Rami Katz

Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Emilia Fridman
email: emilia@eng.tau.ac.il

Stability by averaging via time-varying Lyapunov functions

Keywords: stability, averaging, time-varying

We study linear continuous-time systems with fast-varying almost periodic coefficients that are piecewise-continuous in time. Recently, a constructive time-delay approach to periodic averaging of systems with a single fast time-scale was introduced and employed to averaging of systems with small time-varying delays (of the order of the small parameter). In this paper we present a novel transformation of the fast varying coefficients. This transformation is suitable for averaging over multiple time-scales, and is applicable to averaging of systems with constant delays, where the value of delay is not small (i.e. essentially larger than the small parameter). We carry out stability analysis by employing time-varying Lyapunov functions (or functionals for the delayed case). The analysis leads to LMI conditions that are always feasible for small enough parameters. Numerical examples demonstrate the efficiency of the proposed approach and its conservatism.

INTRODUCTION

Systems with almost periodic coefficients and/or excitations are widely used in modern engineering applications (see e.g. [START_REF] Cheng | On robustness analysis of linear vibrational control systems[END_REF], [START_REF] Sandberg | Periodic modelling of power systems[END_REF], [START_REF] Xie | Guaranteed cost control of periodic piecewise linear time-delay systems[END_REF] and the references therein). Such systems often include rapidly time-varying coefficients on multiple time-scales. Perturbation theory-based techniques have been useful in the analysis of systems with rapidly time-varying coefficients and have led to important qualitative results [START_REF] Bogoliubov | Asymptotic methods in the theory of non-linear oscillations[END_REF], [START_REF] Khalil | Nonlinear systems third edition[END_REF], [START_REF] Kokotovic | Singular perturbations in systems and control[END_REF], [START_REF] Teel | A unified framework for input-to-state stability in systems with two time scales[END_REF], [START_REF] Moreau | Practical stability and stabilization[END_REF] .

The method of averaging has been especially efficient in the study of stability of systems with oscillatory control inputs [START_REF] Bullo | Averaging and vibrational control of mechanical systems[END_REF], [START_REF] Krstić | Stability of extremum seeking feedback for general nonlinear dynamic systems[END_REF], [START_REF] Meerkov | Principle of vibrational control: theory and applications[END_REF]. The underlying idea behind asymptotic averaging is that stability of the original rapidly-varying system is guaranteed for small enough values of the parameter provided the averaged system is stable. However, by using such methods it is often difficult to find an explicit bound on the small parameter that preserves stability of the perturbed system. Direct Lyapunov techniques can be used to derive explicit bounds on the small parameter. For singularly perturbed systems, such bounds were derived in, e.g., [START_REF] Kokotovic | Singular perturbations in systems and control[END_REF] and [START_REF] Fridman | Effects of small delays on stability of singularly perturbed systems[END_REF], via a direct Lyapunov approach.

Recently, a constructive time-delay approach to periodic averaging of a system with a single fast time-scale was introduced in Fridman and Zhang [2020]. It was suggested to use backward integration of the system, thereby pre-senting it as a delayed neutral-type system, with the delay magnitude equal to the time-scale parameter. Stability of the delayed system then guarantees the stability of the original one. A direct Lyapunov method was used to obtain LMI-based conditions that allow to find an explicit upper bound on the small parameter which preserves the stability and ISS of the original system. This approach was also employed for averaging of systems with timevarying delays, where the delay bound which preserved the stability of the system was of the order of the time-scale parameter. These results were later extended to L 2 -gain analysis for periodic averaging and to stochastic systems in [START_REF] Zhang | L 2 -gain analysis via time-delay approach to periodic averaging with stochastic extension[END_REF]. However, the approach was confined to a single time-scale and the results for timedelay systems were limited to delays on the order of the small parameter.

In this paper we consider linear systems with rapidlyvarying almost periodic coefficients that are piecewisecontinuous in time. We begin by considering an ISS-like property of a system with two fast time-scales. Differently from [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF], we employ a novel system transformation. This transformation leads to a new system whose ISS-like bounds guarantees the ISS-like bounds of the original one. For ISS-like analysis of the transformed system, we adopt a direct Lyapunov method leading to LMIs. In comparison to [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF], our LMIs are significantly simpler. Feasibility of the LMIs provides quantitative estimates on the small parameters, which guarantee ISS-like estimates for the system. The LMIs are accompanied by theoretical guarantees on their feasibility for small values of the time-scale parameters.

We further extend our approach to rapidly time-varying systems with constant delay. Results on strict Lyapunov functions for rapidly time-varying nonlinear systems were presented in [START_REF] Mazenc | Further results on strict Lyapunov functions for rapidly time-varying nonlinear systems[END_REF] and [START_REF] Mazenc | Extensions of Razumikhin's theorem and Lyapunov-Krasovskii functional constructions for time-varying systems with delay[END_REF]. Differently from [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF], the novel transformation suggested in this paper decouples the effects of the delay bound and time-scale parameter on the stability of the system, thereby allowing to obtain stability for non-small delay, relative to the time-scale parameter. We carry out stability analysis via a timevarying Lyapunov functional, leading to LMIs and provide theoretical guarantees on their feasibility.

Notations: Throughout the paper R n denotes the ndimensional Euclidean space with the vector norm |•|, R n×m is the set of all n×m real matrices with the induced matrix norm ∥•∥. We also denote Z + = {0, 1, 2, . . . }. The superscript ⊤ denotes matrix transposition, and the notation P > 0, for P ∈ R n×n means that P is symmetric and positive definite. The symmetric elements of the symmetric matrix are denoted by * . For 0 < P ∈ R n×n and x ∈ R n , we write |x| 2 P = x ⊤ P x. We denote by W ([-h, 0]) the Banach space of a.e differentiable functions ϕ : [-h, 0] → R n with square integrable derivative. The norm on W ([-h, 0]) is given by the norm ∥ϕ∥ W = ∥ϕ∥ W + ∥ϕ ′ ∥ L 2 .

ISS OF RAPIDLY TIME-VARYING SYSTEMS

In this section we consider the fast-varying system ẋ(t) = A (t/ϑ) x(t) + B (t/ϵ) d(t), t ≥ 0.

(2.1) Here x(t) ∈ R n for t ≥ 0, the small independent parameters ϑ, ϵ > 0 define two fast time-scales, d ∈ C 1 ([0, ∞)) is a disturbance and A, B : R → R n×n are uniformly bounded, piecewise continuous matrix functions (i.e. having at most a finite number of jump discontinuities on any interval in R). The aim of this section is to derive explicit conditions, which guarantee ISS-like estimates for (2.1).

Assumption 1: A and B are almost periodic, meaning that there exist 0 < T and A av , B av , where A av is Hurwitz, such that for all t ∈ R:

T -1 t+T t L(τ )dτ = L av + ∆L(t), L ∈ {A, B} (2.2)
where ∆A, ∆B : R → R n×n are functions satisfying sup

τ ∈R ∥∆B(τ )∥ ≤ ∆ B,M , sup τ ∈R ∥∆A(τ )∥ ≤ ∆ A,M (2.3) for some ∆ A,M , ∆ B,M > 0.
Assumption 2: The time-scale parameters ϑ, ϵ satisfy max(ϑA

M , ϵB M ) • T < 2 (2.4) where L M := sup τ ∈R ||L(τ )|| < ∞, L ∈ {A, B} .
(2.5)

Inspired by [START_REF] Mazenc | Further results on strict Lyapunov functions for rapidly time-varying nonlinear systems[END_REF], we introduce

ω ϵ (t) = - 1 ϵT t+ϵT t (t + ϵT -τ ) B (τ /ϵ) dτ, ϱ ϑ (t) = - 1 ϑT t+ϑT t (t + ϑT -τ ) A (τ /ϑ) dτ, t ≥ 0.
(2.6) Note that piecewise continuity of A and B implies that ω ϵ (t) and ϱ ϑ (t) are piecewise continuously differentiable. Indeed, both are continuously differentiable at any t ≥ 0 for which t/ϵ + jT, j = 0, 1 and t/ϑ + kT, k = 0, 1 are points of continuity of B and A, respectively. Hence, any interval in R can be partitioned into finitely many subintervals with both ω ϵ (t) and ϱ ϑ (t) piecewise continuously differentiable in the interior of each subinterval. Using (2.5), (2.2) and ( 2.3), it can be easily verified that sup

τ ∈R ∥ω ϵ (τ )∥ ≤ 0.5ϵT B M , sup τ ∈R ∥ϱ ϑ (τ )∥ ≤ 0.5ϑT A M
(2.7) and for t ≥ 0 (outside of a nowhere dense set):

ωϵ (t) = B (t/ϵ) -B av -∆B (t/ϵ) , ρϑ (t) = A (t/ϑ) -A av -∆A (t/ϑ) , (2.8)
whereas by Assumption 2 and Neumann series, I n -ϱ ϑ (t) and I n -ω ϵ (t) are invertible for all t ∈ R and

(I n -ϱ ϑ (t)) -1 = ∞ k=0 ϱ ϑ (t) k (2.7) ≤ 2 2 -ϑT A M , (I n -ω ϵ (t)) -1 = ∞ k=0 ω ϵ (t) k (2.7) ≤ 2 2 -ϵT B M .
(2.9)

Using (2.6)-(2.8), the system (2.1) can be presented as

ẋ(t) = [ ρϑ (t) + A av + ∆A (t/ϑ)] x(t) + [ ωϵ (t) + B av + ∆B (t/ϵ)] d(t), t ≥ 0.
(2.10)

To eliminate the terms ρϑ (t)x(t) and ωϵ (t)d(t) in (2.10), we introduce the following transformation:

z(t) = x(t) -ϱ ϑ (t)x(t) -ω ϵ (t)d(t).
(2.11) Note that z(t) is continuous and piecewise continuously differentiable. In the proof of Theorem 2.1 we show that ISS-like estimates (with respect to d and ḋ) on z(t) imply ISS-like estimates for the system (2.1). Remark 2.1. For the case of a single time-scale (i.e., ϵ = ϑ), differently from transformation (2.11), the time-delay transformation in [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF] has a form z(t) = x(t) -G(t) (2.12) with

G(t) = 1 ϵT t t-ϵT (τ -t + ϵT ) [A(τ )x(ϵτ ) + B(τ )d(ϵτ )] dτ,
which leads to a neutral-type system. The transformation (2.12) allows for ISS of a larger class of essentially bounded disturbances than in the present paper.

Substituting (2.11) into (2.10), we obtain the following expression for ż(t) (outside of a nowhere dense set):

ż(t) = A av z(t) + B av d(t) + C 1 (t)x(t) +C 2 (t)d(t) -ω ϵ (t) ḋ(t) (2.13)
where

C 1 (t) := A av ϱ ϑ (t) -ϱ ϑ (t)A (t/ϑ) + ∆A (t/ϑ) , C 2 (t) := A av ω ϵ (t) -ϱ ϑ (t)B (t/ϵ) + ∆B (t/ϵ) . (2.14)
Note that by (2.3) and (2.7), we have

sup τ ∈R ∥C 1 (τ )∥ ≤ ϑT A M [A M + ∥A av ∥] 2 + ∆ A,M =: C 1 , sup τ ∈R ∥C 2 (τ )∥ ≤ T B M (ϵ ∥A av ∥ + ϑA M ) 2 + ∆ B,M =: C 2 .
(2.15) Remark 2.2. Differently from [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF], where a fast-varying system over a single time-scale was considered and a neutral-type system was obtained by integrating both sides of the ODE, here (2.11) allows us to treat the system (2.1), containing two time-scales. Furthermore, the approach presented in this manuscript allows us to avoid the introduction of a neutral-type system, which is more challenging for analysis.

