Local sensitivity analysis for the Cucker-Smale model with random inputs

We present pathwise flocking dynamics and local sensitivity analysis for the Cucker-Smale(C-S) model with random communications and initial data. For the deterministic communications, it is well known that the C-S model can model emergent local and global flocking dynamics depending on initial data and integrability of communication function. However, the communication mechanism between agents are not a priori clear and needs to be figured out from observed phenomena and data. Thus, uncertainty in communication is an intrinsic component in the flocking modeling of the C-S model. In this paper, we provide a class of admissible random uncertainties which allows us to perform the local sensitivity analysis for flocking and establish stability to the random C-S model with uncertain communication.


Introduction
Emergent phenomena in complex systems are ubiquitous in our nature, to name a few, flocking of birds, aggregation of bacteria and swarming of fish [5,8,37,45] etc. In this paper, we use the terminology "flocking" to denote some concentration phenomena in velocity where individual particles are organized into an ordered motion using the simple rules and environment information through hidden communication mechanism. Modeling of such collective dynamics has received lots of attention in control theory community due to recent applications in the unmanned aerial vehicles (e.g., drons), unmanned cars and sensor networks [34,39,40]. Several mathematical models were proposed in [12,13,38,45,46] and have been studied in relation to the emergent dynamics. Among them, our main interest in this paper lies on the particle model proposed by Cucker and Smale about a decade ago. In [12], Cucker and Smale proposed a Newton-like second-order model and provided some analytical results for the emergent dyanmics of the C-S model. More precisely, let x i ∈ R d and v i be the position and velocity of the i-th C-S particle with unit mass. Then, the dynamics of (x i , v i ) is governed by the following second-order model: d dt x i (t) = v i (t), t > 0, i = 1, · · · , N, , (1.1) where ψ = ψ(x) is a communication weight function depending on the relative distances between C-S particles. After Cucker-Smale's seminal work in [12], the C-S model (1.1) has been extensively studied in many aspects, e.g., collision-avoidance [1,10], stochastic noise effects [2,11,17], emergence of local and global flockings [7,12,18,19,42], kinetic and hydrodynamic descriptions [4,14,16,30], generalized C-S models [9,31,32,33,38]. The fcuntion ψ is an empirical function, and in real applications, the determination of ψ is based on the phenomenology or modeler's free will. Thus, the uncertainty in ψ is intrinsic. Then, a natural question is how uncertainty in communication and initial data can affect dynamic features such as flocking and stability. The stochastic noise effect in the C-S flocking was partly addressed in [2], where the communication function ψ is decomposed into a deterministic part and a white noise part. This decomposition turns the deterministic system (1.1) into a stochastic C-S model with a multiplicative noise. Several elementary stochastic estimates for the velocity process have been studied using Ito's calculus in [2,17].
In this paper, we consider a general communciation weight function ψ with uncertainty, and we do not assume any specific form for ψ unlike the ansatz in [2] so that our model can cover a bounded noises [46] as well. With this motivation in mind, we consider a random C-S model with uncertain communication ψ = ψ(x, z) and uncertain initial data depending on z, where z is a random variable defined on sample space Ω with a probability density function (pdf) π(z). The random processes (x i (t, z), v i (t, z)) is governed by the following random C-S model: ∂ t x i (t, z) = v i (t, z), t > 0, i = 1, · · · , N, ψ(x j (t, z) − x i (t, z), z)(v j (t, z) − v i (t, z)), (1.2) where the communication function ψ = ψ(x, z) =ψ(|x|, z) is radially symmetric in the first argument, We also assume that ψ(·, z) satisfies several structural properties such as the positivity, boundedness monotonicity and Lipschitz continuity in the first argument: for z ∈ Ω, The purpose of this paper is to study theoretical certain aspects of the C-S model (1.2) -(1.3) under the effect of random communications. Similar analysis has been done recently for kinetic equations with random uncertainties from initial data and/or collison kernels, see [20,21,22,25,36,24,27,28,29,35,43] for uncertainty quantification in kinetic and hyperbolic models. Recently, the works [3,6] addressed uncertainty quantification for the swarming models from a numerical point of view. Thus, our work can be viewed as a theoretical justification of their works (see [47,48] for related works on uncertainty quantification).
The main results of this paper are as follows. First, we give conditions on the random communication function such that the random dynamical system (1.2) exhibits the same asymptotic flocking dynamics as the deterministic C-S model [12] along the sample path. We also provide average dynamics of position and velocity processes. Second, we provide a local sensitivity analysis for (1.2) -(1.3). In particular, we show that the regularity in random space is propagated along the C-S flow and it is stable with respect to initial data.
The rest of this paper is organized as follows. In Section 2, we review a theoretical minimum for (1.2) -(1.3). In Section 3, we provide a pathwise flocking and stability dynamics of the random C-S model. In Section 4, we provide two local sensivity analysis such as the propagation of regularity and its stability analysis with respect to initial data. Finally Section 5 is devoted to the brief summary of our main results and future works. In Appendix A and Appendix B, we present a proof of Gronwall type lemma and chain rule for higher derivatives of composition of two functions, which are crucially used in Section 3 and Section 4.
Gallery of Notation: Let π : Ω → R + ∪ {0} and ϕ = ϕ(z) be nonnegative pdf function and scalar-valued random function defined on the sample space Ω, respectively. Then, we define the expected value as and a weighted L 2 -space: with an inner product and norm: Moreover, as long as there is no confusion, we suppress π and Ω dependence in L 2 π (Ω)-norm and H k π (Ω)-norm: . For a vector-valued function y(z) = (y 1 (z), · · · , y d (z)) ∈ R d , we set We also set Finally, we use f g to denote that there exists a positive generic constant C such that f ≤ Cg.

