Contribution à l'étude du trafic routier sur réseaux à l'aide des équations d'Hamilton-Jacobi

Contribution à l’étude du trafic routier sur réseaux
à l’aide des équations d’Hamilton-Jacobi
Guillaume Costeseque
(with supervisors R. Monneau & J-P. Lebacque)
Université Paris Est, Ecole des Ponts ParisTech & IFSTTAR
PhD defense, Champs-sur-Marne,
September 12, 2014
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks Champs, Sept. 12, 2014 1 / 86

Motivation
Traﬃc ﬂows on a network

Motivation
Traﬃc ﬂows on a network
Road network ≡ graph made of edges and vertices

Motivation
Breakthrough in traffic monitoring
Traffic monitoring
“old”: loop detectors at fixed locations (Eulerian)
“new”: GPS devices moving within the traffic (Lagrangian)
Data assimilation of Floating Car Data
[Mobile Millenium, 2008]

Motivation
Outline
1 Introduction to traﬃc
2 Micro to macro in traﬃc models
3 Variational principle applied to GSOM models
4 HJ equations on a junction
5 Conclusions and perspectives

Introduction to traﬃc
Outline

Introduction to traﬃc Microscopic models
Notations: microscopic
Speed
Time
(i − 1)Space
x
Spacing
Headway
t
(i) vehicle position xi (t),
instantaneous speed ˙xi (t),
acceleration ¨xi (t),
spacing
Si (t) = xi+1(t) − xi (t) > 0
relative speed ˙Si (t)
...

Introduction to traffic Microscopic models
Microscopic models
Proposition (Models classification)
Assume the following classification of microscopic models
General expression Examples
(1) ˙xi (t) = F(Si (t)) Pipes; Forbes
(2) ˙xi (t + τ) = F(Si (t), ˙xi (t)) Chandler; Gazis; Newell
(3) ˙xi (t + τ) = F(Si (t), ˙xi (t), ˙xi−1(t)) Kometani; Gipps; Krauss
(4) ¨xi (t + τ) = F(Si (t), ˙Si (t), ˙xi (t), ¨xi (t)) Bando; Helly; Treiber
where τ is a time delay.

Introduction to traﬃc Macroscopic models
Convention for vehicle labeling
N
x
t
Flow

Notations: macroscopic
N(t, x) vehicle label at (t, x)
the ﬂow Q(t, x) = lim
∆t→0
N(t + ∆t, x) − N(t, x)
∆t
,
x
N(x, t ± ∆t)
the density ρ(t, x) = lim
∆x→0
N(t, x) − N(t, x + ∆x)
∆x
,
x
∆x
N(x ± ∆x, t)
the stream speed (mean spatial speed) V (t, x).

Macroscopic models
Hydrodynamics analogy
Two main categories: first and second order models
Two common equations:



∂tρ(t, x) + ∂x Q(t, x) = 0 conservation equation
Q(t, x) = ρ(t, x).V (t, x) definition of flow speed
(1)
x x + ∆x
ρ(x, t)∆x
Q(x, t)∆t Q(x + ∆x, t)∆t

Introduction to traﬃc Focus on LWR model
First order: the LWR model
Scalar one dimensional conservation law
ρt(t, x) + Fx(ρ(t, x)) = 0 (2)
[Lighthill and Whitham, 1955], [Richards, 1956]

Three representations of traﬃc ﬂow
Moskowitz’ surface
Flow
x
t
N
x
See also [Makigami et al, 1971], [Laval and Leclercq, 2013]

Overview: conservation laws (CL) / Hamilton-Jacobi (HJ)
Eulerian Lagrangian
t − x t − n
Variable (CL) Density ρ Spacing r
Equation (CL) ∂tρ + ∂x F(ρ) = 0 ∂tr + ∂x V (r) = 0
Variable (HJ) Label N Position X
N(t, x) =
+∞
x
ρ(t, ξ)dξ X(t, n) =
+∞
n
r(t, η)dη
Equation (HJ) ∂tN + H (∂x N) = 0 ∂tX + V (∂x X) = 0
Hamiltonian H(p) = −F(−p) V(p) = −V (−p)