For ISS-like analysis of (2.13), let 0 < α and 0 < P ∈ R n×n . We introduce the Lyapunov function

V (t) = |z(t)| 2 P .
(2.16) Differentiating V along the solution to (2.13), we have

V + 2αV = z(t) ⊤ P A av + A ⊤ av P + 2αP z(t) +2z(t) ⊤ P B av d(t) + 2z(t) ⊤ P C 1 (t)x(t) +2z(t) ⊤ P C 2 (t)d(t) -2z(t) ⊤ P ω ϵ (t) ḋ(t).
(2.17) Note that using Young's inequality, we have

|C 1 (t)x(t)| 2 (2.11) ≤ C 2 1 (I -ϱ ϑ (t)) -1 [z(t) + ω ϵ (t)d(t)] 2 (2.7) ≤ 2C 2 1 (2 -ϑT A M ) 2 4 |z(t)| 2 + (ϵT B M ) 2 |d(t)| 2 , |C 2 (t)d(t)| 2 (2.15) ≤ C 2 2 |d(t)| 2 .
(2.18) Let 0 < λ i ∈ R, i = 1, 2. Taking into account (2.18), we apply the S-procedure by introducing

W 1 (t) = λ 1 2C 2 1 (2 -ϑT A M ) 2 4 |z(t)| 2 + (ϵT B M ) 2 |d(t)| 2 -|C 1 (t)x(t)| 2 + λ 2 C 2 2 |d(t)| 2 -|C 2 (t)d(t)| 2 ≥ 0. (2.19) Let 0 < γ i ∈ R, i = 1, 2 and introduce for t ≥ 0 η(t) = col z(t), d(t), C 1 (t)x(t), C 2 (t)d(t), ω ϵ (t) ḋ(t) .
From (2.17) and (2.19) we obtain

V + 2αV -γ 2 1 |d(t)| 2 -γ 2 2 ω ϵ (t) ḋ(t) 2 + W 1 ≤ η(t) ⊤ Φ ϵ,ϑ η(t) ≤ 0, (2.20) provided Φ ϵ,ϑ =     Φ (1) ϵ,ϑ P 0 P 0 -P 0 * -λ 1 In 0 0 * * -λ 2 In 0 * * * -γ 2 2 In     < 0, Φ (1)
ϵ,ϑ =

P Aav + A ⊤ av P + 2αP P Bav * -γ 2 1 + Λ ϵ,ϑ , Λ ϵ,ϑ =     8λ 1 C 2 1 (2 -ϑT A M ) 2 In 0 * λ 2 C 2 2 + 2λ 1 ϵT B M C 1 2 (2 -ϑT A M ) 2     .
(2.21) Summarizing, we arrive at Theorem 2.1. Consider the system (2.1) subject to Assumptions 1-2. Given the matrices A av , B av ∈ R n×n and constants 0 < α, ϵ, ϑ, T, A M , B M , ∆ A,M , ∆ B,M , let there exist 0 < P ∈ R n×n and scalars 0 < λ i , γ 2 i , i = 1, 2, where λ i , i = 1, 2 are given in (2.19), such that the LMI (2.21) holds. Then (2.1) satisfies the following ISS-like estimate:

|x(t)| 2 ≤ β 2 1 e -2αt |x(0)| 2 + β 2 2 max s∈[0,t] |d(s)| 2 +β 2 3 max s∈[0,t] ḋ(s) 2 , t ≥ 0 (2.22)
for some β i > 0, i = 1, 2, 3. The LMI (2.21) is always feasible for small enough 0 < α, ϵ, ϑ, ∆ A,M , ∆ B,M and large enough 0 < γ 2 i , i = 1, 2. Feasibility of (2.21) for α, ϵ, ϑ, T, A M , B M , ∆ A,M , ∆ B,M , P, λ i , γ 2 i , i = 1, 2 implies its feasibility with any ϵ ′ < ϵ, ϑ ′ < ϑ and the same α, T, A M , B M , ∆ A,M , ∆ B,M , P, λ i , γ 2 i , i = 1, 2, whence β i in (2.22) are independent of ϵ ′ ∈ (0, ϵ) and ϑ ′ ∈ (0, ϑ).

Proof : Fix τ > 0. Feasibility of (2.21) implies that for all t ∈ [0, τ ]

V + 2αV -γ 2 1 |d(t)| 2 -γ 2 2 ω ϵ (t) ḋ(t) 2 ≤ 0 ⇒ V (t) ≤ e -2αt V (0) + t 0 e -2α(t-s) γ 2 1 |d(s)| 2 + γ 2 2 ω ϵ (s) ḋ(s) 2 ds.
Since

σ min (P ) |z(t)| 2 ≤ V (t) ≤ σ max (P ) |z(t)| 2 for all t ≥ 0, we obtain for t ≥ 0 |z(t)| 2 (2.7) ≤ σ max (P ) σ min (P ) e -2αt |z(0)| 2 + γ 2 1 2σ min (P )α × max s∈[0,τ ] |d(s)| 2 + γ 2 2 (ϵT B M ) 2 8σ min (P )α max s∈[0,τ ] ḋ(s) 2 ,
(2.23) meaning that (2.13) satisfies an ISS-like estimate. Recalling (2.11), we have

|x(t)| 2 (2.11) = (I -ϱ ϵ (t)) -1 [z(t) + ω ϵ (t)d(t)] 2 (2.7) ≤ 8 (2 -ϑT A M ) 2 |z(t)| 2 + 2 (ϵT B M ) 2 (2 -ϑT A M ) 2 |d(t)| 2 .
(2.24) From (2.7) and (2.11), we find

|z(0)| 2 ≤ 2 1 + ϑT A M 2 2 |x(0)| 2 + (ϵT B M ) 2 2 |d(0)| 2 .
(2.25) Finally, (2.23)-(2.25) imply (2.22), where β i , i = 1, 2, 3 can be computed explicitly. We omit the formulas for {β i } 3 i=1 due to space constraints.

Next, set ϵ = ϑ = ∆ A,M = ∆ B,M = 0 and consider Φ 1 in (2.21) with Λ ϵ,ϑ = 0 (see (2.19)). By Assumption 1, for 0 < α small enough and 0 < γ 2 1 large enough, we find that there exists 0 < P ∈ R n×n such that Φ 1 < 0. Fixing this P , applying Schur complement in Φ with respect to diag -λ 1 I n , -λ 2 I n , -γ 2 2 I n and taking 0 < λ i , γ 2 2 , i = 1, 2 large enough we find that (2.21) holds. By continuity of eigenvalues, (2.21) remains true for the same values of α, P, λ i , γ 2 i , i = 1, 2 and small enough 0 < ϵ, ϑ, ∆ A,M , ∆ B,M . Assume (2.21) holds for α, ϵ, ϑ, T, A M , B M , ∆ A,M , ∆ B,M , P, λ i , γ 2 i , i = 1, 2. Replacing ϵ and ϑ by ϵ ′ < ϵ and ϑ ′ < ϑ, respectively, and recalling (2.19), we have Λ ϵ ′ ,ϑ ′ ≤ Λ ϵ,ϑ . Hence, we obtain Φ

(1)

ϵ ′ ,ϑ ′ ≤ Φ (1)
ϵ,ϑ in (2.21). Taking the latter into account and applying Schur complement to Φ ϵ ′ ,ϑ ′ in (2.21) we find that Φ ϵ,ϑ < 0 implies Φ ϵ ′ ,ϑ ′ < 0. 2

RAPIDLY TIME-VARYING SYSTEMS WITH CONSTANT DELAY

We consider the system

ẋ(t) = Ax(t) + B (t/ϵ) x(t -h), t ≥ 0, x(θ) = ϕ(θ), θ ∈ [-h, 0] (3.1)
where x(t) ∈ R n for t ≥ 0, A ∈ R n×n , h, ϵ > 0 and ϕ ∈ W ([-h, 0]). Here, the delayed term x(t-h) is multiplied by a fast-varying coefficient B (t/ϵ), where B : R → R n×n is a piecewise continuous function subject to Assumption 1 and inequality (2.5). We further make the following assumption Assumption 3: The matrix A 0 = A + B av is Hurwitz.

Remark 3.1. We assume that A in (3.1) is constant for simplicity of presentation only. Our approach can be easily extended to the case of a system of the form ẋ(t) = A (t/ϑ) x(t) + B (t/ϵ) x(t -h), t ≥ 0 where ϵ, ϑ > 0 and A : R → R n×n is a piecewise continuous function satisfying Assumption 1 and A av +B av is Hurwitz. In this case, the transformation (3.3) below will be replaced by the following transformation:

z(t) = (I -ϱ ϑ (t)) x(t) -ω ϵ (t)
x(t -h), t ≥ 0 with ϱ ϑ (t) and ω ϵ (t) given in (2.6). In this case A av need not by Hurwitz.

Using ω ϵ (t) in (2.6), ωe (t) in (2.8) and (2.2), (3.1) can be presented as

ẋ(t) = Ax(t) + [ ωϵ (t) + B av + ∆B (t/ϵ)] x(t -h) (3.2) Introduce z(t) = x(t) -ω ϵ (t)x(t -h), t ≥ 0 (3.
3) which is again continuous and piecewise continuously differentiable. Differentiating z(t) for t ≥ h (outside of a nowhere dense set), we obtain the following:

ż(t) = A 0 z(t) -B av ζ h (t) + ∆B (t/ϵ) x(t -h) +H ϵ (t)x(t -h) + G ϵ (t)x(t -2h), t ≥ h (3.4)
where A 0 is given in Assumption 3, ζ h (t) = z(t) -z(t -h), t ≥ h, (3.5) is state error induced by the delay and

H ϵ (t) = Aω ϵ (t) -ω ϵ (t)A, G ϵ (t) = B av ω ϵ (t -h) -ω ϵ (t)B ((t -h)/ϵ) .
(3.6) By using (2.6), (2.7) and (2.8), we see that

sup τ ∈R ∥H ϵ (τ )∥ ≤ ϵT B M ∥A∥ =: H ϵ , sup τ ∈R ∥G ϵ (τ )∥ ≤ 0.5ϵT B M (∥B av ∥ + B M ) =: G ϵ . (3.7)
For exponential stability of z(t) subject to (3.4), let 0 < P, S, R ∈ R n×n and 0 < α, q i , i = 1, . . . , 4. We consider the Lyapunov functional

