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

We study a Cucker-Smale-type flocking model with distributed time delay where individuals interact with each other through normalized communication weights. Based on a Lyapunov functional approach, we provide sufficient conditions for the velocity alignment behavior. We then show that as the number of individuals N tends to infinity, the N-particle system can be well approximated by a delayed Vlasov alignment equation. Furthermore, we also establish the global existence of measure-valued solutions for the delayed Vlasov alignment equation and its large-time asymptotic behavior.


Introduction
In the last years the study of collective behavior of multi-agent systems has attracted the interest of many researchers in different scientific fields, such as biology, physics, control theory, social sciences, economics. The celebrated Cucker-Smale model has been proposed and analyzed in [20,21] to describe situations in which different agents, e.g. animals groups, reach a consensus (flocking), namely they align and move as a flock, based on a simple rule: each individual adjusts its velocity taking into account other agents' velocities.
It is natural to introduce a time delay in the model, as a reaction time or a time to receive environmental information. The presence of a time delay makes the problem more difficult to deal with. Indeed, the time delay destroys some symmetry features of the model which are crucial in the proof of convergence to consensus. For this reason, in spite of a great amount of literature on Cucker-Smale models, only a few papers are available concerning Cucker-Smale model with time delay [22,13,14,35]. Cucker-Smale models with delay effects are also studied in [33,34] when a hierarchical structure is present, namely the agents are ordered in a specific order depending on which other agents they are leaders of or led by.
Here we consider a distributed delay term, i.e. we assume that the agent i, i = 1, . . . , N, changes its velocity depending on the information received from other agents on a time interval [t − τ (t), t]. Moreover, we assume normalized communication weights (cf [13]). Let us consider a finite number N ∈ N of autonomous individuals located in R d , d ≥ 1. Let x i (t) and v i (t) be the position and velocity of ith individual. Then, in the current work, we will deal with a Cucker-Smale model with distributed time delays. Namely, let τ : [0, +∞) → (0, +∞), be the time delay function belonging to W 1,∞ (0, +∞). Throughout this paper, we assume that the time delay function τ satisfies for some positive constant τ * . Then, denoting τ 0 := τ (0), we have It is clear that the constant time delay τ (t) ≡τ > 0 satisfies the conditions above.
Our main system is given by where φ(x k (s), x i (t)) are the normalized communication weights given by We consider the system subject to the initial datum i.e., we prescribe the initial position and velocity trajectories . For the particle system (1.3), we will first discuss the asymptotic behavior of solutions in Section 2. Motivated from [13,25,30], we derive a system of dissipative differential inequalities, see Lemma 2.2, and construct a Lyapunov functional. This together with using Halanay inequality enables us to show the asymptotic velocity alignment behavior of solutions under suitable conditions on the initial data.
We next derive a delayed Vlasov alignment equation from the particle system (1.3) by sending the number of particles N to infinity: where f t = f t (x, v) is the one-particle distribution function on the phase space R d × R d and the velocity alignment force F is given by We show the global-in-time existence and stability of measure-valued solutions to (1.6) by employing the Monge-Kantorowich-Rubinstein distance. As a consequence of the stability estimate, we discuss a mean-field limit providing a quantitative error estimate between the empirical measure associated to the particle system (1.3) and the measure-valued solution to (1.6). We then extend the estimate of large behavior of solutions for the particle system (1.3) to the one for the delayed Vlasov alignment equation (1.6). For this, we use the fact that the estimate of large-time behavior of solutions to the particle system (1.3) is independent of the number of particles. By combining this and the mean-field limit estimate, we show that the diameter of velocity-support of solutions of (1.6) converges to zero as time goes to infinity. Those results will be proved in Section 3.