Fundamental diagram (FD)
Flow-density fundamental diagram F : ρ → ρI(ρ)
Empirical function with
ρmax the maximal or jam density,
ρc the critical density
Flux is increasing for ρ ≤ ρc: free-ﬂow phase
Flux is decreasing for ρ ≥ ρc: congestion phase
ρmax
Density, ρ
ρmax
Density, ρ
0
Flow, F
0
Flow, F
0 ρmax
Flow, F
Density, ρ
[Garavello and Piccoli, 2006]

Introduction to traﬃc Second order models
Motivation for higher order models
Experimental evidences
fundamental diagram: multi-valued in congested case
[S. Fan, U. Illinois], NGSIM dataset

Motivation for higher order models
Experimental evidences
fundamental diagram: multi-valued in congested case
phenomena not accounted for: bounded acceleration, capacity drop...
Need for models able to integrate measurements of diﬀerent traﬃc
quantities (acceleration, fuel consumption, noise)

GSOM family [Lebacque, Mammar, Haj-Salem 2007]
Generic Second Order Models (GSOM) family



∂tρ + ∂x (ρv) = 0 Conservation of vehicles,
∂t(ρI) + ∂x (ρvI) = ρϕ(I) Dynamics of the driver attribute I,
v = I(ρ, I) Fundamental diagram,
(3)
Speciﬁc driver attribute I
the driver aggressiveness,
the driver origin / destination,
the vehicle class,
...
Flow-density fundamental diagram
F : (ρ, I) → ρI(ρ, I).

Micro to macro in traﬃc models
Outline

Micro to macro in traﬃc models Homogenization
Setting
Let i be the discrete position index
Assume a pseudo continuum of vehicles
Deﬁnition of continuous variables n and t
n = iε and t = εs

Setting
Let i be the discrete position index
Assume a pseudo continuum of vehicles
Deﬁnition of continuous variables n and t
n = iε and t = εs
Proposition (Rescaling)
Let consider
xi (s) =
1
ε
Xε
(εs, iε) (4)
with ε > 0 representing a scale factor such that Xε describes the rescaled
position of the vehicle. It implies that
Xε
(t, n) = ε.x⌊n
ε
⌋
t
ε
(5)

General result
Let consider
the simplest microscopic model;
˙xi (t) = F(xi−1(t) − xi (t)) (6)
the LWR macroscopic model (HJ equation in Lagrangian):
∂tX0
= F(−∂nX0
) (7)

General result
Let consider
the simplest microscopic model;
˙xi (t) = F(xi−1(t) − xi (t)) (6)
the LWR macroscopic model (HJ equation in Lagrangian):
∂tX0
= F(−∂nX0
) (7)
Theorem (Convergence to the viscosity solution)
If Xε(t, n) := εx⌊n
ε
⌋
t
ε
with (xi )i∈Z solution of (6) and X0 the unique
solution of HJ (7), then under suitable assumptions,
|Xε
− X0
|L∞(K) −→
ε→0
0, ∀K compact set.

Micro to macro in traﬃc models Multi-anticipative traﬃc
Toy model
Vehicles consider m ≥ 1 leaders
First order multi-anticipative model
˙xi (t + τ) = max

0, Vmax −
m
j=1
f (xi−j(t) − xi (t))

 (8)
f speed-spacing function
non-negative
non-increasing
Speed-spacing Fundamental Diagram
F {xk − xi }i−1≥k≥i−m := Vmax −
m
j=1
f (xi−j(t) − xi (t))

Homogenization
Proposition ((R. Monneau) Convergence)
Assume m ≥ 1 ﬁxed.
If τ is small enough, if Xε(t, n) := εx⌊n
ε
⌋
t
ε
with (xi )i∈Z solution of (8)
and if X0(n, t) solves
∂tX0
= ¯V −∂nX0
, m (9)
with
¯V (r, m) = max

0, Vmax −
m
j=1
f (jr)

 ,
then under suitable assumptions,
|Xε
− X0
|L∞(K) −→
ε→0
0, ∀K compact set.

