
Multi Stakeholders collaboration becomes imperative when a single agency such as the NNPC or the Oil Companies alone cannot adequately address a recurring menace such as oil pipeline disasters. Thus, agencies such as National Emergency Management Agency (NEMA), NOSDRA, FMHDSD, Fire Service, Oil Companies, Health and Security agencies, the Media and Academia, as well the Community must seek to promote cooperation in order to achieve successful oil pipeline disaster policy implementation. To achieve this goal efficiently and effectively, a framework for stakeholders* collaboration in the management and mitigation of oil pipeline disasters in Nigeria was developed in this study, following a logical path and adopting the use of interviews conducted among the main stakeholders and with industry experts. In a bid to achieve an un-biased opinion, questionnaires and document analysis of data obtained from secondary sources was carried out.
A framework for the mitigation of oil pipeline disasters before, during and after disaster occurrence was developed. The framework captures the relevant stakeholders as well as their roles in disaster mitigation.
Citation: Francis I. Johnson, Richard Laing, Bassam Bjeirmi, Marianthi Leon. Developing a framework for stakeholders collaboration in the management and mitigation of oil pipeline disasters in Nigeria[J]. AIMS Energy, 2022, 10(6): 1230-1260. doi: 10.3934/energy.2022058
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Multi Stakeholders collaboration becomes imperative when a single agency such as the NNPC or the Oil Companies alone cannot adequately address a recurring menace such as oil pipeline disasters. Thus, agencies such as National Emergency Management Agency (NEMA), NOSDRA, FMHDSD, Fire Service, Oil Companies, Health and Security agencies, the Media and Academia, as well the Community must seek to promote cooperation in order to achieve successful oil pipeline disaster policy implementation. To achieve this goal efficiently and effectively, a framework for stakeholders* collaboration in the management and mitigation of oil pipeline disasters in Nigeria was developed in this study, following a logical path and adopting the use of interviews conducted among the main stakeholders and with industry experts. In a bid to achieve an un-biased opinion, questionnaires and document analysis of data obtained from secondary sources was carried out.
A framework for the mitigation of oil pipeline disasters before, during and after disaster occurrence was developed. The framework captures the relevant stakeholders as well as their roles in disaster mitigation.
Individual-based models of collective behavior attracted the interest of researchers in several scientific disciplines. A particularly interesting aspect of the dynamics of multi-agent systems is the emergence of global self-organizing patterns, while individual agents typically interact only locally. This is observed in various types of systems - physical (e.g., spontaneous magnetization and crystal growth in classical physics), biological (e.g., flocking and swarming, [1,2]) or socio-economical [3,4]. The field of collective (swarm) intelligence also found many applications in engineering and robotics [5,6]. The newest developments in the mathematical approaches to the field are captured in, e.g., [7,8,9,10,11,12,13,14,15,16,17].
The Cucker-Smale model is a prototypical model of consensus seeking, or, in physical context, velocity alignment. Introduced in [18,19], it has been extensively studied in many variants, where the main point of interest is the asymptotic convergence of the (generalized) velocities towards a consensus value. In this paper we focus on a variant of the Cucker-Smale model with distributed delay. We consider N∈N autonomous agents described by their phase-space coordinates (xi(t),vi(t))∈R2d, i=1,2,⋯,N, t≥0, where xi(t)∈Rd, resp. vi(t)∈Rd, are time-dependent position, resp. velocity, vectors of the i-th agent, and d≥1 is the physical space dimension. The agents are subject to the following dynamics
˙xi=vi | (1.1) |
˙vi=λNN∑j=1∫∞0ψ(|xi(t−s)−xj(t−s)|)(vj(t−s)−vi(t−s))dP(s), | (1.2) |
for i=1,2,⋯,N, where |⋅| denotes the Euclidean distance in Rd. The parameter λ>0 is fixed and P is a probability measure on [0,∞). For simplicity we consider constant initial datum on (−∞,0] for the position and velocity trajectories,
(xi(t),vi(t))≡(x0i,v0i)for t∈(−∞,0], | (1.3) |
with (x0i,v0i)∈Rd×Rd for i=1,2,⋯,N. The function ψ:[0,∞)→(0,∞) is a positive nonincreasing differentiable function that models the communication rate between two agents i, j, in dependence of their metric distance. For notational convenience, we shall denote
ψij(t):=ψ(|xi(t)−xj(t)|). |
In our paper we shall introduce the following three assumptions on ψ=ψ(r). First, we assume
ψ(r)≤1for all r≥0, | (1.4) |
which clearly does not restrict the generality due to the freedom to choose the value of the parameter λ>0. Moreover, we assume that there exist some γ<1 and c,R>0 such that
ψ(r)≥cr−1+γfor all r≥R, | (1.5) |
and that there exists α>0 such that
ψ′(r)≥−αψ(r)for all r>0. | (1.6) |
The prototype rate considered by Cucker and Smale in [18,19] and many subsequent papers is of the form
ψ(r)=1(1+r2)β, | (1.7) |
with the exponent β≥0. The assumption (1.5) is verified for (1.7) if β<1/2, while assumption (1.6) is satisfied for all β≥0 by choosing α:=2β. Let us point out that the results of our paper are not restricted to the particular form (1.7) of the communication rate.