2.
Asymptotic flocking behavior of the particle system (1.3) We start with presenting a notion of flocking behavior for the system (1.3), and for this we introduce the spatial and, respectively, velocity diameters as follows: (2.1) Definition 2.1. We say that the system with particle positions x i (t) and velocities v i (t), i = 1, . . . , N and t ≥ 0, exhibits asymptotic flocking if the spatial and velocity diameters satisfy where d X and d V denote, respectively, the spatial and velocity diameters defined in (2.1).
Then the solution of the system (1.3)-(1.5) is global in time and satisfies for a suitable positive constant γ independent of t. The constant γ can be choosen also independent of N.
Remark 2.1. Note that our theorem above gives a flocking result when the number of agents N is greater than two. The result for N = 2 is trivial for our normalized model.
For the proof of Theorem 2.1, we will need several auxiliary results. Inspired by [13], we first show the uniform-in-time bound estimate of the maximum speed of the system (1.3).
Then the solution is global in time and satisfies Proof. Let us fix ǫ > 0 and set By continuity, S ǫ = ∅. Let us denote T ǫ := sup S ǫ > 0. We want to show that T ǫ = +∞. Arguing by contradiction, let us suppose T ǫ finite. This gives, by continuity, This is in contradiction with (2.4). Therefore, T ǫ = +∞ . Being ǫ > 0 arbitrary, the claim is proved.
In the lemma below, motivated from [13,37] we derive the differential inequalities for d X and d V . We notice that the diameter functions d X and d V are not C 1 in general. Thus we introduce the upper Dini derivative to consider the time derivative of these functions: for a given function F = F (t), the upper Dini derivative of F at t is defined by h .
Note that the Dini derivative coincides with the usual derivative when the function is differentiable at t.
Proof. Due to the continuity of the velocity trajectories v i (t), there is an at most countable system of open, mutually disjoint intervals {I σ } σ∈N such that Then, by using the simplified notation i := i(σ), j := j(σ) (we may assume that i = j), we have for every t ∈ I σ , Note that, from the definition (1. Observe that a k ij (s, t) ≥ 0 and N k=1 a k ij (s, t) = 1. Thus, if we consider the convex hull of a finite velocity point set {v 1 (t), . . . , v N (t)} and denote it by Ω(t), then

This gives
which, used in (2.7), implies Then, Lemma 2.1 gives and due to the monotonicity property of the influence function ψ, for k = i, we deduce On the other hand, we find Then, from (2.9), we obtain Using the last estimate in (2.8), we have that, used in (2.6), concludes the proof. Lemma 2.3. Let u be a nonnegative, continuous and piecewise C 1 -function satisfying, for some constant 0 < a < 1, the differential inequality for almost all t > 0. (2.10) Then we have where H is the product logarithm function, i.e., H satisfies z = H(z) exp(H(z)) for any z ∈ R.
We are now ready to proceed with the proof of Theorem 2.1.
Proof of Theorem 2.1. For t > 0, we introduce the following Lyapunov functional for the system (1.3)-(1.5): where R v is given by (2.2) and the diameters d X (t), d V (t) are defined in (2.1). We first estimate L 1 as for almost all t ≥ 0, due to (1.1) and the first inequality in Lemma 2.2. Analogously, we also get for almost all t ≥ 0. Then, from Lemma 2.2, (2.11) and (2.12), we have, for almost all t > 0, On the other hand, since h ′ (t) = α(τ (t))τ ′ (t) ≤ 0, we have D + (h(t)L(t)) ≤ 0, namely This, together with (2.13) and (1.2), implies (2.14) Note that Combining this and (2.14), we obtain Since the left hand side of the above inequality is positive, we have We then again use (2.15) to find and t ≥ 0. Hence, by Lemma 2.2 together with the monotonicity of ψ we have for almost all t > 0, where ψ * = ψ(d * + 4R v τ 0 ). We finally apply Lemma 2.3 to complete the proof. Note that, in order to have an exponential decay rate of d V independent of N, it is sufficient to observe that β N ≥ 1/2 for N ≥ 3.

