Conservation laws with point constraints on the flow and their applications

Conservation laws with point constraints on the flow and their applications

Massimiliano Daniele Rosini

Instytut Matematyki Uniwersytet Marii Curie-Sk lodowskiej

Lublin, Poland mrosini@umcs.lublin.pl

ANCONET

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This talk is concerned with the recent results on conservation laws with point constraints presented in

B.Andreianov,C.Donadello,M.D.Rosini

“A second order model for vehicular traffics with local point constraints on the flow”, to appear on Mathematical Models and Methods in Applied Sciences B.Andreianov,C.Donadello,U.Razafison,J.Y.Rolland,M.D.Rosini

“Solutions of the Aw-Rascle-Zhang system with point constraints”, to appear on Networks and Heterogeneous Media

B.Andreianov,C.Donadello,U.Razafison,M.D.Rosini

“Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks”, to appear on ESAIM: Mathematical Modelling and Numerical Analysis

B.Andreianov,C.Donadello,U.Razafison,M.D.Rosini

“Riemann problems with non–local point constraints and capacity drop”, Mathematical Biosciences and Engineering 12 (2015) 259–278

B.Andreianov,C.Donadello,M.D.Rosini

1 The motivating problem

2 Modelling traffic flows

3 Constrained LWR

4 Non-local constraint

5 ARZ

6 Constrained ARZ

7 Two phase model (M.Benyahia, GSSI L’Aquila)

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Section 1

The motivating problem

The motivating problem

Study of the traffic flow through bottlenecks, i.e. locations with reduced capacity.

Capacity of the bottleneck: the maximum number of pedestrians or vehicles that can flow through the bottleneck in a given time interval.

Typical phenomena at the bottlenecks are:

Capacity drop Braess’ paradox Faster Is Slower effect

Possible outcomes of this study are:

Optimization of the traffic(green waves, stop-and-go, mean travel time, etc.).

Reducing traffic accidents.

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Crowd Accidents

YEAR DEAD CITY NATION

1711 245 Lyon France

1876 278 Brooklyn USA 1883 180 Sunderland England 1896 1,389 Moscow Russia 1903 602 Chicago USA 1913 73 Michigan USA 1941 4,000 Chongqing China 1942 354 Genoa Italy 1943 173 London England 1956 124 Yahiko Japan 1971 66 Glasgow England 1982 66 Moscow Russia 1985 39 Brussels Belgium 1988 93 Tripureswhor Nepal 1989 96 Sheffield England 1990 1,426 Al-Mu’aysam Saudi Arabia 1991 40 Orkney South Africa 1991 42 Chalma Mexico

1993 73 Madison USA

1994 270 Mecca Saudi Arabia 1994 113 Nagpur India 1996 82 Guatemala City Guatemala 1998 70 Kathmandu Nepal 1998 118 Mecca Saudi Arabia

YEAR DEAD CITY NATION

2001 43 Henderson USA

2001 126 Accra Ghana

2003 100 West Warwick USA 2004 194 Buenos Aires Argentina

2004 251 Mecca Saudi Arabia

2005 300 Wai India

2005 265 Maharashtra India

2005 1,000 Baghdad Iraq

2006 345 Mecca Saudi Arabia

2006 74 Pasig City Philippines

2006 51 Ibb Yemen

2008 147 Jodhpur India

2008 162 Himachal Pradesh India

2008 147 Jodhpur India

2010 71 Kunda India

2010 63 Amsterdam Netherlands 2010 347 Phnom Penh Cambodia

2011 102 Kerala India

2012 79 Port Said Egypt

2013 61 Abidjan Ivory Coast

2013 242 Rio Grande do Sul Brazil 2013 36 Uttar Pradesh India 2013 115 Datia Madhya Pradesh India 2014 18 Maharashtra India

Crowd Accidents

1985, Heysel Stadium in Brussels (Belgium):39fans died as they tried to escape Liverpool supporters.

2010, Phnom Penh (Cambodia):347people died in a bridge stampede at the Khmer Water Festival.