V (t) = |z(t)| 2 P + V S (t) + V R (t) + 4 i=1 V qi (t),
where

V S (t) = t t-h e -2α(t-s) |z(s)| 2 S ds, V R (t) = h 0 -h t t+θ e -2α(t-s) | ż(s)| 2 R dsdθ, are introduced to compensate ζ h (t) and V q1 (t) = q 1 t t-2h e -2α(t-s) |G ϵ (s + 2h)x(s)| 2 ds, V q2 (t) = q 2 t t-h e -2α(t-s) |ω ϵ (s + h)x(s)| 2 ds, V q3 (t) = q 3 t t-h e -2α(t-s) |H ϵ (s + h)x(s)| 2 ds, V q4 (t) = q 4 t t-h e -2α(t-s) |∆B ((s + h)/ϵ) x(s)| 2 ds (3.8) will compensate ∆B t ϵ x(t -h), H ϵ (t)x(t -h) and G ϵ (t)x(t -2h).
Differentiating V S and V R along the solution to (3.4) gives

VS + 2αV S = (1 -e -2αh ) |z(t)| 2 S -e -2αh |ζ h (t)| 2 S +2e -2αh z(t) ⊤ Sζ h (t), VR + 2αV R ≤ h 2 | ż(t)| 2 R -e -2αh |ζ h (t)| 2 R (3.9) whereas d dt |z(t)| 2 P + 2α |z(t)| 2 P = z(t) ⊤ P A 0 + A ⊤ 0 P + 2αP z(t) -2z(t) ⊤ P B av ζ h (t) + 2z(t) ⊤ P ∆B (t/ϵ) x(t -h) +2z(t) ⊤ P H ϵ (t)x(t -h) + 2z(t) ⊤ P G ϵ (t)x(t -2h).
(3.10)

Differentiation of V q1 (t) gives Vq1 + 2αV q1 = q 1 |G ϵ (t + 2h)x(t)| 2 -q 1 e -4αh |G ϵ (t)x(t -2h)| 2 (3.3) = q 1 G 2 ϵ |z(t)| 2 +q 1 G 2 ϵ |ω ϵ (t)x(t -h)| 2 + 2q 1 G 2 ϵ × z(t) ⊤ ω ϵ (t)x(t -h) -q 1 e -4αh |G ϵ (t)x(t -2h)| 2 (3.11) whereas similar arguments for V qi , i = 2, 3, 4 yield Vq2 + 2αV q2 ≤ q 2 (ϵT B M ) 2 4 |z(t)| 2 + q 2 (ϵT B M ) 2 2 z(t) ⊤ × ω ϵ (t)x(t -h) -q 2 e -2αh - (ϵT B M ) 2 4 |ω ϵ (t)x(t -h)| 2 , Vq3 + 2αV q3 ≤ q 3 H 2 ϵ |z(t)| 2 + q 3 H 2 ϵ |ω ϵ (t)x(t -h)| 2 +2q 3 H 2 ϵ z(t) ⊤ ω ϵ (t)x(t -h) -q 3 e -2αh |H ϵ (t)x(t -h)| 2 , Vq4 + 2αV q4 ≤ q 4 ∆ 2 B,M |z(t)| 2 + q 4 ∆ 2 B,M |ω ϵ (t)x(t -h)| 2 +2q 4 ∆ 2 B,M z(t) ⊤ ω ϵ (t)x(t -h) -q 4 e -2αh × |∆B (t/ϵ) x(t -h)| 2 . (3.12)
Finally,

h 2 | ż(t)| 2 Rz = h 2 η ⊤ (t)Λ ⊤ RΛη(t), Λ = [A 0 , -B av , I n , I n , I n , 0] , η(t) = col {z(t), ζ h (t), ∆B (t/ϵ) x(t -h), H ϵ (t)x(t -h) G ϵ (t)x(t -2h), ω ϵ (t)x(t -h)} . (3.13) From (3.10)-(3.13) we have V + 2αV ≤ η(t) ⊤ Ψ ϵ,h η(t) ≤ 0 (3.14) provided Ψ ϵ,h =       Ψ (1) ϵ P 0 P 0 P 0 σϵIn 0 * -q 4 e -2αh In 0 0 0 * * -q 3 e -2αh In 0 0 * * * -q 1 e -4αh In 0 * * * * -βϵIn       +h 2 Λ ⊤ RΛ < 0, Ψ (1) ϵ
= ψ -P Bav + e -2αh S * -e -2αh (S + R) + σ ϵ In 0 0 0 , ψ = P A 0 + A ⊤ 0 P + 2αP + 1 -e -2αh S, σ ϵ = q 2 (ϵT B M ) 2 4 + q 1 G 2 ϵ + q 3 H 2 ϵ + q 4 ∆ 2 B,M , β ϵ = q 2 4e -2αh -(ϵT B M ) 2 4 -q 1 G 2 ϵ -q 3 H 2 ϵ -q 4 ∆ 2 B,M .
(3.15) We are now ready to state the main theorem of this section. Theorem 3.1. Consider the system (3.1) subject to Assumptions 1 and 3. Given the matrices A 0 , A, B av ∈ R n×n , constants 0 < α, ϵ, T, B M , h and tuning parameter 0 < ∆ B,M , let there exist matrices 0 < P, S, R ∈ R n×n and scalars 0 < q i , i = 1, . . . , 4 such that κ ϵ := ϵT B M 2 < e -αh and the LMI (3.15) hold. Then the system (3.1) is exponentially stable with decay rate α > 0. The LMI (3.15) is always feasible for small enough 0 < α, ϵ, h, ∆ B,M . Furthermore, let (3.15) hold for some ϵ, h, α, T, B M , ∆ B,M , P, S, R, q i , i = 1, . . . , 4. Then, (3.15) is feasible for any ε < ϵ and the same h, α, T, B M , ∆ B,M , P, S, R, q i , i = 1, . . . , 4.

Proof : Due to space constraints, we only provide a sketch. First, Feasibility of the LMI (3.15) implies that there exists a constant 0 < M such that |z(t)| ≤ M e -α(t-h) , t ≥ h.

(3.16) Next, for any k ∈ N, we denote X k = sup τ ∈[kh,(k+1)h) |x(τ )|. From (3.3) and (3.16), we find that X k+1 ≤ M e -αkh + κ ϵ X k , k ∈ N. Comparing with the solution of the linear difference equation α(t-h) .

Y k+1 = M e -αkh + κ ϵ Y k , k ∈ N. (3.17) with X 1 = Y 1 it follows that |x(t)| ≤ X k ≤ κ -1 ϵ (X 1 -µ ϵ,h ) + µ ϵ,h e αh e -
Consider (3.15) with ϵ = ∆ B,M = 0, which implies σ ϵ = 0 and β ϵ = q 2 e -2αh . By Assumption 3, there exist 0 < P, S, R ∈ R n×n and small 0 < α ∈ R such that

Ψ 1 + h 2 A ⊤ 0 -B ⊤ av R [A 0 -B av ] < 0, (3.18)
provided h > 0 is small enough. Fixing such 0 < α, h, taking q i = q, i = 1, . . . , 4 for q > 0 large enough and applying Schur complement in Ψ ϵ,h , we find that (3.15) holds. By continuity of eigenvalues, we have that κ ϵ < e -αh and (3.15) hold for the same α, h and small enough 0 < ϵ, ∆ B,M .

If (3.15) hold with ϵ, h, α, T, B M , ∆ B,M , P, S, R, q i , i = 1, . . . , 4 then Schur complement of (3.15) and monotonicity of κ ϵ < e -αh yield feasibility of (3.15) with ε < ϵ and the same h, T, B M , ∆ B,M , P, S, R, q i , i = 1, . . [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF], the delay bound which preserved the stability of the system was of the order of the time-scale ϵ. Here (3.3) decouples the effects of the delay bound and ϵ on the stability of the system, thereby allowing to obtain stability of (3.1) for non-small delay h, relative to ϵ.

NUMERICAL EXAMPLE

We show that our method is complementary to that of [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF] and [START_REF] Zhang | L 2 -gain analysis via time-delay approach to periodic averaging with stochastic extension[END_REF].

Rapidly varying persistently excited system

We consider stability of the following persistently excited system [Fridman and Zhang 2020, Example 3.1]

ẋ(t) = -p (t/ϑ) p (t/ϑ) ⊤ x(t), t ≥ 0,
where x(t) ∈ R n and there exists constants 0 < ρ < M such that

ρI n ≤ 1 0 p (τ -θ) p (τ -θ) ⊤ ≤ M 2 I n , τ ≥ 1.
Recalling (2.1), here we have A (t/ϑ) = -p (t/ϑ) p (t/ϑ) ⊤ , whereas B ≡ 0.

We consider the case n = 1, A av = 0.5(M 2 + ρ), ∆A = 0.5(M 2 -ρ), where M = 1 and ρ = 0.55, as in [Fridman and Zhang 2020, Example 3.1]. Choosing further α = 0.5, we find that feasibility of the LMI of Theorem 2.1 is preserved for ϑ max = 0.1544. In comarison to [Fridman and Zhang 2020, Example 3.1], where ϑ max = 0.0645, our approach performs better in this example.

Stabilization by fast switching

Consider stabilization by fast switching of a linear system [Fridman and Zhang 2020, Example 4.2]. Let ϵ > 0 and A 1 = 0.1 0.3 0.6 -0.2 , A 2 = -0.13 -0.16 -0.33 0.03 and set

A(τ ) = A 1 , τ ∈ [k, k + 0.4), k ∈ Z + A 2 , τ ∈ [k + 0.4, k + 1), k ∈ Z + . (4.1)
Given τ ∈ [k, k + 1), (4.1) can be written as

A (τ ) = χ [k,k+0.4) (τ ) A 1 + 1 -χ [k+0.4,k+1) (τ ) A 2 ,
where χ ⊤ and B M = 1.1. We fix ϑ = 0.025 and ϵ ∈ 1 π • [0.0002, 0.5] with grid of step side 0.0001. Note that A (t/ϑ) and B (t/ϵ) do not have a common period. Hence, the approach in [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF] cannot be used in this system. Choosing α = 0.005 and restricting the bound on the ISS gain, β 2 in (2.22) (computed from (2.23)-(2.25)), to be at most 1.079 × 10 2 , we verify the LMI of Theorem 2.1 to obtain the maximal value of ϵ which preserves feasibility of the LMI. The LMI was found feasible for ϵ max = 0.178. Comparing to the previous simulation with ϑ max = ϵ max = 0.0376, we see that for two independent time-scale parameters, the magnitude of ϑ is a larger constraint for feasibility of the LMI of Theorem 2.1 than the magnitude of ϵ in this example.

Next, we consider the delayed system (3.1), where we choose A = 0 and B(t) = A(t) from (4.1). We fix the decay rate α = 0.005. Then, the LMIs of Theorem 3.1 were checked for feasibility twice. First, for ϵ ∈ [0, 0.0475] the LMIs were verified to find the maximal value of h, which preserves feasibility. Second, for h ∈ [0, 0.356] the LMIs were verified to find the maximal value of ϵ, which preserves feasibility. The results are given in Figure 1. Differently from [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF], where ϵ and the delay bound were of the same order, it can be seen that as ϵ → 0 + , the value of h * (ϵ) (see (3.19)) grows.