Multi-anticipatory macroscopic model
χj the fraction of j-anticipatory vehicles
traffic flow ≡ superposition of traffic of j-anticipatory vehicles
χ = (χj )j=1,...,m , with 0 ≤ χj ≤ 1 and
m
j=1
χj = 1
GSOM model with driver attribute I = χ



∂tρ + ∂x (ρv) = 0,
v :=
m
j=1
χj
¯V (1/ρ, j) = W (1/ρ, χ),
∂t(ρχ) + ∂x (ρχv) = 0.
(10)

Micro to macro in traﬃc models Numerical schemes
Numerical resolution
Godunov-like schemes with (∆t, ∆xk) steps ⇒ CFL condition
Variational formulation and dynamic programming techniques [2]
Particle methods in the Lagrangian framework (t, n)
xt+1
n = xt
n + ∆tW (rt
n, χt
n)
rt
n =
xt
n−1 − xt
n
∆n
χt+1
n = χt
n
(11)

Numerical example
0 5 10 15 20 25 30 35 40
0
5
10
15
20
25
Spacing r (m)
Speed−spacing diagrams
Speedv(m/s)
1
2
3
4
5

Numerical example: (χj)j

Numerical example: Lagrangian trajectories

Numerical example: Eulerian trajectories

Variational principle applied to GSOM models
Outline

Variational principle applied to GSOM models LWR model
LWR in Eulerian (t, x)
Cumulative vehicles count (CVC) or Moskowitz surface N(t, x)
f = ∂tN and ρ = −∂x N
If density ρ satisfies the scalar (LWR) conservation law
∂tρ + ∂x F(ρ) = 0
Then N satisfies the first order Hamilton-Jacobi equation
∂tN − F(−∂x N) = 0 (12)

Legendre-Fenchel transform with F concave (relative capacity)
M(q) = sup
ρ
[F(ρ) − ρq]
M(q)
u
w
Density ρ
q
q
Flow F
w u
q
Transform M
−wρmax

(continued)
Lax-Hopf formula (representation formula) [Daganzo, 2006]
N(T, xT ) = min
u(.),(t0,x0)
T
t0
M(u(τ))dτ + N(t0, x0),
˙X = u
u ∈ U
X(t0) = x0, X(T) = xT
(t0, x0) ∈ J
(13) Time
Space
J
(T, xT )˙X(τ)
(t0, x0)
Viability theory [Claudel and Bayen, 2010]

(Historical note)
Dynamic programming [Daganzo, 2006] for triangular FD
(u and w free and congested speeds)
Flow, F
w
u
0 ρmax
Density, ρ
u
x
w
t
Time
Space
(t, x)
Minimum principle [Newell, 1993]
N(t, x) = min N t −
x − xu
u
, xu ,
N t −
x − xw
w
, xw + ρmax(xw − x) ,
(14)

LWR in Lagrangian (n, t)
Consider X(t, n) the location of vehicle n at time t ≥ 0
v = ∂tX and r = −∂nX
If the spacing r := 1/ρ satisfies the LWR model (Lagrangian coord.)
∂tr + ∂nV(r) = 0
with the speed-spacing FD V : r → I (1/r) ,
Then X satisfies the first order Hamilton-Jacobi equation
∂tX − V(−∂nX) = 0. (15)

LWR in Lagrangian (n, t)
(continued)
Dynamic programming for triangular FD
1/ρcrit
Speed, V
u
−wρmax
Spacing, r
1/ρmax
−wρmax
n
t
(t, n)
Time
Label
Minimum principle ⇒ car following model [Newell, 2002]
X(t, n) = min X(t0, n) + u(t − t0),
X(t0, n + wρmax (t − t0)) + w(t − t0) .
(16)

Variational principle applied to GSOM models GSOM family
GSOM in Lagrangian (n, t)
From [Lebacque and Khoshyaran, 2013], GSOM in Lagrangian



∂tr + ∂N v = 0 Conservation of vehicles,
∂tI = ϕ(N, I, t) Dynamics of I,
v = W(N, r, t) := V(r, I(N, t)) Fundamental diagram.
(17)
Position X(N, t) :=
t
−∞
v(N, τ)dτ satisﬁes the HJ equation
∂tX − W(N, −∂N X, t) = 0, (18)
And I(N, t) solves the ODE
∂tI(N, t) = ϕ(N, I, t),
I(N, 0) = i0(N), for any N.