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Crowd Accidents

2010, Duisburg (Germany):21people died near anovercrowded tunnelleading into the festival.

2014, Shanghai (China):35people died rushing to catch (fake) money thrown from a nightclub.

Crowd Accidents

Jamarat Bridge (Saudi Arabia) Year Dead Injured

2015 1,470 700

2006 363 289

2004 249 252

2001 35 179

1998 118 434

1997 22 43

1994 266 98

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Panic management

Is it possible to avoid or to mitigate

crowd accidents?

Panic management

Crowd behaviour is not always the primary cause of accidents.

Architects and city planners try to provide a safe environment for places of public assembly by using mathematical models to determine and prevent congestions.

Example

Nowadays, all public entertainment venues are equipped withdoors that open outwards using crash bar latchesthat open when pushed.

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Panic management

Crowd behaviour is not always the primary cause of accidents.

Architects and city planners try to provide a safe environment for places of public assembly by using mathematical models to determine and prevent congestions.

Example

A tall column placed in the upstream of the door exit may reduce the inter-pedestrian pressure, decreasing the magnitude of clogging and making the overall outflow higher and more regular (Braess’s paradox).

Section 2

Modelling traffic flows

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LWR

The prototype of first order macroscopic models for traffic flows is the Lighthill, Whitham, Richards model

(ρt+ [ρ v ]x = 0,

v = V (ρ), (LWR)

where V : [0, ρ_max] → [0, v_max] is non-increasing, V (0) = v_max and V (ρ_max) = 0.

Lighthill, Whitham, “On kinematic waves. II. A theory of traffic flow on long crowded roads”, Royal Society of London.

Series A, Mathematical and Physical Sciences 1955

Richards, “Shock waves on the highway”, Operations Research

ARZ

The most celebrate second order model for traffic flows is the Aw, Rascle, Zhang model

(ρt+ (ρ v )x = 0,

[ρ (v + p(ρ))]_t+ [ρ (v + p(ρ)) v ]_x = 0, (ARZ) e.g. p(ρ) = ρ^γ, γ > 0.

Aw, Rascle, “Resurrection of “Second Order” Models of Traffic Flow”, SIAM Journal on Applied Mathematics 2000

Zhang, “A non-equilibrium traffic model devoid of gas-like behavior”, Transportation Research Part B: Methodological 2002

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Motivating problem

Both LWR and ARZ assume that the road is homogeneous.

However in real life this assumption is NOT always justified.

Motivating problem

Both LWR and ARZ assume that the road is homogeneous.

Toll gates

Traffic lights

Construction sites

Speed bumps

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Motivating problem

Both LWR and ARZ assume that the road is homogeneous.

Escalators and stairs

Traffic lights

Turnstiles

Doors and revolving doors

Motivating problem

Both LWR and ARZ assume that the road is homogeneous.

Definition (roughly speaking)

By point constraint we mean a pointwise bottleneck that hinders the flow.

Applicability

We apply the theory for point constraints when we are interested in the effects on the upstreamanddownstreamand not in the dynamics insidethe constraint location.

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Motivating problem

The concept of point constraintson the flow was introduced in the framework of vehicular traffic to model the presence of toll gates, construction sites, traffic lights, speed bumps, etc.

Colombo, Goatin, “A well posed conservation law with a variable unilateral constraint”, Journal of Differential Equations, 2007

in the framework of crowd dynamics to model the presence of doors, revolving doors, turnstiles, escalators, traffic lights, stairs, etc.

Colombo, Rosini, “Pedestrian flows and non-classical shocks”, Mathematical Methods in the Applied Sciences,

Section 3 Constrained LWR

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Constrained LWR

If in x = 0 there is a constraint, then the corresponding problem is ρt+ f (ρ)x = 0, ρ(0, x) = ρ0(x), f (ρ(t, 0^±)) ≤ F. (cLWR)

The expression of F depends on the situation at hand:

F is constant if we have a construction site or stairs.

F depends on time if we have a traffic light.

F depends non-locally on ρ if we have a toll gate or door.