CONCLUSION

This paper presented a novel quantitative approach to stability of linear continuous-time systems. Differently from the time-delay approach to averaging, our method is efficient for systems with fast-varying almost periodic coefficients with multiple independent time-scales, as well as to such systems in the presence of constant delay, which does not vanish when the small parameter approaches zero. Moreover, our approach leads to essentially simpler single reduced-order LMI with fewer decision parameters. Future work may include improvement of the method and its extension to time-varying delays.

  Fig. 1. Theorem 3.1: feasibility preserving (ϵ, h). Top: X = h, Y = max ϵ. Bottom: X = ϵ, Y = max h

  . , 4. 2 Corollary 1. Fix α, T, B M , ∆ B,M and let ϵ * > 0 be such that there exists h > 0 for which the LMI (3.15) is feasible with (ϵ * , h). Define h * : (0, ϵ * ] → (0, ∞) by h * (ϵ) = sup {h > 0 | (3.15) is feasible for (ϵ, h)} . (3.19) Then h * (ϵ) is positive and non-increasing on (0, ϵ * ]. Remark 3.2. In

⋆ Supported by Israel Science Foundation (grant no. 673/19) and by C. and H. Manderman Chair at Tel Aviv University


Quentin Acher
email: quentin.acher@inria.fr

Claudia-Lavinia Ignat

Shadi Ibrahim
email: shadi.ibrahim@inria.fr

Quantifying the Performance of Conflict-free Replicated Data Types in InterPlanetary File System

Keywords: CCS Concepts:, Information systems → Distributed storage, Data replication tools, • Networks → Network measurement P2P Storage Systems, Replication, Consistency, CRDT, IPFS, Performance Evaluation

The continuous growth in data volume increases the interest in using peer-to-peer (P2P) systems not only to store static data (i.e., immutable data) but also to store and share mutable data -data that are updated and modified by multiple users. Unfortunately, current P2P systems are mainly optimized to manage immutable data. Thus, each modification creates a new copy of the file, which leads to a high "useless" network usage. Conflict-free Replicated Data Types (CRDTs) are specific data types built in a way that mutable data can be managed without the need for consensus-based concurrency control. A few studies have demonstrated the potential benefits of integrating CRDTs in the InterPlanetary File System (IPFS), an opensource widely used P2P content sharing system. However, they have not been implemented and evaluated in a real IPFS deployment. This paper tries to fill the gap between theory and practice and provides a quantitative measurement of the performance of CRDTs in IPFS. Accordingly, we introduce IM-CRDT, an implementation of CRDTs in IPFS that focuses on the simple data type (i.e., Set); and carry out extensive experiments to verify whether CRDTs can efficiently be utilized in IPFS to handle mutable data. Experiments on Grid'5000 show that IM-CRDT reduces the data transfer of an update by up to 99.96% and the convergence time by 54.6%-62.6%. More importantly, we find that IM-CRDT can sustain low convergence time under concurrent updates.

Introduction

In current cloud storage services, data are replicated to ensure high availability in case of failures. Specifically, multiple copies of the same data are stored in a few machines normally placed in different geographical locations, so if a failure occurs data are still accessible in other replicas. A key question is how to maintain consistency between all replicas [START_REF] Gilbert | Brewer's Conjecture and the Feasibility of Consistent, Available, Partition-Tolerant Web Services[END_REF][START_REF] Chihoub | Consistency Management in Cloud Storage Systems[END_REF]. Most of the current cloud storage systems use a central server to synchronize replicas. However, this solution does not scale up with the number of replicas and their frequency of modification. To eliminate the disadvantages of a central entity that has to perform the synchronisation and to ensure easy deployment, and resilience to failures and attacks, alternative solutions, using Peer-to-Peer (P2P) technologies, were proposed [START_REF] Rodrigues | Peer-to-Peer Systems[END_REF].

Meant to be "the storage layer of the decentralized web", the InterPlanetary File System (IPFS) [START_REF] Benet | IPFS -Content Addressed, Versioned, P2P File System[END_REF] is an open-source P2P content addressed file system. It allows people to share data, either directly or via programs that use IPFS as a library. In IPFS, data are immutable by design and modifying an object requires creating a new modified object. IPFS does not offer support for merging concurrent changes, i.e., users are not able to update concurrently the replicas of the same data without losing their modifications.

As stated by the CAP theorem [START_REF] Gilbert | Brewer's Conjecture and the Feasibility of Consistent, Available, Partition-Tolerant Web Services[END_REF], high availability, low latency and data consistency are difficult to achieve in distributed systems in the presence of network partitions. In order to ensure high data availability, consistency is relaxed. Eventual consistency [START_REF] Saito | Optimistic Replication[END_REF][START_REF] Chihoub | Consistency Management in Cloud Storage Systems[END_REF] is a weak consistency model where replicas are allowed to diverge and they will converge later after the reception of all updates. One of the main family of replication algorithms for ensuring convergence under eventual consistency is Conflict-free Replicated Data Types (CRDTs) [START_REF] Shapiro | Conflict-Free Replicated Data Types[END_REF]. The main idea is to define data structures where parallel modifications are conflict free. CRDTs respect Strong Eventual Consistency, a property that ensures convergence as soon as every replica has integrated the same modifications without further message exchange among replicas.

Several studies have discussed the feasibility of using CRDTs in P2P systems, including IPFS. LogootSplit [START_REF] André | Supporting adaptable granularity of changes for massive-scale collaborative editing[END_REF] is a sequence CRDT that was applied for building a P2P web based real-time collaborative editor [START_REF] Nicolas | MUTE: a peer-to-peer web-based real-time collaborative editor[END_REF]. In [START_REF] Sanjuan | Merkle-CRDTs: Merkle-DAGs meet CRDTs[END_REF], the authors formalised Merkle-CRDTs and discussed the benefits of integrating them in IPFS. While previous work evaluated CRDTs in the context of collaborative editing [START_REF] Ahmed-Nacer | Evaluating CRDTs for Real-time Document Editing[END_REF], no implementation and evaluation of CRDTs was done in a real IPFS deployment. To this end, in an attempt to fill the gap between theory and practice, this paper provides -to the best of our knowledge -the first quantitative measurement of the performance of CRDTs in IPFS. Specifically, we introduce IM-CRDT, an implementation of CRDTs in IPFS that focuses on the simple data type (i.e., Set). Experiments on Grid'5000 [START_REF] Balouek | Adding Virtualization Capabilities to the Grid'5000 Testbed[END_REF] show that IM-CRDT reduces the data transfer of an update by up to 99.96% and the convergence time by 54.6%-62.6%. More importantly, we find that IM-CRDT can sustain low convergence time under concurrent updates. To summarize, we make the following contributions.

• We present an implementation of CRDTs in IPFS named IM-CRDT, which targets specifically the Set data type.

Background

This section first briefly introduces IPFS, and then presents Merkleclocks and CRDTs. IPFS. IPFS [START_REF] Benet | IPFS -Content Addressed, Versioned, P2P File System[END_REF] is a P2P version-controlled file system that enables data sharing among users. IPFS's protocols define its behavior, such as how to address data through Content IDentifier (CID), how to share and store data and how to manage security issues and replication. A file is presented as a Merkle Directed Acyclic Graph (Merkle-DAG), where each node refers to a shard of the data. A user informs other users that it shares a specific file by publishing a Provider Record on the Distributed Hash Table (DHT) that contains the link between the file CID and the Peer Identifier. The naming protocol allows users to share a file that can be modified only by its creator, hence other peers can only read it. efficient and widely used [START_REF] Viet Doan | Towards Decentralised Cloud Storage with IPFS: Opportunities, Challenges, and Future Directions[END_REF]. However, it is not meant to deal with mutable data that can be concurrently modified, as each modification requires creating a whole new file. Merkle-clocks. In distributed systems, logical clocks are generally used for ordering events [START_REF] Lamport | Time, Clocks, and the Ordering of Events in a Distributed System[END_REF]. A logical clock is a structure 𝑐 that associates to every event 𝑒 a value 𝐶 (𝑒) that creates a partial order respecting the causal order. Merkle-clocks [START_REF] Sanjuan | Merkle-CRDTs: Merkle-DAGs meet CRDTs[END_REF] are Merkle-DAG-based logical clocks where nodes represent events. The order relation between events is defined by the existence of a path between nodes. CRDTs. CRDT is a data structure that ensures Strong Eventual Consistency without explicit coordination. There are two main families of CRDTs: state-based and operation-based [START_REF] Shapiro | Conflict-Free Replicated Data Types[END_REF]. They differ in the way payloads are defined, i.e., how the updates are shared. A payload under state-based CRDT contains the whole data, while the payload under operation-based CRDT carries only a single update. Merkle-CRDTs [START_REF] Sanjuan | Merkle-CRDTs: Merkle-DAGs meet CRDTs[END_REF] are based on Merkle-clocks where each node of the Merkle-DAG refers to CRDT payloads.

Integration of CRDTs in IPFS

In this section, we explain how we implemented the data replication mechanism and the communication mechanism among the replicas. We used the Go implementation of IPFS, named kubo 1 , to implement IPFS as a file sharing system, and used libP2P's communication protocols. In our evaluation, we focus on the Set data type. However, it is important to note that the modules for the definition of Merkle-CRDT nodes and for Merkle-CRDT merging can be easily extended to other CRDTs. 1 https://github.com/ipfs/kubo

Merkle-CRDT's Nodes

Nodes of the Merkle-CRDT's DAG are built to be easily converted into a file which will can be then shared through IPFS. This is done by encoding the node's structure into a JSON file. The file must contain all the necessary information needed by a peer after downloading it. A node contains all the CIDs of its direct sons in the DAG, we refer to them as the direct dependencies. The file also contains the Peer IDentifier (PID) of the peer that emitted the event, and the payload of the event it carries, i.e., the data shared by the event in a CRDT format. Figure 1 shows an operation-based CRDT DAG with data that was modified five times, once per node. The root node 522 refers to a file stored in IPFS containing the following information: the operation of the update, i.e., an 𝑎𝑑𝑑 operation , followed by the data added; the peer identifier of the publisher, 𝑃 1 ; the Direct dependencies which include the root nodes that were known by the publisher 𝑃 1 when publishing the 𝑎𝑑𝑑 operation.

Merkle-CRDT's DAG

The Merkle DAG is defined by a set of root nodes (that do not have parents). The data state can be obtained by means of a recursive traversal of the DAG, starting from these root nodes. Note that when concurrent updates occur, we can have multiple root nodes. Indeed, multiple updates can be concurrently done and will therefore have the same direct dependency. For instance, in Figure 1, before the creation of the node 522, the Merkle-CRDT's DAG had two root nodes: 156 and 203. Once a node is read by IPFS, it is stored locally in a file. Hence, each node in Merkle-CRDT refers to a local file containing the updated data.