In real systems of interacting agents - animals, humans or robots, the agents typically react to the information perceived from their surroundings with positive processing (or reaction) delay, which might have a significant effect on their collective behavior. The system (1.1)–(1.2) represents a model for flocking or consensus dynamics where the reaction (or information processing) delay is distributed in time according to the probability distribution P. Observe that the delay τ>0 is present in both the vi and vj, as well as the xi and xj variables in the right-hand side of (1.2). In contrast, the modeling assumption that the agents receive information from their surroundings with a non-negligible delay (for instance, due to finite speed of signal propagation) would lead to delay present in the vj (and xj) variables only (and not in vi and xi, since these do not involve transmission of information). However, since in typical applications in biology or engineering the agents communicate through light signals, and their distances are small compared to the speed of light, one can assume the signal propagation to be practically instantaneous. Consequently, for this type of applications it is appropriate to assume reaction-type delay as represented by our system (1.1)–(1.2).
The main objective in the study of Cucker-Smale type models is their asymptotic behavior, in particular, the concept of conditional or unconditional flocking. In agreement with [18,19] and many subsequent papers, we say that the system exhibits flocking behavior if there is asymptotic alignment of velocities and the particle group stays uniformly bounded in time.
Definition 1. We say that the particle system (1.1)–(1.2) exhibits flocking if its solution (x(t),v(t)) satisfies
limt→∞|vi−vj|=0,supt≥0|xi−xj|<∞, |
for all i,j=1,2,⋯,N.
The term unconditional flocking refers to the case when flocking behavior takes place for all initial conditions, independently of the value of the parameters λ>0 and N∈N. The celebrated result of Cucker and Smale [18,19] states that the system (1.1)–(1.2) without delay (this corresponds to the formal choice dP(s):=δ(s)ds, with δ the Dirac delta measure) with the communication rate (1.7) exhibits unconditional flocking if and only if β<1/2. For β≥1/2 the asymptotic behavior depends on the initial configuration and the particular value of the parameters λ>0 and N∈N. In this case we speak about conditional flocking. The proof of Cucker and Smale (and its subsequent variants, see [20,21,22]) is based on a bootstrapping argument, estimating, in turn, the quadratic fluctuations of positions and velocities, and showing that the velocity fluctuations decay monotonically to zero as t→∞.
The presence of delays in (1.1)–(1.2) introduces a major analytical difficulty. In contrast to the classical Cucker-Smale system (without delay), the quadratic velocity fluctuations are, in general, not decaying in time, and oscillations may appear. In fact, oscillations are a very typical phenomenon exhibited by solutions of differential equations or systems with delay, see, e.g., [23]. In [24] we developed an analytical approach for the Cucker-Smale model with lumped delay (corresponds to the formal choice dP(s):=δ(s−τ)ds in (1.1)–(1.2), with a fixed τ>0). It is based on the following two-step procedure: first, construction of a Lyapunov functional, which provides global boundedness of the quadratic velocity fluctuations. Second, forward-backward estimates on appropriate quantities that give exponential decay of the velocity fluctuations. The main goal of this paper is to generalize the approach to the case of distributed delays with a general probability measure P. A demonstration of the approach to the scalar negative feedback equation with distributed delay, which can be seen as a special case of (1.1)–(1.2) with ψ≡1 and N=2, was recently given in [25].
Flocking in Cucker-Smale type models with fixed lumped delay and renormalized communication weights was recently studied in [26,27]. Both these papers consider the case where the delay in the velocity equation for the i-th agent is present only in the vj-terms for j≠i, i.e., transmission-type delay. This allows for using convexity arguments to conclude a-priori uniform boundedness of the velocities. Such convexity arguments are not available for our system (1.1)–(1.2). In [28] the method is extended to the mean-field limit (N→∞) of the model. In [31] the authors consider heterogeneous delays both in the xj and vj terms and they prove asymptotic flocking for small delays and the communication rate (1.7). A system with time-varying delays was studied in [15], under the a-priori assumption that the Fiedler number (smallest positive eigenvalue) of the communication matrix (ψij)Ni,j=1 is uniformly bounded away from zero. The same assumption is made in [29] for a Cucker-Smale type system with delay and multiplicative noise. The validity of this relatively strong assumption would typically be guaranteed by making the communication rates ψij a-priori bounded away from zero, which excludes the generic choice (1.7) for ψ. Our approach does not require such a-priori boundedness.
Cucker-Smale systems with distributed delays were studied in [12] and [17]. In both works, the delay is present in the expression for vj only, while vi in (1.2) is evaluated at the present time vi(t). The L∞ analysis in [17] is based on a system of dissipative differential inequalities for the position and velocity diameters, leading to a nonexplicit "threshold on the time delay''. The work [17] introduces hierarchical leadership to the distributed delay system. For the case of free will ultimate leader (i.e., it can change its velocity freely), a flocking result is given under a smallness condition on the leader's acceleration. To our best knowledge, the Cucker-Smale system of the form (1.1)–(1.2), where distributed delay is present in both the vj and vi terms on the right-hand side (1.2), has not been studied before.