Constrained Riemann solver

Consider the Riemann problem for (cLWR) ρt+ f (ρ)x = 0, ρ(0, x) =

(ρ_ℓ if x < 0,

ρr if x > 0, f (ρ(t, 0^±)) ≤ F . If f (RLWR[ρℓ, ρr](0)) ≤ F , then CRLWR[ρℓ, ρr] ≡ RLWR[ρℓ, ρr].

If f (RLWR[ρℓ, ρr](0)) > F , then CRLWR[ρℓ, ρr](x) =

(RLWR[ρ_ℓ, ˆρ](x) if x < 0, RLWR[ˇρ, ρr](x) if x > 0.

ff

F F

ˇ ρˇ

ρ ρˆρˆ ρρ

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Example

Consider the Riemann problem for constrained LWR

ρt+ f (ρ)x = 0, ρ(0, x) ≡ ρℓ = ρr, f (ρ(t, 0^±)) ≤ F . If f (ρ_ℓ) > F , then the solution is

f F

ρ ˇ

ρ ρℓ ρˆ

t

ρℓ

ρℓ

x ˇ

ρ ˆ ρ

Example

Consider the Riemann problem for constrained LWR

ρt+ f (ρ)x = 0, ρ(0, x) ≡ ρℓ = ρr, f (ρ(t, 0^±)) ≤ F . If f (ρ_ℓ) > F , then the solution is

ρ(t) ρℓ

ρℓ

ˆ ρ

ˇ ρ

x

t

ρℓ

ρℓ

x ˇ

ρ ˆ ρ

No maximum principle!

TV bound explosion!

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Well-posedness in BV

constraint ⇒ non-classical shocks ⇒ explosion of TV

Hence BV is not the proper space ⇒ introduction of the space D =ⁿρ ∈ L¹: Ψ(ρ) ∈ BV^o

where

Ψ(ρ) = Z ρ

0

f^′(r )dr.

Cauchy problem

ρt+ f (ρ)x = 0, ρ(0, x) = ρ₀(x), f (ρ(t, 0^±)) ≤ F. (cLWR) Definition (Colombo, Goatin ’07)

ρ ∈Dis a solution to (cLWR) if:

1) ∀φ ∈ C¹_c, φ ≥ 0, and ∀k ∈ [0, ρ_max] ZZ

R₊×R

h|ρ − k|φt+ sgn(ρ − k) [f (ρ) − f (k)] φx

idx dt

+ Z

R

|ρ0(x) − k| φ(0, x) dx +2

Z

R₊



1 −F (t) f_max



f (k) φ(t, 0) dt ≥ 0.

2) f (ρ(t, 0⁻)) = f (ρ(t, 0⁺)) ≤ F (t) for a.e. t > 0.

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Cauchy problem

ρt+ f (ρ)x = 0, ρ(0, x) = ρ₀(x), f (ρ(t, 0^±)) ≤ F. (cLWR) Definition (Andreianov, Goatin, Seguin, ’10)

ρ ∈L^∞ is a solution to (cLWR) if ∃M > 0 s.t. ∀cℓ, cr ∈ [0, ρmax] and ∀φ ∈ C¹_c, φ ≥ 0

ZZ

R₊×R

h|ρ − c| φt+ sgn(ρ − c) [f (ρ) − f (c)] φx

idx dt

+ Z

R

|ρ0(x) − c(x)| φ(0, x) dx +M

Z

R₊

dist^(c_ℓ, c_r), G(F (t))^φ(t, 0) dt ≥ 0, where G is the set of admissible traces along x = 0 and

(

Cauchy problem

ρt+ f (ρ)x = 0, ρ(0, x) = ρ₀(x), f (ρ(t, 0^±)) ≤ F. (cLWR) Definition (Chalons, Goatin, Seguin ’13)

For more general fluxes, ρ ∈ L^∞ is a solution to (cLWR) if:

1) ∀φ ∈ C¹_c, φ ≥ 0, and ∀k ∈ [0, ρ_max] ZZ

R₊×R

h|ρ − k|φt+ sgn(ρ − k) [f (ρ) − f (k)] φx

idx dt

+ Z

R

|ρ0(x) − k| φ(0, x) dx +2

Z

R₊

hf (k) − min {f (k), F (t)}ⁱf (k) φ(t, 0) dt≥ 0.