Preliminaries
In this section, we briefly review a theoretical minimum for the random C-S flocking model with uncertain communication. First, we recall the local sensitivity analysis of the random ODE system and review the propagation of velocity moments.
2.1. The random C-S model. Consider an ensemble consisting of N -identical C-S particles with the same mass in R d under uncertain communications registered by ψ = ψ(x, z).
Let (x i (t, z), v i (t, z)) ∈ R 2d be the position-velocity processes of the i-th particle. Then, their dynamics is governed by the Cauchy problem for the random C-S model: , ). Next, we present definition of pathwise mono-cluster flocking for the C-S ensemble.
has a asymptotic pathwise mono-cluster flocking if the ensemble P satisfies the following two conditions: for z ∈ Ω, (1) (Formation of velocity alignment): (2) (Formation of group): In the following lemma, we study the time-evolution of first and second velocity moments to be used in later sections.
(ii) We take an inner product (1.2) 2 with 2v i (t, z) and sum it over all i to obtain Remark 2.1. We set the first and second velocity moments: As a direct corollary of Lemma 2.1, we have the following assertions: This and Cauchy-Schwarz's inequality yield (2) Uniform boundedness of mean and variance: be a solution process to the system (1.2)-(1.3) with zero total momentum: Then, for any k ≥ 1, we have Proof. Note that the total momentum is conserved along the dynamics (1.
Then, we differentiate the above relation k-times with respect to z to get

2.2.
Local sensivity analysis. Sensivity analysis deals wtih the effects on the output from the input for mathematical and simulation models [41]. Below, we briefly discuss local sensitivity analysis following the presentation in [26]. Consider the Cauchy problem with the random ordinary differential equations with random initial data: where X : R + × Ω → R d is a random field, and z is system parameters taking a value in Ω ⊂ R m . The sensitivity analysis studies the effect on X with respect to the small perturbation dz of z: Then, we define sensivity matrices consisting of coefficients as follows: In the following two sections, we will study pathwise flocking estimates, Sobolev estimates and stability estimate of the above sensivity matrices for the random C-S model (1.2) -(1.3).

Pathwise flocking and stability estimates
In this section, we present two pathwise estimates, the asymptotic mono-cluster flocking estimate and uniform ℓ 2 -stability estimate for the random C-S model.