This paper is organized as follows. In Section 2 we formulate our assumptions and the main flocking result. In Section 3 we provide its proof divided into three steps - uniform bound on the velocities by a Lyapunov functional, forward-backward estimates, and exponential decay of the velocity fluctuation. Finally, in Section 4 we demonstrate the applicability of our theory to particular delay distributions - exponential, uniform and linear. For the exponential distribution, the flocking conditions can be resolved analytically, leading to an explicit formula. For the other two distributions, the satisfiability of the assumptions is tested numerically.
Let us first introduce several relevant quantities. For t∈R we define the quadratic fluctuation of the velocities,
V(t):=12N∑i=1N∑j=1|vi(t)−vj(t)|2 | (2.1) |
and the quantity
D(t):=12N∑i=1N∑j=1∫∞0ψij(t−s)|vj(t−s)−vi(t−s)|2dP(s). | (2.2) |
Moreover, we introduce the moments of the probability measure P. The k-th order moment for k∈N shall be denoted Mk,
Mk:=∫∞0skdP(s), |
the exponential moment (or moment generating function) Mexp[κ] for κ∈R,
Mexp[κ]:=∫∞0eκsdP(s). |
Finally, we shall need the moment K[κ], defined as
K[κ]:=∫∞0seκs−1κdP(s). | (2.3) |
Our main result is the following:
Theorem 1. Let the communication rate ψ=ψ(r) verify the assumptions (1.4)–(1.6). If there exists κ>0 such that the conditions
2λ√K[κ]<1 | (2.4) |
and
4λ√Mexp[κ]+α√2V(0)<κ | (2.5) |
are mutually satisfied, with α>0 given by (1.6), then the solution of the system (1.1)–(1.2) subject to constant initial datum (1.1) exhibits flocking behavior in the sense of Definition 1. Moreover, the quadratic velocity fluctuation V=V(t) decays monotonically (i.e., non-oscillatory) and exponentially to zero as t→∞.
The above theorem deserves several comments. First, the conditions (2.4)–(2.5) relate the value of the parameter λ>0, the moments of the probability measure P and the fluctuation of the initial datum V(0). For fixed λ>0 and P, (2.5) represents a smallness condition on the fluctuation of the initial datum, which is a very natural requirement in the context of asymptotic flocking. On the other hand, the fact that both (2.4) and (2.5) essentially impose an upper bound on λ seems less natural, since one would intuitively expect that increasing the coupling strength should lead to stronger tendency to flocking. This is in general true for small values of λ>0, however, increasing its value beyond a certain threshold induces oscillations into the system, which become even unbounded for large values of λ. This phenomenon is clearly illustrated by considering the simple case N=2 and ψ≡1 with lumped delay τ>0. Then (1.2) reduces to the delay negative feedback equation ˙u(t)=−λu(t−τ) for u(t):=v1(t)−v2(t), subject to constant initial datum. It is well known that if λτ<e−1, solutions tend to zero monotonically as t→∞. However, if λτ becomes larger than e−1 but smaller than π/2, all nontrivial solutions oscillate (i.e., change sign infinitely many times as t→∞), but the oscillations are damped and vanish as t→∞. For λτ>π/2 the nontrivial solutions oscillate with unbounded amplitude as t→∞; see Chapter 2 of [32] and [23] for details. Consequently, it is natural that (2.4)–(2.5) impose an upper bound on λ, in relation to the moments of the probability measure P, in order to obtain nonoscillatory solutions.
As we shall demonstrate in Section 4, for particular choices of the distribution P the conditions (2.4)–(2.5) lead to systems of nonlinear inequalities in terms of the distribution parameters and the fluctuation of the initial datum. These can be sometimes resolved analytically, leading to explicit flocking conditions. This is the case for the exponential distribution, as we shall demonstrate in Section 4.1. However, even if the nonlinear inequalities turn out to be prohibitively complex to be treated analytically, they are well approachable numerically. We show this for the uniform and linear distributions in Sections 4.2 and 4.3. Let us also remark that the formal choice dP(s)=δ(s)ds, i.e., no delay, gives K[κ]=0, so that (2.4) is void, while Mexp[κ]=1, so that (2.5) is always satisfiable just by choosing a large enough κ>0. Theorem 1 then reduces to the classical unconditional flocking result [18,19] for the original Cucker-Smale model.
We admit that the assumption about the constantness of the initial datum on (−∞,0], or on the interval corresponding to the support of the measure P, can be perceived as too restrictive. In fact, the methods we present in this paper can be generalized to the case of nonconstant initial data, as we demonstrated in [24]. However, since this would significantly increase the technicality of our exposition, we elect to focus on the essence of the method and thus restrict ourselves to the constant initial datum.
We note that by the rescaling of time t↦λt, of the velocities vi↦λ−1vi and of the probability measure P, the parameter λ is eliminated from the system (1.1)–(1.2). Nonetheless, for the purpose of compatibility with previous literature, we shall carry out our analysis for the original form (1.1)–(1.2). The scaling invariance shall become evident in Section 4, where we shall formulate the flocking conditions in terms of properly rescaled parameters of the probability distribution P and in terms of V(0)/λ2.
Finally, we note that the symmetry of the particle interactions ψij=ψji implies that the total momentum is conserved along the solutions of (1.2), i.e.,
N∑i=1vi(t)=N∑i=1vi(0)for all t≥0. | (2.6) |
Consequently, if the solution converges to an asymptotic velocity consensus, then its value is determined by the mean velocity of the initial datum.