2) f (ρ(t, 0⁻)) = f (ρ(t, 0⁺)) ≤ F (t) for a.e. t > 0.

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Cauchy problem

ρt+ f (ρ)x = 0, ρ(0, x) = ρ₀(x), f (ρ(t, 0^±)) ≤ F. (cLWR) Theorem (Colombo-Goatin ’07, Andreianov-Goatin-Seguin ’10, Chalons, Goatin, Seguin ’13)

If F ∈ L^∞, then there exists a semigroup S^F: R₊× L^∞→ L^∞ for (cLWR).

If F¹, F², ρ¹₀, ρ²₀ ∈ L^∞ and F¹− F², ρ¹₀− ρ²₀∈ L¹:

S_t^F1ρ¹₀− S_t^F2ρ²₀

L¹≤ ρ¹₀− ρ²₀

L¹+ 2 F¹− F²

L¹. D is an invariant domain and

ρ₀∈ D ⇒





TV(Ψ(S_t^Fρ₀)) ≤ C ,

Ψ(S_t^Fρ₀) − Ψ(S_s^Fρ₀) ≤ C |t − s|,

Initial-boundary value problems

ρt+ f (ρ)x = 0, ρ(0, x) = ρ0(x), f (ρ(t, 0)) = q(t), f (ρ(t, x_c^±)) ≤ F (t).

Definition (Colombo, Goatin, Rosini ’11) ρ ∈ L^∞ is a solution to the above (cIBVP) if:

1) ∀φ ∈ C¹_c, φ ≥ 0, and ∀k ∈ [0, ρ_max] ZZ

R₊×R+

h|ρ − k|φt+ sgn(ρ − k) [f (ρ) − f (k)] φx

idx dt

+ Z

R₊|ρ0(x) − k| φ(0, x) dx +

Z

R₊

sgn^hf_∗⁻¹(q(t)) − k^{i h}f (ρ(t, 0⁺)) − f (k)ⁱφ(t, 0) dt +2

Z

R₊



1 − F (t) fmax



f (k) φ(t, x_c) dt ≥ 0.

2) f (ρ(t, x_c⁻)) = f (ρ(t, x_c⁺)) ≤ F (t) for a.e. t > 0.

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Initial-boundary value problems

ρt+ f (ρ)x = 0, ρ(0, x) = ρ0(x), f (ρ(t, 0)) = q(t), f (ρ(t, x_c^±)) ≤ F (t).

Theorem (Colombo, Goatin, Rosini ’11)

If q, F ∈ BV, then there exists a semigroup S^q,F: R+× D → D for the above (cIBVP).

If q¹, q², F¹, F², ρ¹₀, ρ²₀ ∈ L^∞ and q¹− q², F¹− F², ρ¹₀− ρ²₀ ∈ L¹:

S_t^F1ρ¹₀− S_t^F2ρ²₀

L¹ ≤ ρ¹₀− ρ²₀

L¹+ q¹− q²

L¹+ 2 F¹− F²

L¹. Moreover

Z T 0

f (S_t^q1,F1ρ¹₀)(x⁻) − f (S_t^q2,F2ρ¹₀)(x⁻) dt ≤

Synchronizing gateways

Colombo-Goatin-Rosini ’11:

0 2 4 6 8 10 12 14

0 0.5 1 1.5 2

t

tau Mean Arrival Time

0 2 4 6 8 10 12 14

0 0.5 1 1.5 2

t

tau Exit Time

The lower graphs corresponding to the lower inflows.

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Section 4

Non-local constraint

Non-local constraint: empirical evidences

Pedestrian inflow and outflow patterns at the London Under- ground station

E.M.Cepolina.

Phased evacuation: An optimisation model which takes into account the capacity drop phenomenon in pedestrian flows.

