holds.
Remark 3.1.Let us note that Theorem 1.1 holds true, in particular, for weights a ij of the form where ψ : IR → IR is a positive, bounded and continuous function.In this case, we can estimate from below ψ in terms of the distance between the initial opinions, namely where D 0 is as in Definition 2.1.Then, the proof of Theorem 1.1 follows with the same arguments we have employed in the general case of weights of the type (1.2). 4 The continuum HK-model with pointwise time delay In this section, we consider the continuum model obtained as mean-field limit of the particle system when N → ∞.Let M(IR d ) be the set of probability measures on the space IR d .Then, the continuum model associated to the particle system (1.1) is given by where the velocity field F is defined as and . For the continuum model, as in [11], we assume that the delay function τ (•) is bounded from below, namely there exists a strictly positive constant τ * > 0 such that τ (t) ≥ τ * , ∀ t ≥ 0.
Moreover, we assume that the potential ψ(•, •) in (4.2) is also Lipschitz continuous, namely for any (x, y), (x ′ , y ′ ) ∈ IR 2d there exists L > 0 such that Before stating the consensus result for solutions to model (4.1), we first recall some basic tools on probability spaces and measures.Definition 4.2.Let µ, ν ∈ M(IR d ) be two probability measures on IR d .We define the 1-Wasserstein distance between µ and ν as where Π(µ, ν) is the space of all couplings for µ and ν, namely all those probability measures on IR 2d having as marginals µ and ν: Let us introduce the space P 1 of all probability measures with finite first-order moment.It is well-known that (P 1 (IR d ), d 1 (•, •)) is a complete metric space.Now, we define the position diameter for a compactly supported measure g ∈ P 1 (IR d ) as follows: d X [g] := diam(supp g).
Since the consensus result for the particle model (1.1) holds without any upper bounds on the time delay τ (•), one can improve the consensus theorem for the PDE model (4.1) obtained in [11] removing the smallness assumption on the time delay τ (t).We omit the proof since, once we have the result for the particle system (1.1), the consensus estimate for the continuum model is obtained with arguments analogous to the ones in [11] (see also [26]).Theorem 4.1.Let µ t ∈ C([0, T ]; P 1 (IR d )) be a measure-valued solution to (4.1) with compactly supported initial datum g s ∈ C([−τ , 0]; P 1 (IR d )) and let F as in (4.2).Then, there exists a constant C > 0 such that for all t ≥ 0.

Distributed time delay
Here, we study the asymptotic behavior of solutions to system (1.6).As before, one can prove the following crucial lemma.Lemma 5.1.Let {x i } i=1,...,N be a solution to system (1.6) with continuous initial conditions.Then, for each v ∈ IR d and T ≥ 0, we have that for all t ≥ T − τ and i = 1, . . ., N .
Proof.First of all, we note that the inequalities in the statement are satisfied for every t ∈ [T − τ , T ].Now, fix T ≥ 0, a vector v ∈ IR d and a positive constant ǫ.Define the constant M T and the set K ǫ as in the proof of Lemma 2.1.Then, denoted as before S ǫ := sup K ǫ , it holds that S ǫ > T .
We claim that S ǫ = +∞.Indeed, suppose by contradiction that S ǫ < +∞.Note that by definition of S ǫ it turns out that max i=1,...,N and lim For all i = 1, . . ., N and t ∈ (T, S ǫ ), we compute Notice that, being t ∈ (T, S ǫ ), then t − τ 2 (t), t − τ 1 (t) ∈ (T − τ , S ǫ ) and (5.4) Moreover, (5.2) implies that Combining this last fact with (5.4) and by recalling of (1.9), we can write for all t ∈ (T, S ǫ ).Then, the Gronwall's Lemma allows us to conclude the proof of the second inequality arguing analogously to the proof of Lemma 2.1.Also the proof of the first inequality is obtained similarly with respect to the pointwise time delay case.We omit details.
As before, one can define the quantities D n , n ∈ N 0 , and prove the analogous, for solutions to the model with distributed time delay (1.6), of the lemmas in Section 2.Then, the following exponential convergence to consensus holds.The related PDE model is now: .For the continuum model, as in [25], we assume that the delay functions are bounded from below, namely there exists a strictly positive constant τ * > 0 such that τ 2 (t) > τ 1 (t) ≥ τ * , ∀ t ≥ 0.
Moreover, as before, we assume that the potential ψ(•, •) in (5.6) is also Lipschitz continuous with respect to the two arguments.Since the consensus result for the particle model (1.6) holds without any upper bounds on the time delays τ 1 (•), τ 2 (•), one can improve the consensus theorem for the PDE model (5.5) of [25].Indeed, in [25], where the author concentrates in the case τ 1 (t) ≡ 0, the consensus estimate is obtained under a smallness condition on the time delay.The proof is analogous, then we omit it.