The proof of asymptotic flocking for the system (1.1)–(1.2) will be carried out in three steps: First, in Section 3.1 we shall derive a uniform bound on the quadratic velocity fluctuation V=V(t) by constructing a suitable Lyapunov functional. Then, in Section 3.2 we prove a forward-backward estimate on the quantity D=D(t) defined in (2.2), which states that D=D(t) changes at most exponentially locally in time. Finally, in Section 3.3 we prove the asymptotic decay of the quadratic velocity fluctuation and boundedness of the spatial fluctuation and so conclude the proof of Theorem 1.
We first derive an estimate on the dissipation of the quadratic velocity fluctuation in terms of the quantity D=D(t) defined in (2.2).
Lemma 1. For any δ>0 we have, along the solutions of (1.1)–(1.2),
ddtV(t)≤2(δ−1)λD(t)+2λ3δ∫∞0s∫t[t−s]+D(σ)dσdP(s), | (3.1) |
where [t−s]+:=max{0,t−s}.
Proof. We have
ddtV(t)=N∑i=1N∑j=1⟨vi−vj,˙vi−˙vj⟩=2NN∑i=1⟨vi,˙vi⟩−2N∑i=1N∑j=1⟨vi,˙vj⟩=2NN∑i=1⟨vi,˙vi⟩, |
where the last equality is due to the conservation of momentum (2.6). With (1.2) we have
ddtV(t)=2λN∑i=1N∑j=1∫∞0ψij(t−s)⟨vi(t),vj(t−s)−vi(t−s)⟩dP(s)=2λN∑i=1N∑j=1∫∞0ψij(t−s)⟨vi(t−s),vj(t−s)−vi(t−s)⟩dP(s)−2λN∑i=1N∑j=1∫∞0ψij(t−s)⟨vi(t−s)−vi(t),vj(t−s)−vi(t−s)⟩dP(s). |
For the first term of the right-hand side we apply the standard symmetrization trick (exchange of summation indices i↔j, noting the symmetry of ψij=ψji),
2λN∑i=1N∑j=1∫∞0ψij(t−s)⟨vi(t−s),vj(t−s)−vi(t−s)⟩dP(s)=−λN∑i=1N∑j=1∫∞0ψij(t−s)|vj(t−s)−vi(t−s)|2dP(s). |
Therefore, we arrive at
ddtV(t)=−2λD(t)−2λN∑i=1N∑j=1∫∞0ψij(t−s)⟨vi(t−s)−vi(t),vj(t−s)−vi(t−s)⟩dP(s). |
For the last term we use the Young inequality with δ>0 and the bound ψ≤1 by assumption (1.4),
|2λN∑i=1N∑j=1∫∞0ψij(t−s)⟨vi(t−s)−vi(t),vj(t−s)−vi(t−s)⟩dP(s)|≤λδN∑i=1N∑j=1∫∞0ψij(t−s)|vj(t−s)−vi(t−s)|2dP(s)+NλδN∑i=1∫∞0|vi(t−s)−vi(t)|2dP(s). |
Hence,
ddtV(t)≤2(δ−1)λD(t)+NλδN∑i=1∫∞0|vi(t−s)−vi(t)|2dP(s) | (3.2) |
Next, we use (1.2) to evaluate the difference vi(t)−vi(t−s),
vi(t)−vi(t−s)=∫tt−sddσvi(σ)dσ=λNN∑j=1∫t[t−s]+∫∞0ψij(σ−η)(vj(σ−η)−vi(σ−η))dP(η)dσ, | (3.3) |
where [t−s]+:=max{0,t−s} and we used the fact that the initial datum for the velocity trajectories is constant. Taking the square in (3.3) and summing over i we have
N∑i=1|vi(t)−vi(t−s)|2=λ2N2N∑i=1|N∑j=1∫t[t−s]+∫∞0ψij(σ−η)(vj(σ−η)−vi(σ−η))dP(η)dσ|2≤λ2NN∑i=1N∑j=1|∫t[t−s]+∫∞0ψij(σ−η)(vj(σ−η)−vi(σ−η))dP(η)dσ|2≤sλ2NN∑i=1N∑j=1∫t[t−s]+∫∞0ψij(σ−η)|vj(σ−η)−vi(σ−η))|2dP(η)dσ≤2sλ2N∫t[t−s]+D(σ)dσ. | (3.4) |
The first inequality in (3.4) is Cauchy-Schwartz for the sum term, i.e., |N∑i=1ai|2≤NN∑i=1|ai|2, and the second Cauchy-Schwartz inequality for the integral term, together with the bound ψ≤1 imposed by assumption (1.4). Combining (3.2) and (3.4) directly leads to (3.1).
We now define for t>0 the functional
L(t):=V(t)+2λ2√M2∫∞0s∫tt−s∫t[θ]+D(σ)dσdθdP(s), | (3.5) |
where V=V(t) is the quadratic velocity fluctuation (2.1) and D=D(t) is defined in (2.2). Note that L(0)=V(0).