Fire Safety Journal, 2009.

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Non-local constraint: explanation

High density upstream =⇒ Capacity drop

D. Helbing, “Agent-Based Simulation of Traffic Jams, Crowds, and Supply Networks – Reality, Simulation, and Design of

Non-local constraint

The corresponding model is ρt+f (ρ)x= 0, f ρ(t, 0^±)
≤ p

Z

R₋

w (x) ρ(t, x) dx

!

, ρ(0, x) = ρ₀(x) (NL)

ρ_max f

f (¯ρ)

¯ ρ f (¯ρ)

ρ x

w

pi

−iw ρ_maxξ

ˇ ρi ρˆi

p

B.Andreianov, C.Donadello, M.D.Rosini, “Crowd dynamics and conservation laws with non–local constraints and capacity drop”, Mathematical Models and Methods in Applied Sciences 24 (2014) 2685–2722

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Non-local constraint

Theorem (Andreianov, Donadello, Rosini ’14) Assume that:

(F) f ∈ Lip ([0, ρ_max]; [0, f_max]), f (0) = 0 = f (ρ_max) and there exists ¯ρ ∈ ]0, ρmax[ s.t. f^′(ρ) (¯ρ − ρ) > 0 a.e. in [0, ρmax].

(W) w ∈ L^∞(R₋; R₊) is an increasing map, kw k_L1(R−;R+)= 1 and there exists iw > 0 such that w (x) = 0 for any x ≤ −iw.

(P) p is non-increasing and belongs to Lip([0, ρ_max] ; ]0, f (¯ρ)]).

Non-local constraint

Theorem (Andreianov, Donadello, Rosini ’14) Then:

(i) For any initial datum ρ₀ ∈ L^∞(R; [0, ρ_max]), the Cauchy problem (cLWR) admits a unique solution ρ. Moreover, if

˜

ρ = ˜ρ(t, x) is the solution corresponding to the initial datum

˜

ρ₀∈ L^∞(R; [0, ρ_max]), then for all T > 0 and L > iw

kρ(T ) − ˜ρ(T )k_L1([−L,L];R)≤ e^CTkρ0− ˜ρ₀k_L1({|x |≤L+MT };R), where M = Lip(f ) and C = 2Lip(p)kw k_L^∞_(R−;R).

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Non-local constraint

Theorem (Andreianov, Donadello, Rosini ’14) Then:

(ii) If ρ₀ belongs to D, then the unique solution of (cLWR) verifies ρ(t, ·) ∈ D for a.e. t > 0, and it satisfies

TV(Ψ (ρ(t))) ≤ Ct = TV (Ψ (ρ₀)) + 4f_max+ C t , moreover, for a.e. t, s in ]0, T [ we have

kΨ (ρ(t, ·)) − Ψ (ρ(s, ·))k_L1(R;R)≤ |t − s| Lip(Ψ) CT .

Non-local constraint

Proof (Wave-front tracking and operator splitting methods).

Introduce fⁿ∈ PLC approximation of f , ρⁿ₀ ∈ PC approximation of ρ0 and p^h∈ PC approximation of p. Define recursively

F^nh[ρⁿ₀](t) = S [ρⁿ₀, Θ [ρⁿ₀]] (t)

if t ∈ [0, ∆t], and, if t ∈ ](ℓ + 1)∆t, (ℓ + 2)∆t], ℓ ∈ N, then F^nh[ρⁿ₀](t) = S^hF^nh[ρⁿ₀] ((ℓ + 1)∆t) , Θ^hF^nh[ρⁿ₀] ((ℓ + 1)∆t)ⁱⁱ(t).

The existence of a limit for F^nh[ρⁿ₀] as n and h go to infinity is ensured by the choice

∆th= 1

2^h+1 w (0−) Lip(p) .