3.1.
Mono-cluster flocking. In this subsection, we present a pathwise flocking estimate in Definition 2.1 using the Lyapunov functional approach [1,18]. Although the arguments are similar to the case of deterministic C-S model (1.1), for reader's convenience, we briefly sketch the pathwise flocking estimate in the sequel. For X = ( 3) with zero initial total momentum. Then, the functionals ||X|| and ||V || satisfy the system of dissipative differential inequalities (SDDI): for a.e. t ∈ (0, ∞) and z ∈ Ω, Proof. For fixed z ∈ Ω, system (1.2) is the deterministic C-S system (1.1). Hence, we can employ the same argument in Lemma 3.1 of [1]. Thus, we omit its detailed proof.
Next, we introduce two Lyapunov type functionals L ± (t, z): Then, we have the following stability estimates in the following lemma.
3) with zero initial total momentum. Then, the functionals L ± satisfy stability estimates: for t ∈ (0, ∞) and z ∈ Ω, Proof. The proof is essentially the same as the deterministic case in Lemma 3.2 of [1]. Hence we omit its proof here.
As a direct application of Lemma 3.2, we obtain the emergence of mono-cluster flocking estimate as follows.
Theorem 3.1. (Pathwise flocking estimate) For z ∈ Ω, suppose that the initial data and ψ satisfy the following relation: where x M (z) is defined to be the unique value satisfying the following implicit relation: Proof. The proof can be split into two steps. • Step A (uniform bound of ||X(·, z)||): Note that the positivity of ψ implies that the unique determination of x M (z) via the relation (3.3). For such x M (z), we claim: Suppose not, i.e., there exists t * ∈ (0, ∞) such that On the other hand, it follows from (ii) in Lemma 3.2 that This and the relation (3.3) imply which gives a contradiction. Hence we have the uniform boundedness (3.4) for ||X(·, z)||. • Step B (exponential decay of ||V (·, z)||): We use (1.3), (3.1) 2 and (3.4) to obtain Then, Gronwall's lemma yields the desired exponential decay estimate of ||V (·, z)||.
Remark 3.1. For a given ψ, x M (z) can be found explicitly or implicitly via the relation (3.3). Thus, we can write As a direct corollary of Theorem 3.1, we have estimates for the mean of the modulus of X(t, z) and V (t, z), when ψ has a positive lower bound.
Corollary 3.1. Suppose that ψ = ψ(x, z), and the strength and the initial data satisfy Proof. Note that the relation (3.5) yields Then, the relation (3.2) 1 holds trivially for any initial data with zero total momentum. Therefore, we have a pathwise mono-cluster flocking: • (Estimate of E||V (t, z)||): We use the positive lower bound (3.5) for ψ to get We multiply π(z) to the above relation and integrate it over Ω to get • (Estimate of E||X(t, z)||): We multiply π(z) to (3.6) 1 and integrate it over Ω to obtain Remark 3.2. Note that the results of Corollary 3.1 imply In this subsection, we study the uniform ℓ 2 -stability of the random C-S model with respect to initial data along the sample path. First, we recall definition of the uniform ℓ 2 -stability as follows.
Before we present the uniform stability estimate along the sample path, we introduce several handy notation in the sequel.
3) satisfying zero total momentum and (3.2). Then, we have where the random variable ψ m is defined by the following relation: Proof. (i) We take an inner product (3.8) 1 with 2∆ i x (t, z), sum it over all i to get

This yields
This yields the first differential inequality (3.9) 1 .
(ii) Similarly, we have =: I 11 + I 12 . (3.11) • Case A (Estimate of I 11 ): We use the upper bound for x ji (t, z): and zero total momentum to obtain (3.12) • Case B (Estimate on I 12 ): For each z ∈ Ω, it follows from the Lipschitz continuity of ψ and Theorem 3.1 that we have Here ||ψ(z)|| Lip is the Lipschtz constant of ψ. This yields (3.13) In (3.11), we combine estimates (3.12) and (3.13) to obtain This and the relation (3.10) yield the desired second inequality: Lemma 3.4. [15] Suppose that two nonnegative Lipschitz functions X and V satisfy the coupled differential inequalities: where α and γ are positive constants, and f : R + ∪ {0} → R is a differentiable, nonnegative, nonincreasing function decaying to zero as its argument goes to infinity and it is integrable. Then, X and V satisfy the uniform bound and decay estimates: there exists a positive constant B ∞ (α, γ) such that where B ∞ (α, γ) is a positive constant defined by the following relation: Proof. We leave its proof in Appendix A.
As a direct application of Lemma 3.3 and Lemma 3.4, we obtain the uniform ℓ 2 -stability with respect to initial data pathwise. Proof. We set Then, the constantB ∞ becomes where || · || Lip is the Lipschitz semi-norm. Then, Lemma 3.4 yields (3.14) These yield Finally, we set to get the desired estimate.