Lemma 2. Let the parameter λ>0 satisfy
2λ√M2≤1. | (3.6) |
Then along the solutions of (1.1)–(1.2) the functional (3.5) satisfies
L(t)≤V(0)for all t>0. |
Proof. Taking the time derivative of the second term in L(t) yields
ddt∫∞0s∫tt−s∫t[θ]+D(σ)dσdθdP(s)=D(t)∫∞0s2dP(s)−∫∞0s∫t[t−s]+D(σ)dσdP(s)=M2D(t)−∫∞0s∫t[t−s]+D(σ)dσdP(s). |
Therefore, with the choice δ:=λ√M2 in (3.1) we eliminate the integral term and obtain,
ddtL(t)≤2λ(−1+2λ√M2)D(t). | (3.7) |
We observe that the right-hand side is nonpositive if (3.6) is satisfied, and therefore, L(t)≤L(0)=V(0) for t≥0.
Consequently, if (3.6) holds, then the velocity fluctuation V(t)≤L(t) is uniformly bounded from above by V(0) for all t≥0.
Remark 1. Having established the decay estimate (3.7), one might attempt to apply Barbalat's lemma [30] to prove the desired asymptotic consensus result, assuming merely the validity of (3.6). Indeed, with the uniform bound on velocities provided by Lemma 2 and the properties of the interaction rate ψ, one can prove that the second-order derivative d2dt2L(t) is uniformly bounded in time, which implies that ddtL(t)→0 as t→∞. This in turn gives D(t)→0 as t→∞. However, since ψ is not a priori bounded from below (and the uniform velocity bound allows for a linear in time expansion of the group in space), this does not imply that the velocity fluctuation V(t) decays asymptotically to zero.
Lemma 3. Let the communication rate ψ=ψ(r) satisfy assumption (1.6) and assume that (3.6) holds. Then along the solutions of (1.1)–(1.2), the quantity D(t) defined by (2.2) satisfies for any fixed ε>0 the inequality
|ddtD(t)|≤(2ε+α√2V(0))D(t)+2λ2ε∫∞0D(t−s)dP(s), | (3.8) |
for all t>0, with α>0 given in (1.6).
Proof. For better legibility of the proof, let us adopt the notational convention that all quantities marked with a tilde are evaluated at time point t−s, i.e., ˜vi:=vi(t−s), ˜xi:=xi(t−s), ˜ψij=ψ(|˜xi−˜xj|), etc. With this notation, we have
D(t)=12N∑i=1N∑j=1∫∞0˜ψij|˜vi−˜vj|2dP(s), |
and differentiation in time and triangle inequality gives, for t>0,
|ddtD(t)|≤12N∑i=1N∑j=1∫∞0|˜ψ′ij⟨˜xi−˜xj|˜xi−˜xj|,˜vi−˜vj⟩||˜vi−˜vj|2dP(s)+|N∑i=1N∑j=1∫∞0˜ψij⟨˜vi−˜vj,d˜vidt−d˜vjdt⟩dP(s)|, | (3.9) |
where ψ′ij=ψ′(|˜xi−˜xj|). By assumption (1.6), |ψ′(r)|≤αψ(r) for r≥0, we have for the first term of the right-hand side
12N∑i=1N∑j=1∫∞0|˜ψ′ij⟨˜xi−˜xj|˜xi−˜xj|,˜vi−˜vj⟩||˜vi−˜vj|2dP(s)≤α2N∑i=1N∑j=1∫∞0˜ψij|˜vi−˜vj||˜vi−˜vj|2dP(s)≤α√2V(0)2N∑i=1N∑j=1∫∞0˜ψij|˜vi−˜vj|2dP(s)=α√2V(0)D(t), |
where in the second inequality we used the bound
|˜vi−˜vj|≤√2V(t−s)≤√2V(0), |
provided for t−s>0 by Lemma 2. For t−s≤0 it holds trivially due to the constantness of the initial datum.
For the second term of the right-hand side of (3.9) we apply the symmetrization trick,
N∑i=1N∑j=1∫∞0˜ψij⟨˜vi−˜vj,d˜vidt−d˜vjdt⟩dP(s)=2N∑i=1N∑j=1∫∞0˜ψij⟨˜vi−˜vj,d˜vidt⟩dP(s), |
and estimate using the Cauchy-Schwartz inequality with some ε>0 and the bound ψ≤1 imposed by assumption (1.4),
2|N∑i=1N∑j=1∫∞0˜ψij⟨˜vi−˜vj,d˜vidt⟩dP(s)|≤εN∑i=1N∑j=1∫∞0˜ψij|˜vi−˜vj|2dP(s)+NεN∑i=1∫∞0|d˜vidt|2dP(s). |
The first term of the right-hand side is equal to 2εD(t), while for the second term, for t−s>0, we have with (1.2), the Jensen inequality and the bound ψ≤1,
N∑i=1|d˜vidt|2=λ2N2N∑i=1|N∑j=1∫∞0ψij(t−s−σ)(vj(t−s−σ)−vi(t−s−σ))dP(σ)|2≤λ2NN∑i=1N∑j=1∫∞0ψij(t−s−σ)|vj(t−s−σ)−vi(t−s−σ)|2dP(σ)=2λ2ND(t−s). |
For t−s<0 we have d˜vidt≡0 due to the constant initial datum.