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Non-local constraint

The proof is quite technical, also because solutions of a Riemann problem in the case of a non-decreasing piecewise constant p may fail to be unique,L¹_loc–continuousandconsistent, see

B.Andreianov, C.Donadello, M.D.Rosini, U.Razafison,

“Riemann problems with non–local point constraints and capacity drop”, Mathematical Biosciences and Engineering 12 (2015) 259–278

More in general, in the above paper we investigate how the regularity of p impacts the well–posedness of the problem.

We show that the regularity of p plays a central role in the well- posedness result.

While existence still holds for the Cauchy problem when p is merely continuous, it is difficult to justify uniqueness in this case.

Comparison: local VS non-local

I

E

A B

L

C D M

O N

R.M.Colombo, Rosini.

Pedestrian Flows and non-classical shocks.

Mathematical Methods in the Applied Sciences 2005

20 40 60 80

−3

−4

−5 B −iw

A

C D EF G HI

L

NM O

Andreianov, Donadello, Rosini.

Crowd dynamics and conservation laws with non-local constraints.

Mathematical Models and Methods in Applied Sciences 2014

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Numerical approximation

Andreianov, Donadello, Razafison, Rosini, “Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks”, to appear on ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we investigate numerically the model (NL). We prove the convergence of a scheme based on a constraint finite volume method. We then perform ad hoc simulations to qualitatively validate the model by proving its ability to reproduce typical phenomena at the bottlenecks, such as the Braess’ paradoxand

Braess’s paradox

24 25 26 27 28 29 30 31 32 33 34

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

evacuationtime

with obstacle without obstacle

d

Evacuation time as a function of the position of the obstacle.

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Faster is slower effect

20 40 60 80 100 120 140 160 180 200

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

¯ ρ

¯ ρ1

¯ ρ2

v_max

evacuationtime

Evacuation time as a function of vmax for different initial densities

Faster is slower effect

20 40 60 80 100 120 140 160 180 200

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

β = 1 β = 0.9 β = 0.8

v_max

evacuationtime

Evacuation time as a function of vmax for different exit efficiencies p_β(ξ) = p(β ξ), β = 1, β = 0.9, β = 0.8

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Faster is slower effect

20 40 60 80 100 120 140 160 180 200

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

¯ ρ

¯ ρ3

¯ ρ4

v_max

evacuationtime

Evacuation time as a function of vmax for different initial locations

General non-local constraints

We are considering now a more general class of point constraints f (ρ(t, 0^±)) ≤ q(t) = Q[ρ](t) for a.e. t ∈ [0, T ],

where

Q : C⁰([0, T ]; L¹(R; R)) → L¹([0, T ]; [0, f (¯ρ)])

so that the operator Q may be non-local both in time and space.

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Example

Consider vehicular traffic through a toll booth, where the number of open gates is decided according to on line data.

If the data are collected by a video camera, then we can take Q[ρ](t) = p

Z

R₋

Z t 0

w (x) κ(t − s) ρ(s, x) ds dx

! ,

beingsupp(w )the area registered by the video camera andsupp(κ) the period of time the data are taken into account.

A time delay in adapting the number of open gates to the available data can be easily modelled.

Example

Consider vehicular traffic through a toll booth, where the number of open gates is decided according to on line data.

If the data come from a photo camera that shoots photos at times ti of the area given bysupp(w ), then we can take

Q[ρ](t) = pe ^X

ti<t

Z

R−

w (x) κ(t − ti) ρ(ti, x)dx

! ,

being supp(κ) the period of time the data are taken into account.

A time delay in adapting the number of open gates to the available data can be easily modelled.

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Example

Consider vehicular traffic through a toll booth, where the number of open gates is decided according to on line data.

If the data come from local sensorslocated inyi, then we can take

Q[ρ](t) = p



 XM i=1



 w (yi) κ(t) ρ0(yi) +

Z t 0

w^′(y_i) κ(t − s) f (ρ)(s, y_i) ds







,

being supp(κ) the period of time the data are taken into account.

A time delay in adapting the number of open gates to the available data can be easily modelled.

General non-local constraints

We introduced a concept of entropy solutions.

By using fixed point arguments we are able to addresssolvability of non-locally constrained problems.