Local sensitivity analysis
In this section, we provide a local sensitivity analysis for position and velocity processes. For the simplicity of presentation, we assume that the random space Ω is one-dimensional, i.e., Ω ⊂ R, and we also recall notation: we set (for k ≥ 1), In the following two subsections, we will derive the following local sensitivity estimates: • (Propagation of regularity in random space): H k z -regularity of initial data is propagated along the random C-S model (1.2). For any T ∈ (0, ∞), Low-order derivative estimate. In this part, we study the propagation of H 1 z -regularity in random space. Suppose that Then, it follows from Theorem 3.1 that for t ≥ 0, z ∈ Ω, we have By the same argument as in Lemma 3.1, we have the following estimates.
Then, we have Proof. (i) We take an inner product (4.1) 1 with 2∂ z x i (t, z) and sum it over all i to get This yields the desired estimate.
(ii) Similarly, we have (4.2) Below, we estimate the term I 2i separately.
As a direct corollary of Theorem 3.1, we have the following H 1 z -estimates. Corollary 4.1. Suppose that the communication weight function ψ satisfies and let {(x i (t, z), v i (t, z))} be a solution process to the system (1.2)-(1.3) with zero total momentum: Then, for t ≥ 0, z ∈ Ω, we have

4.1.2.
Higher-order derivative estimates. In this part, we present higher-order H m z -estimates for the propagation of regularity in random space. (4.11) Note that the terms K contain only lower order terms in (∂ k z v j (t, z) − ∂ k z v i (t, z)) with k ≤ m − 1. As can be seen in Theorems 3.1 and 4.1, the lower-order terms V and ∂ z V decays exponentially. Thus, by the induction and Gronwall's type estimate in Lemma 3.4, we can expect the exponential decay of higher-order derivatives ∂ k z V (t, z) with k ≥ 2. This will be made rigorous in the sequel.  sup and let {(x i (t, z), v i (t, z))} be a solution process to system (1.2). Then, we have Note that the term K 00 can be estimated using the assumption (4.12): On the other hand, the term K 1n needs some care to show that it contains the highest derivative term ∇ n x (x j (t, z) − x i (t, z)) and lower-order terms. For this, we have to expand K 1n using the formula in [23] (for reader's convenience, we stated it in Appendix B): where the sum is over all nonnegative integer solutions of the Diophantine equation: k 1 + 2k 2 + · · · + nk n = n and k = k 1 + k 2 + · · · + k n . Note that the system of Diophantine equation has a solution (k 1 , · · · , k n−1 , k n ) = (0, · · · , 0, 1), k = 1, which yields the highest derivative term: y)) + lower-order terms like (∂ k y x j (t, y) − ∂ k y x i (t, y)) with k < n. (4.14) We now combine (4.13) and (4.14) to obtain y)) + lower order terms.
We denote the lower order terms as a polynomial This yields the desired estimate.
Then, there exists a positive constantC such that where C is a positive constant appearing in Lemma 4.2.
Proof. (i) The first inequality follows from the same argument as in (i) Lemma 4.1.
(ii) We take an inner product (4.11) 2 with 2∂ m z v i (t, z) and sum it over all i to get (4.16) Next, we estimate the terms I 3i , i = 1, 2 separately.
• (Estimate of I 31 ): By the exchange transformation i ↔ j and zero total momentum, we have (4.17) • (Estimate of I 32 ): We use Lemma 4.3 to obtain to obtain and zero total momentum to obtain (4.28) • Case B (Estimate of I 42 ): First, we rewrite I 42 as follows. (4.29) Then, we use the same arguments as in (4.18) to find and to obtain This and (4.2) 1 imply We set to get the desired estimate. Other higher-order estimates can be made inductively using the differential inequalities (4.26) in Lemma 4.4. We omit its details.

Conclusion
In this paper, we presented local sensitivity analysis for the Cucker-Smale model with random communications. More precisely, we have presented two results. First, we provided conditions on the random commununications for the pathwise flocking estimates along the sample path which give rise to deterministic flocking asymptotically, and obtained uniform stability analysis with respect to initial data. Second, we performed a local sensitivity analysis for the random Cucker-Smale model. To the best of our knowledge, this is the first theoretical work on the interplay between flocking dynamics and uncertainty quantification. One can also conduct such analysis for the mean field kinetic equations for flockings [19], and other related models for collective dynamics, decision making and self-organization in complex systems coming from biology and social sciences [44]. This will be the subject of future resarch.
Appendix A. Proof of Lemma 3.4 In this appendix, we present a proof of Lemma 3.4 which is a slight generalization of Gronwall's inequality appearing in [15].
Then, we claim: (A.2) First, we will show that the uniform bound of V, and then we use this uniform bound to derive the exponential decay of V in two steps. • Step A (Uniform boundedness of V): In this step, we derive V(τ ), t > 0.
It follows from the first differential inequality that We substitute this into the second differential inequality to get dV dt ≤ −αV + γe −αt X + f ≤ −αV + γe −αt X 0 +