Combining the above estimates in (3.9), we finally arrive at
|ddtD(t)|≤(2ε+α√2V(0))D(t)+2λ2ε∫∞0D(t−s)dP(s), |
which is (3.8).
The following lemma constitutes the core of the forward-backward estimate method and was proved in [24, Lemma 3.5]. We present it here for the sake of the reader.
Lemma 4. Let y∈C(R) be a nonnegative function, continuously differentiable on (0,∞) and constant on (−∞,0]. Let the differential inequality
|˙y(t)|≤C1y(t)+C2∫∞0y(t−s)dP(s)for all t>0 | (3.10) |
be satisfied with some constants C1,C2>0.
If there exists some κ>0 such that
κ>max{|˙y(0+)|y(0),C1+C2Mexp[κ]}, | (3.11) |
then the following forward-backward estimate holds for all t>0 and s>0
e−κsy(t)<y(t−s)<eκsy(t). | (3.12) |
Proof Due to the assumed continuity of y(t) and ˙y(t) on (0,∞), (3.11) implies that there exists T>0 such that
−κ<˙y(t)y(t)<κfor all t<T. | (3.13) |
We claim that (3.13) holds for all t∈R, i.e., T=∞. For contradiction, assume that T<∞, then again by continuity we have
|˙y(T)|=κy(T). | (3.14) |
Integrating (3.13) on the time interval (T−s,T) with s>0 yields
e−κsy(T)<y(T−s)<eκsy(T). |
Using this with (3.10) gives
|˙y(T)|≤C1y(T)+C2∫∞0y(T−s)dP(s)<(C1+C2∫∞0eκsdP(s))y(T)=(C1+C2Mexp[κ])y(T). |
Assumption (3.11) gives then
|˙y(T)|<κy(T), |
which is a contradiction to (3.14). Consequently, (3.13) holds with T:=∞, and an integration on the interval (t−s,t) implies (3.12).
We now apply the result of Lemma 4 to derive a backward-forward estimate on the quantity D=D(t) defined in (2.2).
Lemma 5. Let (3.6) be verified and let κ>0 be such that (2.5) holds, i.e.,
4λ√Mexp[κ]+α√2V(0)<κ. |
Then, along the solution of the system (1.1)–(1.2), we have for all t>0 and s>0,
e−κsD(t)<D(t−s)<eκsD(t). | (3.15) |
Proof We shall combine Lemma 3 with Lemma 4 for y:=D, where we use formula (3.10) with
C1:=2ε+α√2V(0),C2:=2λ2ε. |
Clearly, we want to choose ε>0 to minimize the expression C1+C2Mexp[κ] in (3.11), which leads to ε:=λ√Mexp and
C1+C2Mexp[κ]=4λ√Mexp[κ]+α√2V(0). |
Therefore, condition (3.11) reads
κ>max{|˙D(0+)|D(0),4λ√Mexp[κ]+α√2V(0)}. | (3.16) |
To estimate the expression |˙D(0+)|D(0), we apply Lemma 3 again, this time with t:=0 and the optimal choice ε:=λ. Using the constantness of the initial datum, we have D(s)≡D(0) for all s<0, and (3.8) gives then
|˙D(0+)|≤(4λ+α√2V(0))D(0). |
Since, by definition, Mexp[κ]≥1 for κ>0, condition (3.16) reduces to (2.5), and we conclude.
In order to bound D=D(t) from below by the quadratic velocity fluctuation V=V(t), we introduce the minimum interparticle interaction φ=φ(t),
φ(t):=mini,j=1,⋯,Nψ(|xi(t)−xj(t)|), | (3.17) |
and the position diameter
dX(t):=maxi,j=1,⋯,N|xi(t)−xj(t)|. | (3.18) |
We then have the following estimate:
Lemma 6. Let the parameter λ>0 satisfy
2λ√M2≤1. |
Then along the solutions of (1.1)–(1.2) we have
φ(t)≥ψ(dX(0)+√2V(0)t)for all t>0. | (3.19) |
Proof. Since, by assumption, ψ=ψ(r) is a nonincreasing function, we have
φ(t)=mini,j=1,⋯,Nψ(|xi(t)−xj(t)|)=ψ(dX(t)), | (3.20) |
with dX=dX(t) defined in (3.18). Moreover, we have for all i,j=1,⋯,N,
ddt|xi−xj|2≤2|xi−xj||vi−vj|, |
and Lemma 2 gives
|vi(t)−vj(t)|2≤2V(t)≤2V(0)for all t>0. |
Consequently,
ddt|xi−xj|2≤2√2V(0)|xi−xj|, |
and integrating in time and taking the maximum over all i,j=1,⋯,N yields
dX(t)≤dX(0)+√2V(0)t, |
which combined with (3.20) directly implies (3.19).
We are now in position to provide a proof of Theorem 1.