(continued)
Legendre-Fenchel transform of W according to r
M(N, c, t) = sup
r∈R
{W(N, r, t) − cr}
M(N, p, t)
pq
W(N, q, t)
W(N, r, t)
q r
p
p
u
c
Transform M

(continued)
Lax-Hopf formula
X(NT , T) = min
u(.),(N0,t0)
T
t0
M(N, u, t)dt + ξ(N0, t0),
˙N = u
u ∈ U
N(t0) = N0, N(T) = NT
(N0, t0) ∈ J
(19)

(continued)
Optimal trajectories = characteristics
˙N = ∂r W(N, r, t),
˙r = −∂N W(N, r, t),
(20)
System of ODEs to solve
Diﬃculty: not straight lines in the general case

Variational principle applied to GSOM models Methodology
General ideas
First key element: Lax-Hopf formula
Computations only for the characteristics
X(NT , T) = min
(N0,r0,t0)
T
t0
M(N, ∂r W(N, r, t), t)dt + ξ(N0, t0),
˙N(t) = ∂r W(N, r, t)
˙r(t) = −∂NW(N, r, t)
N(t0) = N0, r(t0) = r0, N(T) = NT
(N0, r0, t0) ∈ K
(21)
K is the set of initial/boundary values

Variational principle applied to GSOM models Methodology
General ideas
(continued)
Second key element: inf-morphism prop. [Aubin et al, 2011]
Consider a union of sets (initial and boundary conditions)
K =
l
Kl ,
then the global minimum is
X(NT , T) = min
l
Xl (NT , T), (22)
with Xl partial solution to sub-problem Kl .

Variational principle applied to GSOM models Numerical example
Fundamental Diagram and Driver Attribute
0 20 40 60 80 100 120 140 160 180 200
0
500
1000
1500
2000
2500
3000
3500
Density ρ (veh/km)
FlowF(veh/h)
Fundamental diagram F(ρ,I)
0 5 10 15 20 25 30 35 40 45 50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Label N
Initial conditions I(N,t
0
)
DriverattributeI
0

Initial and Boundaries Conditions
0 5 10 15 20 25 30 35 40 45 50
5
10
15
20
25
30
35
40
45
50
Label N
Initial conditions r(N,t
0
)
Spacingr0
(m)
0 5 10 15 20 25 30 35 40 45 50
−1400
−1200
−1000
−800
−600
−400
−200
0
Label N
PositionX(m)
Initial positions X(N,t
0
)
0 20 40 60 80 100 120
12
14
16
18
20
22
24
26
28
30
Time t (s)
Spacing r(N
0
,t)
Spacingr
0
(m.s
−1
)
0 20 40 60 80 100 120
0
500
1000
1500
2000
2500
Time t (s)
PositionX0
(m)
Position X(N
0
,t)

Numerical result (Initial cond. + ﬁrst traj.)

Numerical result (Initial cond. + ﬁrst traj.)
0 20 40 60 80 100 120
−1500
−1000
−500
0
500
1000
1500
2000
2500
Location(m)
Time (s)
Vehicles trajectories

Numerical result (Initial cond.+ 3 traj.)

Numerical result (Initial cond. + 3 traj.)
0 20 40 60 80 100 120
−1500
−1000
−500
0
500
1000
1500
2000
2500
Location(m)
Time (s)
Vehicles trajectories

Numerical result (Initial cond. + 3 traj. + Eulerian data)

HJ equations on a junction
Outline

HJ equations on a junction
Motivation
Classical approaches:
Macroscopic modeling on (homogeneous) sections
Coupling conditions at (pointwise) junction
For instance, consider



ρt + (Q(ρ))x = 0, scalar conservation law,
ρ(., t = 0) = ρ0(.), initial conditions,
ψ(ρ(x = 0−, t), ρ(x = 0+, t)) = 0, coupling condition.
(23)
See Garavello, Piccoli [3], Lebacque, Khoshyaran [6] and Bressan et al. [1]

HJ equations on a junction HJ junction model
Star-shaped junction
JN
J1
J2
branch Jα
x
x
0
x
x

Junction model
Proposition (Junction model [IMZ, ’13])
That leads to the following junction model (see [5])



uα
t + Hα(uα
x ) = 0, x > 0, α = 1, . . . , N
uα = uβ =: u, x = 0,
ut + H(u1
x , . . . , uN
x ) = 0, x = 0
(24)
with initial condition uα(0, x) = uα
0 (x) and
H(u1
x , . . . , uN
x ) = max
α=1,...,N
H−
α (uα
x )
from optimal control
.