A delayed Vlasov alignment equation
In this section, we are interested in deriving the mean-field equation of the particle system (1.3). More precisely, at the formal level, we can derive the following delayed Vlasov alignment equation from (1.3) as the number of particles N goes to infinity: (3.1) with the initial data: Here the interaction term F [f s ] is given by where P 1 (R d × R d ) denotes the set of probability measures on the phase space R d × R d with bounded first-order moment and we adopt the notation f t−τ 0 :≡ g t−τ 0 for t ∈ [0, τ 0 ]. We next introduce the 1-Wasserstein distance.
Definition 3.2. Let ρ 1 , ρ 2 ∈ P 1 (R d ) be two probability measures on R d . Then 1-Waserstein distance between ρ 1 and ρ 2 is defined as where Π(ρ 1 , ρ 2 ) represents the collection of all probability measures on R d ×R d with marginals ρ 1 and ρ 2 on the first and second factors, respectively.
Theorem 3.1. Let the initial datum g t ∈ C([−τ 0 , 0]; P 1 (R d × R d )) and assume that there exists a constant R > 0 such that where B 2d (0, R) denotes the ball of radius R in R d × R d , centered at the origin. Then for any T > 0, the delayed Vlasov alignment equation (3.1) admits a unique measure-valued solution f t ∈ C([0, T ); P 1 (R d × R d )) in the sense of Definition 3.1.
Proof. The proof can be done by using a similar argument as in [13, Theorem 3.1], thus we shall give it rather concisely. Let f t ∈ C([0, T ]; P 1 (R d × R d )) be such that for some positive constant R > 0. Then we use the similar estimate as in [13,Lemma 3.1] to have that there exists a constant C > 0 such that where supp x f t and supp v f t represent x-and v-projections of suppf t , respectively. We also set We first construct the system of characteristics where we again adopt the notation f t−τ 0 :≡ g t−τ 0 for t ∈ [0, τ 0 ]. The system (3.3) is considered subject to the initial conditions Then, arguing as in the proof of Lemma 2.1, we get Using again a similar argument as in the proof of Lemma 2.1 and the comparison lemma, we obtain This completes the proof.
3.2. Stability & mean-field limit. In this subsection, we discuss the rigorous derivation of the delayed Vlasov alignment equation (3.1) from the particle system (1.3) as N → ∞. For this, we first provide the stability of measure-valued solutions of (3.1).
, i = 1, 2, be two measure-valued solutions of (3.1) on the time interval [0, T ], subject to the compactly supported initial data g i s ∈ C([−τ 0 , 0]; P 1 (R d × R d )). Then there exists a constant C = C(T ) such that Proof. Again, the proof is very similar to [13,Theorem 3.2], see also [7,9]. Indeed, we can obtain Then we have for t ∈ [0, T ).
, is a solution to the equation (3.1), we can use Theorem 3.2 to have the following mean-field limit estimate: where C is a positive constant independent of N .

3.3.
Asymptotic behavior of the delayed Vlasov alignment equation (3.1). In this part, we provide the asymptotic behavior of solutions to the equation (3.1) showing the velocity alignment under suitable assumptions on the initial data. For this, we first define the position-and velocity-diameters for a compactly supported measure g ∈ P 1 (R d × R d ), where supp x f denotes the x-projection of suppf and similarly for supp v f .
be a weak solution of (3.1) on the time interval [0, T ) with compactly supported initial data g s ∈ C([−τ 0 , 0]; P 1 (R d × R d )). Furthermore, we assume that Then the weak solution f t satisfies where C is a positive constant independent of t.
Let us point out that the flocking estimate at the particle level, see Section 2 and Remark 2.2, is independent of the number of particles, thus we can directly use the same strategy as in [11,13,25]. However, we provide the details of the proof for the completeness.
Proof of Theorem 3.3. We consider an empirical measure {g N s } N ∈N , which is a family of N -particle approximations of g s , i.e., Note that we can choose (x 0 i , v 0 i ) i=1,...,N such that the condition (2.3) is satisfied uniformly in N ∈ N due to the assumption (3.5). Let us denote by (x N i , v N i ) i=1,...,N the solution of the N -particle system (1.3)-(1.5) subject to the initial data (x 0 i , v 0 i ) i=1,...,N constructed above. Then it follows from Theorem 2.1 that there exists a positive constant C 1 > 0 such that with the diameters d V , d X defined in (2.1), where C 1 > 0 is independent of t and N . Note that the empirical measure is a measure-valued solution of the delayed Vlasov alignment equation (3.1) in the sense of Definition 3.1. On the other hand, by Theorem 3.2, for any fixed T > 0, we have the following stability estimate where the constant C 2 > 0 is independent of N . This yields that sending N → ∞ implies d V [f t ] = d V (t) on [0, T ) for any fixed T > 0. Thus we have Since the uniform-in-t boundedness of d X [f t ] just follows from the above exponential decay estimate, we conclude the desired result.