Proof. Let us recall the estimate (3.1) of Lemma 1,
ddtV(t)≤2(δ−1)λD(t)+2λ3δ∫∞0s∫t[t−s]+D(σ)dσdP(s). |
Moreover, note that for any κ>0,
M2=∫∞0s2dP(s)<∫∞0seκs−1κdP(s)=K[κ]. |
{{Therefore}}, if assumption (2.4) of Theorem 1 is satisfied, i.e., if there exists κ>0 such that 2λ√K[κ]<1, then condition (3.6) holds and we may apply the forward-backward estimate (3.15) of Lemma 5 to the integral term
∫t[t−s]+D(σ)dσ≤∫s0D(t−σ)dσ<D(t)∫s0eκσdσ=D(t)eκs−1κ. |
Consequently, we have
ddtV(t)≤2λ[δ−1+λ2δ∫∞0seκs−1κdP(s)]D(t)=2λ[δ−1+λ2δK[κ]]D(t). |
Optimizing in δ>0 gives δ:=λ√K[κ], so that
ddtV(t)≤2λ[2λ√K[κ]−1]D(t). | (3.21) |
By the definition (3.17) of the minimal interaction φ=φ(t) we have the estimate
D(t)=12N∑i=1N∑j=1∫∞0ψij(t−s)|vj(t−s)−vi(t−s)|2dP(s)≥12N∑i=1N∑j=1∫∞0φ(t−s)|vj(t−s)−vi(t−s)|2dP(s)≥12ψ(dX(0)+√2V(0)t)N∑i=1N∑j=1∫∞0|vj(t−s)−vi(t−s)|2dP(s)=ψ(dX(0)+√2V(0)t)∫∞0V(t−s)dP(s), |
where for the last inequality we used (3.19) and the monotonicity of ψ,
φ(t−s)≥ψ(dX(0)+√2V(0)(t−s))≥ψ(dX(0)+√2V(0)t). |
Now, if assumption (2.4) is verified, (3.21) implies that V=V(t) is nonincreasing. Thus we have V(t−s)≥V(t) for all s>0, and, consequently,
D(t)≥ψ(dX(0)+√2V(0)t)V(t). | (3.22) |
Inserting into (3.21) yields
ddtV(t)≤2λ[2λ√K[κ]−1]ψ(dX(0)+√2V(0)t)V(t). |
Denoting ω:=−2λ[2λ√K[κ]−1]>0 and integrating in time, we arrive at
V(t)≤V(0)exp(−ω∫t0ψ(dX(0)+√2V(0)s)ds). | (3.23) |
Consequently, if ∫∞ψ(s)ds=∞, we have the asymptotic convergence of the velocity fluctuation to zero, limt→∞V(t)=0.
By assumption 1.5, namely that ψ(r)≥Cr−1+γ for all r>R, we have, asymptotically for large t>0,
∫tψ(dX(τ)+√2V(0)s)ds≳tγ. |
Therefore, from (3.23),
V(t)≲exp(−ωtγ). |
A slight modification of the proof of Lemma 6 gives
dX(t)≤dX(0)+∫t0√V(s)ds≲dX(0)+∫t0exp(−ωsγ/2)dsfor t≥0. |
The integral on the right-hand side is uniformly bounded, implying the uniform boundedness of the position diameter dX(t)≤ˉdX<+∞ for all t>0, with some ˉdX>0. This in turn implies φ(t)≥ψ(dX(t))≥ψ(ˉdX), so that (3.21) is replaced by the sharper estimate
D(t)≥ψ(ˉdX)V(t). |
Thus we finally have, for all t>0,
ddtV(t)≤−ωψ(ˉdX)V(t), |
and conclude the exponential decay of the velocity fluctuations.
In this section we demonstrate how the flocking conditions (2.4)–(2.5) of Theorem 1 are resolved for particular delay distributions - exponential, uniform on a compact interval and linear. The conditions (2.4)–(2.5) lead to systems of nonlinear inequalities in terms of the distribution parameters. For the exponential distribution they can be resolved analytically, leading to an explicit flocking condition. For the uniform and linear distributions they can be recast as nonlinear minimization problems and easily resolved numerically, using standard matlab procedures.