Basic assumptions
For all α = 1, . . . , N,
(A0) The initial condition uα
0 is Lipschitz continuous.
(A1) The Hamiltonians Hα are C1(R) and convex such that:
p
H−
α (p) H+
α (p)
pα
0

HJ equations on a junction Numerical scheme
Presentation of the scheme
Proposition (Numerical Scheme)
Let us consider the discrete space and time derivatives:
pα,n
i :=
Uα,n
i+1 − Uα,n
i
∆x
and (DtU)α,n
i :=
Uα,n+1
i − Uα,n
i
∆t
Then we have the following numerical scheme:



(DtU)α,n
i + max{H+
α (pα,n
i−1), H−
α (pα,n
i )} = 0, i ≥ 1
Un
0 := Uα,n
0 , i = 0, α = 1, ..., N
(DtU)n
0 + max
α=1,...,N
H−
α (pα,n
0 ) = 0, i = 0
(25)
With the initial condition Uα,0
i := uα
0 (i∆x).
∆x and ∆t = space and time steps satisfying a CFL condition

HJ equations on a junction Numerical scheme
CFL condition
The natural CFL condition is given by:
∆x
∆t
≥ sup
α=1,...,N
i≥0, 0≤n≤nT
|H′
α(pα,n
i )| (26)

HJ equations on a junction Mathematical results
Gradient estimates
Theorem (Time and Space Gradient estimates)
Assume (A0)-(A1). If the CFL condition (26) is satisﬁed, then we have
that:
(i) Considering Mn = sup
α,i
(DtU)α,n
i and mn = inf
α,i
(DtU)α,n
i , we have the
following time derivative estimate:
m0
≤ mn
≤ mn+1
≤ Mn+1
≤ Mn
≤ M0
(ii) Considering pα
= (H−
α )−1(−m0) and pα = (H+
α )−1(−m0), we have
the following gradient estimate:
pα
≤ pα,n
i ≤ pα, for all i ≥ 0, n ≥ 0 and α = 1, ..., N
Proof

Stronger CFL condition
−m0
pα
p
Hα(p)
pα
As for any α = 1, . . . , N, we have that:
pα
≤ pα,n
i ≤ pα for all i, n ≥ 0
Then the CFL condition becomes:
∆x
∆t
≥ sup
α=1,...,N
pα∈[pα
,pα]
|H′
α(pα)| (27)

Existence and uniqueness
(A2) Technical assumption (Legendre-Fenchel transform)
Hα(p) = sup
q∈R
(pq − Lα(q)) with L′′
α ≥ δ > 0, for all index α

Existence and uniqueness
(A2) Technical assumption (Legendre-Fenchel transform)
Hα(p) = sup
q∈R
(pq − Lα(q)) with L′′
α ≥ δ > 0, for all index α
Theorem (Existence and uniqueness [IMZ, ’13])
Under (A0)-(A1)-(A2), there exists a unique viscosity solution u of (24) on
the junction, satisfying for some constant CT > 0
|u(t, y) − u0(y)| ≤ CT for all (t, y) ∈ JT .
Moreover the function u is Lipschitz continuous with respect to (t, y).

Convergence
Theorem (Convergence from discrete to continuous [CML, ’13])
Assume that (A0)-(A1)-(A2) and the CFL condition (27) are satisﬁed.
Then the numerical solution converges uniformly to u the unique viscosity
solution of (24) when ε → 0, locally uniformly on any compact set K:
lim sup
ε→0
sup
(n∆t,i∆x)∈K
|uα
(n∆t, i∆x) − Uα,n
i | = 0
Proof

HJ equations on a junction Traﬃc interpretation
Setting
J1
JNI
JNI +1
JNI +NO
x < 0 x = 0 x > 0
Jβ
γβ Jλ
γλ
NI incoming and NO outgoing roads