Merkle-CRDT's communication

The communication module among peers was implemented by means of a publish-subscribe mechanism. A publish-subscribe topic is created by a specific peer named the bootstrap peer. The bootstrap peer opens a socket where any peer can connect in order to join the publish-subscribe topic and subscribe to it. When a peer applies an update to the data, it creates a file corresponding to the update's node. Then the file is added to IPFS and the peer registers the corresponding CID that will be sent to all other peers through the publish-subscribe topic. On the receiver side, when a peer receives a CID from the topic, it saves the CID in a file, so later an active loop can read it. If the CID is unknown, it will search for the corresponding data in IPFS, and write it in a new local file. When the corresponding data is received, two cases must be handled. If all the direct dependencies are already known, the node is directly added to the Merkle-DAG. Otherwise, the system will recursively download all the dependencies. Such download might introduce an overhead when the Merkle-DAG becomes large, as confirmed in [START_REF] Sanjuan | Merkle-CRDTs: Merkle-DAGs meet CRDTs[END_REF]. Figure 2 represents an example of an update managed by our system. The system applies updates through three remote steps, and a fourth local step. At the beginning, Peer 1 prepares the node's information in a file including the performed operation, the peer creating it, and the CID of the direct dependencies. In step 1, the updater Peer 1 sends these data to IPFS. IPFS computes the data's CID and announces it to the other peers. In step 2, libP2P's pubsub protocol is used to send a message containing the node's CID to the other peers. In step 3, each peer that receives the CID retrieves the data from an IPFS provider, stores it in a new local file, and locally

Evaluation

In this section, we evaluate the performance of integrating simple string Set CRDTs in IPFS. Our evaluation demonstrates that:

• Compared to the default IPFS implementation, IM-CRDT improves convergence time and reduces network usage. IM-CRDT reduces the data transfer of an update by up to 99.96% and the convergence time by 54.6%-62.6%. • IM-CRDT can sustain low convergence time under concurrent updates. However, sequential downloading becomes a bottleneck when the update rate is high. • IM-CRDT does not introduce high overhead (compute time)

when integrating files into Merkle-CRDTs, under concurrent updates.

Performance metric

We quantify the feasibility and effectiveness of our system based on the convergence time, i.e., the time period between an update is issued and when it is integrated in the Merkle-CRDT DAGs of all replicas. We denote the time period for an update 𝑥 issued by a peer 𝑝 𝑗 to be successfully received by a peer 𝑝 𝑖 as update Latency

𝑝 𝑗 →𝑝 𝑖 (𝑥).
It is composed of the time required by the publisher to issue the corresponding file in IPFS (𝑡 𝑥 𝑎𝑑𝑑 ), the time for transmitting the CID from the publisher to the receiver (𝑡 𝑥 𝑝𝑢𝑏𝑠𝑢𝑏 ), the time to retrieve the IPFS file and unknown dependencies (𝑡 𝑥 𝑟𝑒𝑡𝑟𝑖𝑒𝑣𝑒 ), and the time to integrate the file in the receiver's local Merkle-CRDT DAG (𝑡 𝑥 𝑐𝑜𝑚𝑝𝑢𝑡𝑒 ). update Latency 𝑝 𝑗 →𝑝 𝑖 (𝑥) can be computed as: update Latency

𝑝 𝑗 →𝑝 𝑖 (𝑥) = 𝑡 𝑥 𝑎𝑑𝑑 + 𝑡 𝑥 𝑝𝑢𝑏𝑠𝑢𝑏 + 𝑥 𝑟𝑒𝑡𝑟𝑖𝑒𝑣𝑒 + 𝑡 𝑥 𝑐𝑜𝑚𝑝𝑢𝑡𝑒 = 𝑡 𝑥 𝑝 𝑖 -𝑡 𝑥 𝑝 𝑗
where 𝑡 𝑥 𝑝 is the time at which the peer 𝑝 creates/receives 𝑥. We for an update 𝑥 issued by 𝑝 𝑖 0 as: maximum latency(𝑥) = 𝑚𝑎𝑥

1≤𝑖 ≤𝑛; 𝑖≠𝑖 0 𝑡 𝑥 𝑝 𝑖 -𝑡 𝑥 𝑝 𝑖 0 = 𝑚𝑎𝑥 1≤𝑖 ≤𝑛; 𝑖≠𝑖 0 update Latency 𝑝 𝑗 →𝑝 𝑖 (𝑥)
where (𝑝 𝑖 ) {0≤𝑖 ≤𝑛-1}\{𝑖 0 } are the peers hosting the replica.

Compared solutions.

We evaluated two approaches: the default implementation of IPFS, and IM-CRDT ( InterPlanetary Merkle-CRDT).

• IPFS is used as a baseline approach when updates are issued sequentially. IPFS sends the whole file to all other peers after each update. We do not evaluate IPFS under concurrent updates because this requires to implement a locking mechanism in IPFS which may introduce significant overhead and scalability limitations. • IM-CRDT is our implementation which integrates CRDTs with IPFS. We implemented string Set CRDTs based on the grow-Only Set structure [START_REF] Shapiro | Conflict-Free Replicated Data Types[END_REF], supporting string addition (Add) and removal (Remove). When adding a string 𝑥 in the set 𝑆 (𝑆 = (𝑆 𝐴𝑑𝑑 , 𝑆 𝑅𝑒𝑚𝑜𝑣𝑒 )), the addition operation is applied 𝑆 𝐴𝑑𝑑 = 𝑆 𝐴𝑑𝑑 ∪ {𝑥 }; and when removing a string, the removal operation is applied 𝑆 𝑅𝑒𝑚𝑜𝑣𝑒 = 𝑆 𝑅𝑒𝑚𝑜𝑣𝑒 ∪ {𝑥 }. Grow-Only Sets maintain Strong Eventual Consistency, ensuring replicas convergence regardless of the order of execution of (𝐴𝐷𝐷, 𝑥) and (𝑅𝐸𝑀𝑂𝑉 𝐸, 𝑥) operations. We use operationbased CRDTs, as they can easily manage incremental modifications. A unique identifier is associated to every element added into the Set.

Experimental setup

Experimental testbed. Our experiments were conducted on the French scientific testbed Grid'5000 [START_REF] Balouek | Adding Virtualization Capabilities to the Grid'5000 Testbed[END_REF] at the site of Nantes. We use the econome and ecotype clusters with 66 machines. Deployment. In each experiment, we start by initiating 𝑛 nodes/peers with 𝑛 ∈ {2, 5, 10, 20, 50} nodes/peers. Their clocks are then synchronized using the NTP protocol. We select one peer to act as a bootstrap peer. We wait for 1 minute (this time is set to 5 minutes in the case of 5 replicas) -to make sure that all peers are connected -and then we start sending out the updates. In the case of sequential updates, we configured the bootstrap peer to send the updates. And in the case of concurrent updates, the bootstrap peer and 𝑛 𝑝𝑒𝑒𝑟𝑈 𝑝𝑑𝑎𝑡𝑖𝑛𝑔 ∈ {1, 4, 9, 19} peers are configured to send updates. We save the 𝑡 𝑥 𝑎𝑑𝑑 in the updating nodes; and the 𝑡 𝑥 𝑟𝑒𝑡𝑟𝑖𝑒𝑣𝑒 and 𝑡 𝑥 𝑐𝑜𝑚𝑝𝑢𝑡𝑒 in the receiving nodes. For the 𝑡 𝑥 𝑝𝑢𝑏𝑠𝑢𝑏 , we compute it using the following equation:

𝑡 𝑥 𝑝𝑢𝑏𝑠𝑢𝑏 = update Latency 𝑝 𝑗 →𝑝 𝑖 (𝑥) -𝑡 𝑥 𝑎𝑑𝑑 -𝑡 𝑥 𝑟𝑒𝑡𝑟𝑖𝑒𝑣𝑒 -𝑡 𝑥 𝑐𝑜𝑚𝑝𝑢𝑡𝑒
Workload. We start the experiment with a 1 𝑀𝐵 file. Then, each updater sends one update per second. In each update, a unique string is added. This string is a growing integer signed by the date and the peer identifier. IM-CRDT vs. IPFS. Figure 3 shows how IM-CRDT compares with IPFS with respect to convergence time (maximum latency). We vary the number of replicas from 2 to 50, and the number of updates from 10 to 1000. We can see that IM-CRDT can reduce the maximum latency by 54.6%-62.6% compared to IPFS, on average. The reason behind that is the reduction in the retrieval time under IM-CRDT compared to IPFS, as shown in Figure 4, which is due to the significant reduction in the size of updates. The average size of an update is only 350 bytes with IM-CRDT, while it is 1 MB with IPFS -IPFS sends the whole file to all replicas after each update.

The impact of concurrency in IM-CRDT. Figure 5 shows the changes in maximum latency of IM-CRDT under concurrent updates. Interestingly, for most cases, IM-CRDT can sustain low maximum latency when increasing the number of concurrent updates. For example, the latency increases by only 6%, from 249 ms to 264 ms when increasing the number of concurrent updates from 1 to 20, with 10 updates. However, we noticed that the maximum latency increases significantly, by 133% and 205%, when the number of concurrent updates is set to 20 and the total number of updates is 100 and 1000, respectively. Taking a closer look at the results, Figure 6 shows the breakdown of the different steps for the updates, with 20 replicas. We observe that pubsub time is considerably higher when having 20 concurrent updates. As a reminder, the pubsub time (𝑡 𝑥 𝑝𝑢𝑏𝑠𝑢𝑏 ) is the time from when the updater starts sending the update's CID to the time when the receivers start downloading the update. This increase in pubsub time is not caused by libP2P's pubsub mechanism, but it is the result of the sequential downloading of updates in IPFS. An update takes about 50 -100 ms so when 20 updates arrive at the same time, it takes more than one second for the retrieve step to be completed, during which the next batch of updates should wait. Overhead of IM-CRDT: Compute time. Figure 7 shows the compute time when varying the number of concurrent updates. IM-CRDT introduces a slight increase in the compute time (by up to 7%), reaching 60.8 𝜇s for 20 updates.

Conclusion and Future Work

This paper presents IM-CRDT, an implementation of CRDTs in IPFS for the Set data type. Our evaluation results show that IM-CRDT can reduce the amount of data transfer and achieve fast convergence time. In future work, we plan to evaluate our implementation using more complex replicated data types including sequences [START_REF] André | Supporting adaptable granularity of changes for massive-scale collaborative editing[END_REF] and relational databases [START_REF] Yu | Conflict-Free Replicated Relations for Multi-Synchronous Database Management at Edge[END_REF]. We also plan to investigate how to handle efficiently concurrent downloads under a high rate of concurrent updates.

Figure 3 .

 3 Figure 3. Maximum latency for each update using IM-CRDT and IPFS

Figure 4 .Figure 5 .

 45 Figure 4. Mean retrieve time for an update using IM-CRDT and IPFS

Figure 6 .

 6 Figure 6. Breakdown of the different steps for the updates in IM-CRDT with 20 Replicas

Figure 7 .

 7 Figure 7. Evolution of compute time when the number of concurrent updates grows

Acknowledgments

This work is supported by the "Alvearium" Inria and hive partnership. Experiments presented in this paper were carried out using the Grid'5000 testbed, supported by a scientific interest group hosted by Inria and including CNRS, RENATER, and several Universities as well as other organizations (see http://www.grid5000.fr/).