We first consider the exponential distribution dP(s)=μ−1e−s/μds with mean μ>0. We have for κ<μ−1,
Mexp[κ]=11−κμ,K[κ]=2−κμ(1−κμ)2μ2. |
Therefore, conditions (2.4) and (2.5) are satisfied if there exists κ>0 such that
2λμ1−κμ√2−κμ≤1,4λ√11−κμ+α√2V(0)<κ. |
Due to scaling properties, it is more convenient to investigate the flocking conditions in terms of the product λμ and rescale V(0) by λ2. In this form the flocking conditions read
2λμ1−κμ√2−κμ≤1,4λμ√11−κμ+αλμ√2V(0)λ2<κμ. | (4.1) |
The first condition in (4.1) is easily resolved for κμ,
κμ≤1−2(λμ)2−2λμ√(λμ)2+1, | (4.2) |
and gives the necessary condition λμ<(2√2)−1. The second condition in (4.1) we reformulate as
√2V(0)λ2<1αλμ(κμ−4λμ√11−κμ). |
Maximization of the right-hand side in κμ>0 leads to κμ=1−(2λμ)23 and
√2V(0)λ2<1αλμ(1−3(2λμ)23), | (4.3) |
which gives the necessary condition λμ<12(13)32 for positivity of the right-hand side. It is easily checked that for this range of λμ, the choice κμ:=1−(2λμ)23 verifies (4.2). Therefore, we conclude that the flocking condition imposed by Theorem 1 is equivalent to the explicit formula
λμ<12(13)32,V(0)λ2<12[1αλμ(1−3(2λμ)23)]2. |
Our second example is the uniform distribution on the interval [A,B] with 0≤A<B, i.e., dP(s)=1B−Aχ[A,B](s)ds. The relevant moments are
Mexp[κ]=eκB−eκA(B−A)κ,K[κ]=1(B−A)κ2(BeκB−AeκA−eκB−eκAκ). |
Due to the scaling relations, it is convenient to express the flocking conditions in terms of a:=λA, b:=λB and ˉκ:=κλ−1. Condition (2.4) reads then
4(b−a)ˉκ2(beˉκb−aeˉκa−eˉκb−eˉκaˉκ)≤1, | (4.4) |
and condition (2.5) reads
4√eˉκb−eˉκa(b−a)ˉκ+α√2V(0)λ2<ˉκ. | (4.5) |
Deciding satisfiability (in terms of ˉκ>0) of the above conditions seems to be prohibitively complex for the analytical approach. However, the problem is well approachable numerically. For each pair (a,b) the conditions (4.4)–(4.5) can be recast as a minimization problem in ˉκ, and deciding satisfiability accounts to checking if the minimum is negative. The minimization problem can be efficiently solved using the matlab procedure fminbnd if we provide lower and upper bounds on ˉκ. These can be obtained analytically. Indeed, carrying our Taylor expansion of the exponentials in (4.4) we see that
4(b−a)ˉκ2(beˉκb−aeˉκa−eˉκb−eˉκaˉκ)≥2(a+b)ˉκ+43(a2+ab+b2)+ˉκ2(a3+a2b+ab2+b3). |
Combining this estimate with (4.4) gives a necessary condition for its satisfiability in terms of explicit (in a and b) lower and upper bounds on ˉκ, which are roots of the corresponding quadratic polynomial. We do not print the rather lengthy algebraic expressions here; let us just mention that an immediate rough lower bound is ˉκ≥2(a+b).
We carried out two numerical studies. First, we fixed the values of α:=1 and V(0)λ2:=1 and plotted the critical value of the interval length (b−a) in dependence of the value of a>0, see Figure 1. We see that the flocking conditions (4.4)–(4.5) are satisfiable for a at most approx. 0.16, while for a approaching zero, the interval length can go up to approx. 0.26. In the second study, we fixed a:=0 and plotted critical value of the initial fluctuation V(0)/λ2 in dependence on the interval length b>0, Figure 2.
Our third example is the linear distribution on the interval [0,A] with A>0, i.e., dP(s)=2A2[A−s]+ds, where [A−s]+=max{0,A−s}. We have
Mexp[κ]=2κA(eAκ−1Aκ−1),K[κ]=1κ[2(eAκ+1)Aκ2+4(1−eAκ)A2κ3−A3]. |
Due to the scaling relations, it is again convenient to express the flocking conditions in terms of a:=λA, ˉκ:=κλ−1 and V(0)/λ2. Conditions (2.4) and (2.5) take then the form
4ˉκ[2(eaˉκ+1)aˉκ2+4(1−eaˉκ)a2ˉκ3−a3]<1,4√2ˉκa(eaˉκ−1aˉκ−1)+α√2V(0)λ2<ˉκ. | (4.6) |
A necessary condition for satisfiability of (2.4) is a<√3/2. We are interested in the dependence of the critical value of the rescaled initial fluctuation V(0)/λ2 on the parameter value a. We approach the above satisfiability problem numerically, in two steps. First, we observe that for any fixed a∈(0,√3/2), the function
fa(ˉκ):=4ˉκ[2(eaˉκ+1)aˉκ2+4(1−eaˉκ)a2ˉκ3−a3] |
is an increasing function of κ>0; this is easily seen carrying out the Taylor expansion of the exponentials. Moreover, limˉκ→0+fa(ˉκ)=2a2/3<1. Consequently, there exists ˉˉκa>0 such that the first condition of (4.6) is equivalent to ˉκ∈(0,ˉˉκa). The value of ˉˉκa is conveniently calculable using the matlab procedure fminsearch, profiting from the monotonicity of the function fa. In the second step, we numerically solve the maximization problem
maxˉκ∈(0,ˉˉκa)(κ−4√2ˉκa(eaˉκ−1aˉκ−1)), |
employing the matlab procedure fminbnd. This gives the critical value of V(0)/λ2 for validity of the second condition in (4.6). The outcome of this procedure for α:=1 is plotted in Figure (3).
In this paper we derived sufficient conditions for asymptotic flocking in a Cucker-Smale-type system with distributed reaction delays. The conditions are formulated in terms of moments of the delay distribution and the proof of flocking relies on novel backward-forward and stability estimates on the quadratic velocity fluctuations. A significant feature of our approach is that it guarantees exponential decay of velocity fluctuations, i.e., non-oscillatory flocking regime. Moreover, the sufficient conditions are amenable to either analytic or numerical resolution, as we demonstrated for particular delay distributions (exponential, uniform and linear). An interesting and important question is how far our sufficient conditions are from being optimal. We leave this topic, best approached by systematic numerical simulations, for a future work.
JH acknowledges the support of the KAUST baseline funds. IM was funded by the project ARCHERS - Stavros Niarchos Foundation - IACM.
The authors declare no conflicts of interest in this paper.
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