Vehicle densities
The car density ρα solves the LWR equation on branch α:
ρα
t + (Qα
(ρα
))x = 0
By deﬁnition
ρα
= γα
∂x Uα
on branch α
And
uα(t, x) = −Uα(−x, t), x > 0, for incoming roads
uα(t, x) = −Uα(t, x), x > 0, for outgoing roads
where the vehicle index uα solves the HJ equation on branch α:
uα
t + Hα
(uα
x ) = 0, for x > 0
Setting

Flow
Hα(p) :=



−
1
γα
Qα(γαp) for α = 1, ..., NI
−
1
γα
Qα(−γαp) for α = NI + 1, ..., NI + NO
Incoming roads Outgoing roads
ρcrit
γα
ρmax
γα
p
−
Qmax
γα
p
−
Qmax
γα
HαHα
H−
α H−
α H+
αH+
α
−
ρmax
γα
−
ρcrit
γα

Links with “classical” approach
Deﬁnition (Discrete car density)
The discrete vehicle density ρα,n
i with n ≥ 0 and i ∈ Z is given by:
ρα,n
i :=



γαpα,n
|i|−1 for α = 1, ..., NI , i ≤ −1
−γαpα,n
i for α = NI + 1, ..., NI + NO, i ≥ 0
(28)
J1
JNI
JNI +1
JNI +NO
x < 0 x > 0
−2
−1
2
1
0
−2
−2
−1
−1
1
1
2
2
Jβ
Jλ
ρλ,n
1

Traﬃc interpretation
Proposition (Scheme for vehicles densities)
The scheme deduced from (25) for the discrete densities is given by:
∆x
∆t
{ρα,n+1
i − ρα,n
i } =



Fα(ρα,n
i−1, ρα,n
i ) − Fα(ρα,n
i , ρα,n
i+1) for i = 0, −1
Fα
0 (ρ·,n
0 ) − Fα(ρα,n
i , ρα,n
i+1) for i = 0
Fα(ρα,n
i−1, ρα,n
i ) − Fα
0 (ρ·,n
0 ) for i = −1
With



Fα(ρα,n
i−1, ρα,n
i ) := min Qα
D(ρα,n
i−1), Qα
S (ρα,n
i )
Fα
0 (ρ·,n
0 ) := γα min min
β≤NI
1
γβ
Qβ
D(ρβ,n
0 ), min
λ>NI
1
γλ
Qλ
S (ρλ,n
0 )
incoming outgoing
ρλ,n
0ρβ,n
−1ρβ,n
−2 ρλ,n
1
x
x = 0

Supply and demand functions
Remark
It recovers the seminal Godunov scheme with passing ﬂow = minimum
between upstream demand QD and downstream supply QS.
Density ρ
ρcrit ρmax
Supply QS
Qmax
Density ρ
ρcrit ρmax
Flow Q
Qmax
Density ρ
ρcrit
Demand QD
Qmax
From [Lebacque ’93, ’96]

HJ equations on a junction Numerical simulation
Example of a Diverge
An oﬀ-ramp:
J1
ρ1
J2
ρ2
ρ3
J3
with 


γe = 1,
γl = 0.75,
γr = 0.25

Fundamental Diagrams
0 50 100 150 200 250 300 350
0
500
1000
1500
2000
2500
3000
3500
4000
(ρ
c
,f
max
)
(ρ
c
,f
max
)
Density (veh/km)
Flow(veh/h)

Initial conditions (t=0s)
−200 −150 −100 −50 0
0
10
20
30
40
50
60
70
Road n° 1 (t= 0s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 2 (t= 0s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 3 (t= 0s)
Position (m)
Density(veh/km)

Numerical solution: densities

Numerical solution: Hamilton-Jacobi

Trajectories
1
2
3
4
56
7
78
8
9
9
10
10
11
11
12
12
13
13
Trajectories on road n° 1
Position (m)
Time(s)
−200 −150 −100 −50 0
0
5
10
15
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
12
Position (m)
Time(s)
0 50 100 150 200
0
5
10
15
0
0
1
1
2
2
3
3
4
4
5
56
6
7
7
8
8
9
10
11
12
Position (m)
Time(s)
0 50 100 150 200
0
5
10
15

HJ equations on a junction Recent developments
New junction model
Proposition (Junction model [IM, ’14])
From [4], we have



uα
t + Hα(uα
x ) = 0, x > 0, α = 1, . . . , N
uα = uβ =: u, x = 0,
ut + H(u1
x , . . . , uN
x ) = 0, x = 0
(29)
with initial condition uα(0, x) = uα
0 (x) and
H(u1
x , . . . , uN
x ) = max
ﬂux limiter
L , max
α=1,...,N
H−
α (uα
x )
minimum between
demand and supply
.

