Homogenization of a 1D pursuit law with delay

Transcription

1 Homogenization of a 1D pursuit law with delay Jeremy Firozaly (Joint work with R. Monneau) Paris-Est University June 2, 2016 J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

2 Traffic flow modelling: microscopic scale Figure : View of the microsopic scale considered in our study. J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

3 Traffic flow modelling: macroscopic scale Figure : Schematic view of the global traffic flow around Paris from sytadin.fr. J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

4 Homogenization process: micro-macro passage Figure : Schematic description of the homogenization process. Huge reaction times accordion effect (see video) J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

5 Pursuit law Microscopic dynamics (DDE): dx i dt (t) = F (X i+1(t τ) X i (t τ)) with initial condition: (i, t) Z (0, + ) X i (t) = x 0 i (t) t [ 2τ, 0] J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

6 Rescaled equation and expected limit Hyperbolic rescaling: Microscopic model embedded into: { t u ε (x, t) = F u ε (x, t) := εx x ε ( t ε ) ( u ε (x+ε,t ετ) u ε (x,t ετ) ε ) (x, t) R (0, + ) u ε (x, s) = u 0 (x, s) (x, s) R [ 2ετ, 0] Expected macroscopic model for vanishing ε: { t u 0 (x, t) = F ( x u 0 (x, t)) (x, t) R (0, + ) u 0 (x, 0) = u 0 (x, 0) x R J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

7 Outline 1 Introduction 2 Convergence theorem 3 Strict comparison principle 4 Convergence proof J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

8 Introduction Some references Microscopic traffic flow models [Chandler, Hermann, Montroll; (1958)] [Gipps; (1981)] [Costeseque; (2014)] Macroscopic traffic flow models [Lighthill, Whitham; (1955)] [Richards; (1956)] Periodic homogenization for system of ODEs [Forcadel, Imbert, Monneau; (2007)] [Forcadel, Imbert, Monneau; (2008)] Homogenization with a junction condition [Forcadel, Salazar; (2014)] Homogenization of second order models [Forcadel, Salazar; (2015)] J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

9 Introduction Existence and Uniqueness of solutions Assumptions (B0) F is: non-decreasing bounded C F Lipschitz (B1) u 0 is L Lipschitz Microscopic model (ε singular DDE): Incremental construction Macroscopic model (HJ equation): Perron method and comparison principle J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

10 Convergence theorem Convergence theorem Theorem [F-Monneau] For τ ( ) 1 0; ec F and under assumptions (B0)-(B1), u ε converges locally uniformly towards u 0. The proof relies on a strict comparison principle stated on: { t u(x, t) = F (u(x + 1, t τ) u(x, t τ)) (x, t) R (0, + ) u(x, s) = u 0 (x, s) (x, s) R [ 2τ; 0] J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

11 Strict comparison principle Strict comparison principle Theorem [F-Monneau] Under (B0)-(B1), let v be a supersolution and u a subsolution. If: 0 < δ (v u)(x, t τ ) ρ(τ )(v u)(x, t) τ [0, τ], (x, t) R [ τ, 0] where ρ is non decreasing and such that: Then we have: τ 1 + C F ρ(τ) ρ(s)ds < ρ(τ ) τ [0, τ] 0 0 < δe C F ρ(τ)t v(x, t) u(x, t) (x, t) R [0, + ) J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

12 Strict comparison principle Restriction on τ Lemma There exist non decreasing ρ such that: if and only if τ < 1 ec F. τ 1 + C F ρ(τ) ρ(s)ds < ρ(τ ) τ [0, τ] 0 The proof uses comparison principles on affine differential inequalities. The restriction on τ is not sufficient for the strict comparison principle to hold. Vehicles must be suitably spaced out. J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

13 Strict comparison principle Proof of strict comparison principle (sketch) We want to prove that: 0 < δe C F ρ(τ)t d(x, t) := v(x, t) u(x, t) (x, t) R [0, + ) T := sup{s 0/d(x, t τ ) ρ(τ )d(x, t), τ [0, τ], x R, t [ τ, S]} We show by contradiction that T = +. By definition of d, as F is C F Lipschitz and T = + : t d(x, t) C F d(x, t τ) C F ρ(τ)d(x, t) J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