Raphaël Périllat

Sylvain Girard

Irène Korsakissok

Emmanuel Quentric

EMULATORS FOR THE RAPID PREDICTION OF CONSEQUENCES IN CASE OF NUCLEAR HAZARDS

Keywords: Emulator, prediction, gaussian process, accidental release

In the event of an accidental release of radionuclides into the atmosphere, the use of emulators (also known as surrogate models) makes it possible to quickly obtain useful results to help decision-making. However, the predictions of high dimension data is complex and the protective actions must be parametrized by a limited number of variable to be predicted by emulators.

INTRODUCTION

Numerical simulations of the atmospheric dispersion of radioactive particles are used to predict the areas impacted by accidental releases. IRSN use the C3X operational crisis platform to perform these calculations and to propose urgent protective actions (evacuation, sheltering, stable iodine prophylaxis) to decision makers. The protective action guide levels, expressed in terms of dosimetric quantities, are determined by regulation.

For the first response of the emergency center of the IRSN, the maximum distances up to which those protective actions might be proposed, as well as the angular apertures of the territory likely to be concerned, are scalar variables (Figure 1). In emergency response related to a possible or actual atmospheric release, these distances are estimated using the IRSN's operational dispersion models pX (Gaussian puff, used at short distance) and ldX (Eulerian, used at long distance), on the basis of the meteorological situation and source term assessment. In the graded approach applied by the IRSN's emergency center, first response generally relies on pre-calculated scenarios appropriately selected by the responders and for which experts should choose calculation parameters adapted to the situation and to the assessment objective. Precalculated source terms and consequences are made available for the experts for several accidents on several installations. This database has been constructed by calculations made in preparedness phase for a number of accidental scenarios and simple weather situations described by a few parameters (wind direction and speed, atmospheric stability, rain) assumed to be constant and homogeneous over the simulation domain. For each of these typical accidents, pre-calculated data are gathered in an "Accident Type Sheet" (called FAT in french, for "Fiche Accident Type"). Calculations are performed with the Doury Gaussian standard deviations [START_REF] Doury | Le vademecum des transferts atmosphériques[END_REF], for two stability conditions, normal diffusion (ND) and low diffusion (LD), some wind speed values (3, 7, 13 m/s). A meteorological situation with a rainfall of 2 mm/h is also considered.

However, all possible meteorological situations cannot be covered in a single operational sheet, and the weather condition of the day of the accident rarely matches exactly to one of those described in the FAT. Similarly, other parameters may vary slightly from the calculation assumptions made in the FAT (e.g., the quantity released, the height of release, the standard deviations of Gaussian dispersion, deposition velocities,...). It would be impractical to enrich the FAT to cover all possible situations, that is why providing experts with a tool that instantly provides the response of the simulation for a custom set of parameters would allow operational constraints to be met.

Emulators, also known as metamodels, are surrogate models for the original computational model, used to approximate some of its scalar responses. They are statistical functions built from a large number of simulations, with the aim of predicting new responses with a negligible evaluation time. In this study, Gaussian processes [START_REF] Roustant | DiceKriging, DiceOptim: Two R Packages for the Analysis of Computer Experiments by Kriging-Based Metamodeling and Optimization[END_REF] are used for the construction of the emulators based on the pX model.

EMERGENCY PROTECTION ZONES AFTER A NUCLEAR ACCIDENT

We consider here one of the typical accident scenario of a 1300 MWe Pressurized Water Reactor (PWR) of the IRSN FAT catalog, which consists of a break on the primary circuit leading to the total core meltdown within one hour, and to a twenty-four-hours atmospheric release. The pre-calculated projected doses exceeds urgent protective action guide levels over significant distances. Three zones will be considered in this study:

-A sheltering area, where the simulated total effective dose over 24 hours is greater than 10 mSv, -An evacuation area, where the simulated total effective dose over 24 hours is greater than 50 mSv, -A stable iodine prophylaxis area, where the simulated equivalent dose to the thyroid due to inhalation over 24 hours is greater than 50 mSv. The source term, atmospheric dispersion and dose calculation are conducted using conservative assumptions at each step, without taking into account the effect of protective measures, in order to avoid underestimating the impact of the accident. Each zone is described by three parameters: its maximum distance, its angular aperture and its orientation (see Figure 1), which can be calculated from the dose outputs of the pX model run. The orientation of the plume only depends on the wind direction, assumed to be homogeneous and uniform over the simulation domain, it does not need to be emulated since it is obtained directly from the model input value.

Two emulators (maximum distance and angular aperture) are built for each of the 3 protective action zones. In this paper, we focus on the maximum distances of threshold exceedance of protective action guide levels. Five input parameters of the pX model are considered to be uncertain: the release height of the source term, the source term magnitude, the wind module, the rain intensity and the meandering wind factor, which is a factor increasing the lateral spread of the plume to account for meandering wind direction with a characteristic time between 6 minutes and the meteorological time step (usually one hour). These five variables are selected because they are part of the few parameters that are likely to be modified during the first emergency reponse phase, to take into account either additional clarifications or strong uncertainties on their values. For each of these variables, a range of variation is defined in order to delimit the input space.

EMULATION BY GAUSSIAN PROCESSES

The construction of emulators relies on the simulation of thousands cases to cover a wide range of hypotheses. For each input variable, a draw is made over its range of variation, in order to make the emulators reliable over the entire space of possibilities. In our case, it means to randomly draw points in a space of dimension five. A simulation is then performed with pX for each point of this training sample, with the random values of these parameters. The maximum distances as well as the angular apertures corresponding to the three emergency protection zones can be extracted from the results of these simulations. These quantities are then used in the construction of emulators.

A Gaussian process is defined as a collection of a finite number of random variables having a Gaussian joint distribution. Thus, the response surface described by the output when the inputs vary is modeled as the sum of a constant term and the realization of a centered Gaussian process. If the model output pX is denoted 𝑦𝑦 and the vector of its inputs is denoted 𝑥𝑥, the emulator 𝑓𝑓 is written as :

𝑓𝑓(𝑥𝑥) = 𝐶𝐶 + � 𝑤𝑤 𝑗𝑗 (𝑥𝑥)�𝑔𝑔�𝑥𝑥 (𝑗𝑗) � -𝐶𝐶� 𝑁𝑁 𝑗𝑗=1 (1)
where C is the constant term, x(j) is a point in the training sample of size N, and wj (x) are weights. The determination of f by training is performed using the R module DiceKriging [START_REF] Roustant | DiceKriging, DiceOptim: Two R Packages for the Analysis of Computer Experiments by Kriging-Based Metamodeling and Optimization[END_REF]. In our case, N = 2048 simulations are used to build the metamodels in the normal diffusion mode and 2048 additional simulations are used for the low diffusion mode.

The emulators allow to evaluate very quickly the value of the maximum distance and the aperture of the emergency protection zones, for different dose thresholds and two stability classes. A graphical interface has been developed in order to geometrically represent these parameters, for a better visualization of the results.

Theoretically, with an infinite number of simulations used for their construction, the emulators give results identical to those of the model. But in practice, the number of simulations is limited and the emulators thus constructed can deviate from the model results. To check whether they remain reliable, a common approach is to evaluate the emulator on a test sample and compare the results with the initial model. To do this, we draw M new points in our five-dimensional input space and then run the pX model M times with input parameters corresponding to these draws. We then obtain M output results (maximum distances and angular apertures) which we can compare with what the emulators predict for the same parameter values. We thus evaluate the quality of our emulators at several points randomly distributed in space, which allows us to quantify the error committed by the emulator on cases it does not know. For this study, we consider a test sample of M = 1000 points.

CLASSIFICATION METHODS FOR THE IMPROVEMENT OF PREDICTIONS

The distances of threshold exceedance of protective action guide levels may be all zero for some values of the input variables and that creates a discontinuity that is difficult for emulators to reconstruct. To avoid this problem, we use a classification method to determine, prior to emulation, whether or not the distances are going to be zero, using a nearest k-neighbor method [START_REF] Altman | An Introduction to Kernel and Nearest Neighbor[END_REF]. For a given dose threshold, the input space of the model can be divided into two subspaces: a first one in which the distance is zero and a second one in which it is strictly positive. The purpose of the classification is to estimate whether a point P in the space is located in one or the other of these subspaces.

To carry out this classification, we have used the same N points that were used to build the emulator. By then constructing two classes, {𝑑𝑑 = 0} and {𝑑𝑑 > 0}, it is possible to assign to all the points in space one of these classes, depending on the state of its k-nearest neighbors. In our study, we have chosen 𝑘𝑘 = 5 and we have weighted the influence of each of its 5 points by the inverse of the distance separating it from the point to be evaluated. This distance is defined as the Euclidean distance between the points in a 5-dimensional space normalized so that the range of variation of each variable is between 0 and 1.

Once this classification has been done, the evaluation of whether a point in the space will give a distance of zero is almost instantaneous with this method. It can therefore be used upstream of the emulator to predict whether a result will be zero or whether it needs to be evaluated by an emulator.

The main advantage of this method is to be able to build an emulator only on the restricted space of the inputs where the distances are non-zero. Emulators constructed in this way are more accurate, because the points where the distances behave differently are removed for its construction. Then when we want to evaluate a point in space, we first determine whether the classification predicts that it will have a non-zero distance. If this is the case, the algorithm stops, otherwise the emulator allows us to evaluate the distance. However, the classification setup is not perfect and depends on the points that were used for its construction. It is possible that the classification gives erroneous results, especially in the area close to the interface between the two subspaces we are trying to define, as it can be seen in Figure 2 where several points are located on the lines y = 0 and x = 0. To evaluate the results of this classification, we use an uniform test sample of 1000 points to estimate the number of false positives and false negatives predicted by classification. With the combination of classification followed by emulation, there are still some false positive or negative cases (between 1.8 and 8.0%), but these numbers are much lower than when no classification was made (between 3.7 and 46.0%). Moreover, the evaluation of the emulator on areas where distances are non-zero is in most cases much better, as shown by the estimation errors in Table 1.

Table 1. The mean error, quantile 95% of the error, the number of cases where the emulators, associated with a knearest neighbors classification method or not, give a non-zero response when it should be at zero (false positive) and where it gives a zero response when it should be positive (false negative), on a test sample of 1000 simulations, for the threshold exceedance of Total Effective Dose (TED) and Inhalation thyroid dose (ITD). The error rate correspond to the sum of false positive and false negative divided by the number of simulations in the sample. It is important to identify which errors are acceptable and which are not in this context. An error of a few hundred meters over a maximum distance of 1 km may seem less acceptable than an identical error over a maximum distance of 15 km, but in terms of surface, the area of the zone where the emulator gives a bad prediction will be much smaller in the first. In practice, these values are not used as exact values, but it is rather their order of magnitude that determines the actual areas over which actions will be carried out (often over administrative entities). Therefore, the error committed on the estimation of distances must be smaller than the margins that are taken by decision-makers for safety reasons.