Weaker assumptions on the Hamiltonians
For all α = 1, . . . , N,
(A0) The initial condition uα
0 is Lipschitz continuous.
(A1) The Hamiltonians Hα are continuous and quasi-convex i.e.
there exists points pα
0 such that



Hα is non-increasing on (−∞, pα
0 ],
Hα is non-decreasing on [pα
0 , +∞).

Homogenization on a network
Proposition (Homogenization on a periodic network [IM’14])
Assume (A0)-(A1). Consider a periodic network.
If uε satisﬁes (oscillating) HJ equation on network,
then uε converges uniformly towards u0 when ε → 0,
with u0 solution of
u0
t + H ∇x u0
= 0, t > 0, x ∈ Rd
(30)
See [4]

Numerical homogenization on a network
Numerical scheme adapted to the cell problem
Traffic
Traffic eH
γH
eV
γH
γV
γV
i = 0
i =
N
2
i = −
N
2

First example
Proposition (Effective Hamiltonian for fixed coefficients [IM’14])
If (γH , γV ) are fixed, then the
(Hamiltonian) effective Hamiltonian H is given by
H(uH,x , uV ,x ) = max L, max
i={H,V }
H(ui,x ) ,
(traffic flow) effective flow Q is given by
Q(ρH , ρV ) = min −L,
Q(ρH)
γH
,
Q(ρV )
γV
.

First example
Numerics: assume Q(ρ) = 4ρ(1 − ρ) and L = −1.5,

Second example
Two consecutive traﬃc signals on a 1D road
ﬂow
l LL
x1 x2xE
E
xS
S
Homogenization theory by [G. Galise, C. Imbert, R. Monneau, ’14]

Second example
Eﬀective ﬂux limiter −L (numerics only)
0 5 10 15 20 25 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Offset (s)
Fluxlimiter
l=0 m
l=5 m
l=10 m
l=20 m

Conclusions and perspectives
Outline

Personal conclusions
Advantages Drawbacks
Micro-macro link Limited time delay
Homogeneous Explicit solutions Concavity of the FD
link Data assimilation
Exactness
Multilane
Uniqueness of the solution Fixed proportions
Junction Homogenization result
Multilane

Perspectives
Open questions:
Micro-macro: higher time delay? Relaxing comparison principle?
Extend the Lax-Hopf algorithm for non-zero dynamics ϕ(I) = 0
Confront the results with real data (NGSIM/MOCoPo datasets)
Data assimilation on junctions?
“Simple” formula for time/space dependent Hamiltonians?

References
Some references I
A. Bressan, S. Canic, M. Garavello, M. Herty, and B. Piccoli, Flows
on networks: recent results and perspectives, EMS Surveys in Mathematical
Sciences, (2014).
G. Costeseque and J.-P. Lebacque, A variational formulation for higher order
macroscopic traffic flow models: numerical investigation, Transp. Res. Part B:
Methodological, (2014).
M. Garavello and B. Piccoli, Traffic flow on networks, American institute of
mathematical sciences Springfield, MO, USA, 2006.
C. Imbert and R. Monneau, Level-set convex Hamilton–Jacobi equations on
networks, (2014).
C. Imbert, R. Monneau, and H. Zidani, A Hamilton–Jacobi approach to
junction problems and application to traffic flows, ESAIM: Control, Optimisation
and Calculus of Variations, 19 (2013), pp. 129–166.

References
Some references II
J.-P. Lebacque and M. M. Khoshyaran, First-order macroscopic traffic flow
models: Intersection modeling, network modeling, in Transportation and Traffic
Theory. Flow, Dynamics and Human Interaction. 16th International Symposium on
Transportation and Traffic Theory, 2005.

References
Thanks for your attention
guillaume.costeseque@cermics.enpc.fr
guillaume.costeseque@ifsttar.fr

Contribution à l'étude du trafic routier sur réseaux à l'aide des équations d'Hamilton-Jacobi

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