Under general conditions on Q we obtained the uniqueness of entropy solutions.

B.Andreianov, C.Donadello, U.Razafison, M.D.Rosini,

“Analysis and approximation of first order traffic models with general point constraints on the flow”, in preparation

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Section 5

ARZ

ARZ and the vacuum

ARZ is the 2 × 2 system of conservation laws Y~t+ F (~Y )x = ~0, Y =~ ρ

y

!

, F (~Y ) = v ~Y , v = y

ρ − p(ρ).

away from the vacuum:

It is strictly hyperbolic.

It is ofTemple class.

It has thePanov renormalization property.

Existence results are available (wave-front tracking).

Ferreira, Kondo

Glimm method and wave-front tracking for the Aw-Rascle traffic flow model.

Far East Journal of Mathematical Sciences 2010

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ARZ and the vacuum

ARZ is the 2 × 2 system of conservation laws Y~t+ F (~Y )x = ~0, Y =~ ρ

y

!

, F (~Y ) = v ~Y , v = y

ρ − p(ρ).

at the vacuum: F is not defined!

Vacuum (Godvik, Hanche-Olsen)

Even for initial data away from the vacuum, the solutions may attain values in the vacuum in a (possibly strictly positive) finite time.

Existence results are available (compensated compactness).

Godvik, Hanche-Olsen Lu

BV-estimate

Introduce the Riemann invariantscoordinates

W~ = (v , w ) ∈W =ⁿ(v , w ) ∈ R²₊: v ≤ w^o=W0∪W₀^c given by

Y ( ~W ) = p⁻¹(w − v ) 1 w

!

, W (~Y ) =



 y

ρ − p(ρ) y ρ



,

where v is the velocity,

w is the Lagrangian marker.

Crucial for the existence results the BV-estimate TV( ~W (t)) ≤ TV( ~W₀)

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ARZ and the vacuum

Introduce in PC the concept of physically admissible discontinuities involving a vacuumstate:

W (t, x~ ⁻), ~W (t, x⁺)^∈ G where

G .=













( ~W_ℓ, ~W_r) ∈ W²:

W~ℓ ∈ W₀ W~r ∈ W₀

)

⇒ ~Wℓ = ~Wr, W~ℓ ∈ W₀^c

W~r ∈ W0

)

⇒ ~Wr = (wℓ, wℓ)











 .

ARZ and the vacuum

Then define the Riemann solver RARZ

RARZ: G → C⁰(]0, +∞[ ; L^∞(R; W)) , ( ~Wℓ, ~Wr) 7→ RARZ[ ~Wℓ, ~Wr], by introducing first the elementary waves: shocks, rarefactions and contact discontinuities.

Details on elementary waves

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The Riemann solver R

Definition (roughly speaking)

The solution performs either a rarefaction or a shock, followed by a contact discontinuity.

Definition

1 W~_ℓ∈ W₀^c W~r∈ W0



⇒

 RARZ[ ~W_ℓ, ~Wr] ≡ R[ ~W_ℓ, ~Wm] W~m= (wℓ, w_ℓ) ∈ W0

2 W~_ℓ∈ W0

W~r∈ W₀^c



⇒ RARZ[ ~W_ℓ, ~Wr] ^x_t

=

 _~

W_ℓ if x < vrt W~r if x > vrt

3 W~ℓ, ~Wr∈ W₀^c vr< vℓ< wℓ



⇒

 R_ARZ[ ~Wℓ, ~Wr] ≡ S[ ~Wℓ, ~Wm] + C[ ~Wm, ~Wr] W~m= (vr, wℓ) ∈ W₀^c

4 W~ℓ, ~Wr∈ W₀^c vℓ< vr< wℓ



⇒

 R_ARZ[ ~Wℓ, ~Wr] ≡ R[ ~Wℓ, ~Wm] + C[ ~Wm, ~Wr] W~m= (vr, wℓ) ∈ W₀^c

 




R[ ~W, ~W ] ^x

if ^x < v

Section 6 Constrained ARZ

41 / 53

ARZ as extension of LWR

ARZ can be interpreted as a generalization of LWR possessing a family of fundamental diagramcurves, rather than a single one, corresponding to different Lagrangian markers w .