Emulator

To further reduce the number of false positives and false negatives in the classification, it is possible to use a classification by Gaussian process. Then the number of false points is still lower (between 0.3 and 4.3%), but the call to the classification is much slower, which makes the use of the emulator much less reactive: of the order of 2 seconds for an evaluation, compared to only 0.2 seconds with a classification by nearest neighbor. The first classification is thus more interesting in cases where a very large number of evaluations is to be expected.

CONCLUSION

Emulators allow to obtain a quasi-instantaneous estimate of the areas where protective action guide levels might be exceeded for a given accidental scenario and meteorology. As soon as a catalog of emulators is built on the basis of the "typical" accident scenarios, the production of expertise, and therefore the provision of technical advice for decision-makers, can be accelerated, particularly in the early phases of the crisis when uncertainties reign so much. Prior to this operational use, these emulators must be evaluated against test samples to verify their reliability and sometimes complemented by classification methods to increase their performance.

The first emulation phase has highlighted the main sources of inaccuracy. Upstream classification tools to predict a null output result or a 360° zone have greatly improved the output results of the emulators, but also sometimes slow down the emulator thus created.

One perspective to improve the results is to study a different parameterization of the output data, using dimension reduction methods, in order to reduce the dose maps to a few scalar parameters which can then be emulated. This would allow to avoid the use of thresholds and thus to avoid discontinuities related to threshold effects, but also to be able to go beyond the prediction of distance and aperture of a sector-shaped zone and predict 2D dose maps.

Figure 1 .

 1 Figure 1. Parameterization of an action zone by three quantities.

  (a) Emulator without any classification (b) Emulator associated with a classification by k-nearest neighbors

Figure 2 .

 2 Figure 2. Comparison between the maximum distances observed (pX simulations) and the values predicted by the emulator without any classification and associated with a classification by k-nearest neighbors, for the three dose thresholds considered, for a sample of 1000 test points, in the case of normal diffusion stability.

Stability Threshold Mean error Q95% of the error False positive False negative Error rate

	Emulator without any classification	Normal diffusion Low diffusion	TED ≥ 10 mSv TED ≥ 50 mSv ITD ≥ 50 mSv TED ≥ 10 mSv TED ≥ 50 mSv ITD ≥ 50 mSv	109 m 97 m 116 m 419 m 390 m 462 m	189 m 66 m 293 m 874 m 361 m 1391 m	38 233 36 368 460 399	1 3 1 0 0 1	3.9 % 23.6 % 3.7 % 36.8 % 46.0 % 40.0 %
	Emulator associated with a k-	Normal diffusion	TED ≥ 10 mSv TED ≥ 50 mSv ITD ≥ 50 mSv	53 m 60 m 42 m	142 m 78 m 230 m	11 45 50	9 35 4	2.0 % 8.0 % 1.8 %
	nearest neighbors classification	Low diffusion	TED ≥ 10 mSv TED ≥ 50 mSv ITD ≥ 50 mSv	101 m 414 m 67 m	646 m 222 m 492 m	18 20 18	26 27 25	4.4 % 4.7 % 4.3 %


L'artialisation cinématographique des conduites terroristes récentes à partir de Wanted (Bekmambetov, 2008) et d'Inception (Nolan, 2010)

Hugo Clémot L'artialisation désigne un processus complexe de construction sociale d'un phénomène où un médium artistique au moins joue un rôle actif. Évident dans le cas des conduites révolutionnaires récentes (masque Anonymous et printemps arabe, révolution des parapluies et signe des trois doigts des étudiants thaïlandais), un tel processus ne semble pas non plus sans rapport avec les attentats qui ont frappé ou failli frapper la France en 2015, à revoir des films comme Wanted (Bekmambetov, 2008) ou Inception (Nolan, 2010). Si le premier ne paraît pas pouvoir être rangé dans le genre de la science-fiction, on peut néanmoins soutenir que la présence de « techniques du corps » pour l'instant hors de portée y supplée à l'absence d'une possibilité plus évidemment scientifictive. En outre, la proximité du personnage principal et de sa trajectoire avec ce que l'on croit savoir du profil psychologique des tueurs de masse ne fait que renforcer l'aspect prémonitoire de la séquence du train, comme de la scène d'explosion d'une terrasse d'un café parisien dans Inception.

Pour déterminer comment des oeuvres cinématographiques peuvent se transformer en prophéties autoréalisatrices, on peut trouver des éléments de réponse dans ces deux films qui développent une authentique pensée spéculative, non seulement sur le scepticisme ontologique contemporain et sa relation à l'imagerie numérique, mais aussi sur la manière dont le cinéma de science-fiction peut contribuer malgré lui au fanatisme.

Science-fiction, artialisation cinématographique et conduites révolutionnaires récentes

Les propos prémonitoires du philosophe Jean Baudrillard quant aux attentats du World Trade Center1 sont connus, comme est connu le fait que l'un des films de science-fiction les plus importants de la fin du XXe siècle, The Matrix (1999) des Wachowski, est une adaptation cinématographique de l'un de ses ouvrages, Simulacres et simulation (1981). Si l'on se rappelle à quel point certaines des séquences du film semblent décrire des phénomènes contemporains (double vie de Néo, « recrutement » par internet, État libre des rebelles [« Zion »], destruction d'un building par un hélicoptère, etc.), et si l'on admet que l'oeuvre de Baudrillard relève elle-même davantage du genre de la science-fiction que de la philosophie, on comprend que la relation du cinéma de science-fiction au 11 septembre 2001 ne saurait être conçue seulement comme une relation causale où l'événement historique serait la cause antécédente et déterminante des changements du genre cinématographique, mais que les oeuvres cinématographiques en général et scientifictives en particulier ont pu jouer un rôle, dont l'importance exacte reste à déterminer, dans la préparation du 11 septembre 2001. Au-delà du principe d'une armée secrète qui ferait subir à ses membres des rites d'initiation et un entraînement particulièrement sévères, le « métier à tisser du destin » que l'on trouve dans la fabrique de textile où demeurent les membres de la Confrérie, invention scientifictive analogue aux « précogs » de Minority Report (Dick, 1956) et qui ne figure pas dans le comic book, fournit un élément d'analogie supplémentaire avec le modus operandi récent des tueurs se revendiquant de l'État islamique. En effet, ce métier à tisser est censé délivrer un message qui, une fois interprété, donne une liste de noms de personnes que les membres de la confrérie doivent assassiner afin de restaurer l'ordre du monde que ces victimes désignées sont censées menacer par leur existence. Dans la perspective fanatique de terroristes islamistes, le Coran est censé délivrer un message qui, une fois interprété, donne une liste de cibles que ses membres doivent assassiner afin de punir les crimes occidentaux.

Wanted (Bekmambetov, 2008); captures d'écran L'un des intérêts du texte de Mulhall est de mettre en évidence l'existence d'un rapport d'analogie entre la façon dont le métier à tisser, en donnant le nom des victimes, écrit le destin du monde et la façon dont l'imagerie numérique transforme le cinéma. S'il n'est pas possible de rentrer ici dans les détails, Mulhall en conclut que de même que l'interprète du métier à tisser, Sloan, le chef de la Confrérie, peut faire que la machine donne n'importe quel nom propre, de même peut-on simuler n'importe quelle donnée visuelle à l'aide de séquences de nombres. Les images numériquement créées n'ont en effet pas de rapport nécessaire avec les données visuelles captées l'objectif de la caméra, une opération de captation dont on peut d'ailleurs et pour cette raison se passer techniquement. Dans le film, lorsque Wesley prend conscience de la supercherie, sa réaction est d'assassiner Sloan pour prendre sa place et changer la réalité en la façonnant selon ses désirs. Mulhall interprète ce choix comme signifiant que, pour le spectateur de cinéma, l'hyperréalisme de l'image numérique a pour résultat « une certaine perte de foi dans le monde réel, une volonté de se retirer de tout engagement pour la réalité au profit d'un fantasme avec lequel cette réalité peut maintenant beaucoup plus facilement être confondue8 . » Autrement dit, la possibilité de contrôler entièrement les images renforcerait la perte de foi contemporaine dans le monde réel et encouragerait une fuite désespérée et absurde dans le fantasme délétère d'une maîtrise absolue du monde.

Ces éléments d'analyse peuvent être synthétisés selon le réseau conceptuel suivant : l'imagerie numérique constitue le médium privilégié de la diffusion massive des informations, c'est-à-dire du rapport au monde des individus contemporains. Or, le fait que l'on puisse entièrement contrôler le contenu d'une image numérique induit chez le spectateur un doute quant à la vérité des informations ainsi véhiculées. Ce doute sera d'autant plus grand que le spectateur se percevra davantage comme la victime d'un système dont il ne peut en revanche identifier les procédés. Ce scepticisme nourrit un désir de certitude délétère dans la mesure où la seule façon d'être sûr de reprendre le contrôle de sa vie semble alors de choisir quand y mettre un terme. Cependant, la même logique meurtrière peut être retournée contre le monde, puisque la seule façon d'être sûr que le monde ne pourra pas nous échapper est de le mettre à mort et, numérisant pour l'éternité cette mise à mort, de rendre potentiellement éternelle la revanche de la victime. On parle depuis longtemps, et bien avant Peeping Tom (1960) de Michael Powell, d'une violence exercée par la caméra qui, à suivre Eisenstein, arracherait un morceau de monde de son lieu naturel, pour en faire, selon André Bazin, un « masque mortuaire », que confirmerait Alfred Hitchcock quand il pense le cinéma comme un médium taxidermiste. Il n'est donc pas étonnant que la récente et rapide massification des caméras numériques s'accompagne d'une massification des pratiques médiatiques déviantes : vidéos de décapitations de l'EI, caméra « GoPro » de Merah, diffusion « Facebook Live » de Larossi Abballa… En bref, le réseau conceptuel trouvé dans le film Wanted part de l'imagerie numérique pour finalement y revenir : la manipulation numérique hyperréaliste des apparences renforce le scepticisme et induit un désir de contrôle du monde et des autres qui conduit au meurtre et à sa captation vidéo. Si cette description a quelque rapport avec la mentalité des tueurs de masse scolaires, peut-elle en revanche conserver une valeur dans le cas des terroristes fanatiques de l'EI ? 3 L'« inception » du fanatisme L'un des modèles pour penser l'artialisation cinématographique est celui de la contagion des idées : une idée serait comme un virus, biologique ou informatique, qui s'insinue à notre insu dans notre esprit à un niveau infrarationnel qui la rend ainsi très résistante aux démentis de l'expérience. Or, il est un film de science-fiction récent et populaire, construit sur ce raisonnement par analogie et qui porte aussi incontestablement sur le scepticisme, que Kant appelait le « scandale de la philosophie » : Inception (Nolan, 2010). Mulhall lui a consacré un long article9 , dense et complexe où il s'appuie notamment sur une remarque de Stanley Cavell pour faire un lien entre le scepticisme et le fanatisme tels qu'ils sont pensés dans le film.