14 Strict comparison principle Let us show that T = + : By contradiction, if not: Preliminary result: β (0, 1), τ β [0, τ], (x β, t β ) R (T, T + β] : d(x β, t β τ β ) > ρ(τ β )d(x β, t β ) d(x, t τ ) ρ(τ )d(x, t) τ [0, τ], (x, t) R [0, T ] with ρ given by: τ ρ(τ ) = 1 + C F ρ(τ) ρ(s)ds (< ρ(τ )) 0 Proof based on classical comparison principle for the ODE: τ W = C F ρ(τ)ρ(τ )W J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

15 Strict comparison principle Case 1: t β τ β T. d(x β, t β τ β ) = d(x β, T τ β ) with τ β := T t β + τ β [0, τ] Use of preliminary result: d(x β, t β τ β ) ρ(τ β )d(x β, T ) Moreover, d satisfies: d(x, t) d(x, s) 2 F (t s) (x, t) R [0, + ), s [0, t] Hence: d(x β, t β τ β ) ρ(τ β )(d(x β, t β ) + 2 F β) J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

16 Strict comparison principle Then, β < β 1 := δ 2 F +1 e C F ρ(τ)t > 0 implies: d(x β, t β ) d(x β, T ) 2 F β δe C F ρ(τ)t 2 F β 1 = β 1 By considering β < β 2 := β 1 2 F inf [0,τ] ( ρ ρ 1), we conclude that: d(x β, t β τ β ) ρ(τ β )d(x β, t β ) and using the fact that ρ is non decreasing and τ β τ β this contradicts d(x β, t β τ β ) > ρ(τ β )d(x β, t β ) J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

17 Strict comparison principle Case 2: t β τ β > T. This implies that τ β [0, β]. By using the fact that d is semi-lipschitz and we get: d(x β, t β τ β ) > ρ(τ β )d(x β, t β ), (ρ(τ β ) 1)d(x β, T ) < 2 F ρ(τ β )β By using the result that came from Gronwall s lemma, we finally get: 0 < δe C F ρ(τ)t < ρ(τ β) ρ(τ β ) 1 2 F β which implies δ = 0 for vanishing β. J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

18 Strict comparison principle Counter-example Here we choose F such that F = Id in [0, 1]. Let n 0 N such that τ > 2 n 0. Given two sets of drivers (X i ) i Z and (Y i ) i Z such that there exists j Z such that for t [ τ, 0]: y 0 j (t) = j 1 + n 0 n 0 +1 en 0t < x 0 j (t) = j y 0 j+1 (t) = j + n 0 n 0 +1 en 0t < x 0 j+1 (t) = y 0 j+1 (t) + 1 n 0 +1 By direct integration, we find: (Y j X j )(τ) = n 0 n τ 1 n e n0τ n > 0 J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

19 Strict comparison principle Figure : Initial dynamics for the counter-example. J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

20 Convergence proof Convergence proof (sketch) 1 Lipschitz estimates and barriers: u ε is F Lipschitz in time. Strict comparison principle: u ε is 2L Lipschitz in space. Barriers: u ε (x, t) u 0 (x, t) ( F + L)t. 2 (Classical) use of relaxed semi-limits ū and u: If ū is a subsolution of the HJ equation (and u is a supersolution) then by comparison principle: ū u 0 u = convergence J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

21 Convergence proof ū is a subsolution: By contradiction, if not, there exist ( x, t), ϕ C 1 x,t, (r, θ, η) (0, + ) 3 such that: and, ū < ϕ in B 2r ( x, t) \ {( x, t)} ū( x, t) = ϕ( x, t) ū ϕ 2η in B 2r ( x, t) \ B r ( x, t) t ϕ( x, t) = 2θ + F ( x ϕ( x, t)) with B r (y, s) := (y r, y + r) (s r, s + r) J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

22 Convergence proof We define: For ε small enough: M ε := max B 2r ( x, t) (uε ϕ) := (u ε ϕ)(x ε, t ε ) (x ε, t ε ) B r ( x, t) For ϕ ε := ϕ + M ε, we have (for ε small enough): ϕ ε (x ε t ε ) = u ε (x ε, t ε ) ( t ϕ ε (x, t) F ϕ ε (x+ε,t ετ) ϕ ε (x,t ετ) ε ) + θ 2 in B r ( x, t) Use of localised strict comparison principle: ϕ ε > u ε in B r ( x, t) Contradiction with ϕ ε (x ε, t ε ) = u ε (x ε, t ε ) J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23

23 Convergence proof Thank you for your attention J. Firozaly (Paris-Est University) Homogenization of a delay equation June 2, / 23