On le sait : le scandale de la philosophie est qu'elle ne puisse pas prouver que le monde existe, qu'il n'est pas un simple rêve. C'est évidemment le ressort dramatique du film Inception qui utilise, après bien d'autres twist movies, ce vieil argument sceptique du rêve comme principe de ses retournements de situation. À suivre Kant, le scepticisme naîtrait d'un désir déçu d'atteindre l'inconditionné, c'est-à-dire le réel tel qu'il est en lui-même, indépendamment de la connaissance que nous pouvons en avoir, ce qu'on appelle encore le « donné » pur ou l'absolu. Mais, remarque Cavell dans un passage cité par Mulhall10 , si la déception de ce désir d'absolu doit provoquer ce doute sceptique illimité, absolu, alors, puisqu'il ne saurait y avoir moins de réalité dans la cause que dans l'effet, ce désir d'absolu doit lui-même être conçu comme sans limites et absolu, comme une passion pour l'idéal qui vide le monde de sa valeur et justifie qu'on le détruise. Autrement dit, si le scepticisme est la disposition d'esprit de celui qui doute absolument, sans limites, alors le fanatisme, en tant que disposition d'esprit de celui qui aime l'inconditionné absolument, sans limites, n'est que l'autre face du même phénomène psychologique.

Si la difficulté commune aux candidats au massacre scolaire et au massacre terroriste est l'impossibilité de vivre dans un monde qui ne leur réserve que le statut de victime, alors on peut dire que les seconds ont récemment trouvé dans les réseaux sociaux numériques l'idée que la seule vie absolument bonne et pure qui s'offre à eux est celle qui suivra leur mort en martyr, une forme de vie inconditionnée, dont la bonté est absolue et qui n'existe que dans le langage que leur tiennent ceux qui les vampirisent.

Ce choix du suicide par amour de l'inconditionné est, dans le film, celui effectué à deux reprises par Mal (Marion Cotillard) la femme de Cobb (Leonardo DiCaprio), lorsqu'elle préfère, d'abord, la vie potentiellement éternelle du niveau onirique des Limbes, où elle contrôle absolument l'espace et le temps, et lorsqu'elle se jette, ensuite, par la fenêtre d'une chambre d'hôtel, persuadée que cet « acte de foi » est la seule manière de quitter ce mauvais rêve afin de retrouver la réalité. Prendre au sérieux le phénomène de l'artialisation cinématographique des conduites révolutionnaires et terroristes récentes ne permet donc pas seulement, en envisageant la possibilité d'une action réciproque, d'enrichir notre conception des relations entre un événement historique comme le 11 septembre 2001 et un genre cinématographique particulier, celui de la science-fiction. Cela incite en outre à chercher dans certains films, comme Wanted ou Inception, d'authentiques pensées, propres à élargir le cadre théorique dans lequel nous nous efforçons de nous rapporter à des phénomènes contemporains comme la relation du scepticisme à l'imagerie numérique et au fanatisme. L'une de ces pensées consiste à reconnaître l'ambivalence des images animées, qui peuvent nous apprendre à contenir notre scepticisme en faisant du cinéma un instrument de révélation de la beauté et de la bonté de l'ordinaire mais qui peuvent aussi nous présenter un monde si parfait et si séduisant qu'il fait que l'existence ordinaire dans le monde réel paraît trop imparfaite et pénible pour mériter d'être vécue.

  Bangkok, le 1er juin 2014. (Photo Erik de Castro. Reuters) The Hunger Games (Gary Ross, 2012) ; captures d'écran Connu depuis longtemps, puisque Marcel Mauss n'utilise pas un autre exemple que le cinéma pour rendre compte des circonstances dans lesquelles l'idée célèbre de « techniques du corps » lui est venue 2 , ce phénomène de mimétisme cinématographique ne concerne malheureusement pas que les partisans des libertés publiques et de la démocratie : une réflexion semblable sur l'artialisation cinématographique des conduites terroristes à partir du cinéma populaire pourrait en effet s'appuyer sur nombre de ressemblances frappantes entre violence réelle et violence cinématographique. Pour ne s'en tenir qu'à deux exemples, la séquence récente du massacre d'Orlando aux États-Unis (11 juin 2016) suivi des meurtres du couple de policiers à leur domicile de Magnanville en France (13 juin 2016) rappelle la façon dont le Terminator (Cameron, 1984) fait irruption dans un night-club, attaque un commissariat et attend l'appel téléphonique de sa victime au domicile de sa mère, de même que le modus operandi de Mohammed Merah (mars 2012) semble imité de celui du personnage principal de son film préféré 3 : Faster (George Tillman Jr., 2010).

Faster

  (George Tillman Jr., 2010) ; expédition de Merah (mars 2012) ; captures d'écran 2 La trajectoire d'un tueur de masse selon Wanted : imagerie numérique, scepticisme et meurtre De nature interdisciplinaire, cette réflexion pourrait notamment s'appuyer sur une approche méconnue en France, la lecture philosophique des films, qui se distingue des autres en ce qu'elle préfère partir de ce qu'une étude attentive de certains films particuliers pourrait nous apprendre, plutôt que d'assujettir le cinéma aux problèmes philosophiques traditionnels. L'un des partisans de cette approche, Stephen Mulhall, a ainsi proposé une lecture du film Wanted 4 qui a pris une importance nouvelle depuis la tentative de massacre du Thalys du 21 août 2015. Ce projet d'attentat semble en effet imité de sa séquence la plus spectaculaire. À la lumière de ce mimétisme, la trajectoire du personnage principal de Wanted ressemble étonnamment à celle des tueurs de masse telle qu'elle ressort de l'étude des publications numériques sur YouTube des tueurs impliqués dans sept fusillades scolaires entre 1999 et 2015 5 , puisqu'il s'agissait, à travers leurs pratiques médiatiques déviantes, de passer d'une représentation de soi comme victime passive du système à celle d'un individu exceptionnel en tant qu'il est maître de son destin. Wesley Gibson va en effet passer du statut de victime passive, abandonné par son père dès son plus jeune âge, menant la vie monotone d'un petit comptable sans argent, martyrisé par sa chef, trompé par sa petite amie avec son soi-disant meilleur ami, au statut de tueur professionnel, membre d'une société secrète de maîtres du monde. Il semble d'ailleurs que l'état d'esprit des tueurs de masse soit bien décrit par l'adresse finale au spectateur, à savoir que la plupart des hommes sont des esclaves du « système » qui n'ont d'autre choix, pour reprendre en main leur existence, que de rejoindre une élite d'individus ultraviolents 6 . Wanted (Bekmambetov, 2008); captures d'écran De ce point de vue, il y aurait une continuité entre les massacres scolaires et l'engagement djihadiste des Occidentaux, « la radicalisation ne correspond[ant] pas tant à une manière de promouvoir des idéaux ou des aspirations politiques qu'à une façon de glorifier l'individualité, parfois à titre posthume, et d'adresser un message de revanche identitaire à un ennemi imaginaire 7 ».

  Inception (Nolan, 2010) ; captures d'écran Mais le sort de Mal ne fait pas qu'illustrer cinématographiquement la relation « interne », logique entre scepticisme et fanatisme. En effet, une attention supplémentaire aux circonstances dans lesquelles Mal en est venue à adopter cette perspective fanatique sur le monde se révèle instructive. Il est clair que le responsable du suicide de Mal à l'hôtel est Cobb, dans la mesure où, pour la faire quitter les limbes, il a dû lui implanter l'idée que ce que Mal prenait pour le réel n'était en fait qu'un rêve dont on ne peut sortir qu'en se suicidant, une « inception » qui a si bien fonctionné qu'elle l'a empêchée, une fois de retour dans le monde réel, de « s'y retrouver », c'està-dire de ne pas tenir le réel pour un rêve. Autrement dit, pour implanter cette idée dans l'esprit de Mal, Cobb a dû lui faire perdre le sens de la réalité en la privant des mots pour décrire son expérience du monde, afin de leur substituer des mots qui ne peuvent que dévaloriser l'existant au profit de l'idéal que l'on n'atteint que par la mort. Cependant, il est également possible que Cobb soit responsable de l'implantation du premier fanatisme de Mal en tant qu'architecte d'un espace-temps mental si parfait et si séduisant qu'il fait que l'existence ordinaire dans l'espace et le temps réels paraît trop imparfaite et pénible pour mériter d'être vécue. Si l'on accepte de tenir Inception, après The Prestige (2006), comme une nouvelle réflexion de Christopher Nolan sur les pouvoirs du cinéma, alors on pourra envisager que Cobb, l'expert en architecture mentale, soit le tenant-lieu du réalisateur qui s'interroge sur sa responsabilité de montreur d'images susceptibles d'encourager scepticisme comme fanatisme, mais qui pourraient aussi nous apprendre à vivre avec le scepticisme sans fanatisme.

  On peut repérer d'autres exemples historiques contemporains d'un semblable phénomène d'artialisation.

	Ainsi, la dystopie cinématographique V pour Vendetta (McTeigue, 2005), scénarisée et produite par les Wachowski à partir d'un comic book d'Alan Moore et de David Lloyd, n'a-t-elle pas été sans
	effets sur les révolutions du printemps arabe et sur le mouvement des Anonymous, ne serait-ce
	que par la reprise du masque de « V ». De même, les étudiants de Hong Kong ont baptisé leur
	mouvement « la révolution des parapluies » en une référence volontaire au film Sparrow (2008) du réalisateur Johnny To. Enfin, la révolte thaïlandaise contre le coup d'État militaire de mai 2014 s'est approprié le signe des trois doigts du film Hunger Games (G. Ross, 2012).

Voir Jean Baudrillard, L'Échange symbolique et la mort, Paris, Gallimard, 1976, p. 108-109 sur les deux tours du World Trade Center et p. 63-66 sur « l'efficacité symbolique » des attentats terroristes.

Marcel Mauss, Sociologie et anthropologie, Paris, PUF, 1966, p.

3 Farhad Khosrokhavar, Radicalisation, Paris, Éditions de la Maison des sciences de l'homme, collectionInterventions, 2014, p. 108. 4 Stephen Mulhall, "The Loom of Fate: Graphic Origins and Digital Ontology in Wanted," in New Takes in Philosophy ofFilm, Havi Carel et Greg Tuck (dir.), Basingstoke, Palgrave Macmillan, 2011, p. 103-116.

Nathalie Paton, School shooting. La violence à l'ère de YouTube, Paris, Éditions de la Maison des sciences de l'homme, 2015.

Mark Millar & J.G. Jones, Wanted, Top Cow Productions, 2005; tr. fr. Nick Meylaender, Guy Delcourt Productions, 2008, p. 136-137.

Voir Nathalie Paton, « Radicalisation et injonctions à l'individuation », Mondes sociaux. Magazine de sciences humaines et sociales, 17 mai 2016, https://sms.hypotheses.org/7707 (consulté le 17 février 2017).

Mulhall, "The Loom of Fate", op. cit., p. 113.

Mulhall, "Sharing a Dream of Skepticism: Parasitism, Plagiarism and Fanaticism in Christopher Nolan's Inception", The Harvard Review ofPhilosophy, vol. XIX, 2013, p. 118-136.

Stanley Cavell, Disowning Knowledge in Seven Plays of Shakespeare, Cambridge, Cambridge University Press, updated edition, 2003 (1987), p. 18.