ARZ can be interpreted as a multi population version of LWR:

The population corresponding to the La- grangian marker w is characterized by maximal speed w and length 1/p⁻¹(w ).

ARZ as extension of LWR

Can we extend also the concept of point constraints?

Technically YES because ARZ has the Panov renormalization property

F (~Y ) = v ~Y

This ensures the existence of the traces of the solutions at x = 0.

f

ρ F

ˇ

ρ ˙ρ ρˆ

˙f

ρ f

F

˙ρ ˇ ρ ρˆ

˙f

43 / 53

ARZ with constraint

Assume to have at x = 0 a constraint and consider the problem Y~t+ F (~Y )x = ~0, Y (0, x) = ~~ Y₀, f (~Y (t, 0^±)) ≤ F . (cARZ)

The Riemann problem

If f (R_ARZ[ ~W_ℓ, ~W_r]) ≤ F , then CR_ARZ[ ~W_ℓ, ~W_r] ≡ R_ARZ[ ~W_ℓ, ~W_r].

If f (RARZ[ ~Wℓ, ~Wr]) > F , then

CRARZ[ ~W_ℓ, ~Wr](x/t) =

(RARZ[ ~Wℓ, ˆW (wℓ)](x/t) if x < 0, RARZ[ ˇW (w_ℓ), ~Wr](x/t) if x > 0.

Garavello, Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl. 2011

ARZ with constraint

Assume to have at x = 0 a constraint and consider the problem Y~t+ F (~Y )x = ~0, Y (0, x) = ~~ Y₀, f (~Y (t, 0^±)) ≤ F . (cARZ)

The Cauchy problem

The first results in this direction are in

Andreianov, Donadello, Razafison, Rolland, Rosini, Solutions of the Aw-Rascle-Zhang system with point constraints, to appear on Networks and Heterogeneous Media

44 / 53

ARZ with constraint

Assume to have at x = 0 a constraint and consider the problem Y~t+ F (~Y )x = ~0, Y (0, x) = ~~ Y₀, f (~Y (t, 0^±)) ≤ F . (cARZ)

The Cauchy problem

We revisit the wave-front tracking construction of physically admis- sible solutions possibly involvingvacuumstates.

G .=













( ~Wℓ, ~Wr) ∈ W²:

W~_ℓ∈ W0

W~_r ∈ W0

)

⇒ ~Wℓ= ~Wr, W~_ℓ∈ W₀^c

W~_r ∈ W0

)

⇒ ~W_r = (w_ℓ, w_ℓ)













ARZ with constraint

Assume to have at x = 0 a constraint and consider the problem Y~t+ F (~Y )x = ~0, Y (0, x) = ~~ Y₀, f (~Y (t, 0^±)) ≤ F . (cARZ)

The Cauchy problem

We introduce a Kruzhkov-like family of entropies to select the ad- missible shocks.

Ek( ~W ) =







0 if v ≤ k,

1 −p⁻¹(w − v )

p⁻¹(w − k) if v > k, Qk( ~W ) =







0 if v ≤ k,

k − q( ~W )

p⁻¹(w − k) if v > k.

This tool allows to define rigorously the appropriate notion of ad- missible solution and to approximate the solutions of (cARZ).

44 / 53

ARZ with constraint

Assume to have at x = 0 a constraint and consider the problem Y~t+ F (~Y )x = ~0, Y (0, x) = ~~ Y₀, f (~Y (t, 0^±)) ≤ F . (cARZ) The Cauchy problem

Definition (Constrained entropy solution)

W ∈ L~ ^∞(]0, +∞[ ; BV(R; W)) ∩ C⁰(R₊; L¹_loc(R; W)) is a

constrained entropy solution if it is a constrained weak solution and ZZ

R₊×R

hEk( ~W ) φt+Qk( ~W ) φx

idx dt + Z

R₊Nk( ~W , q₀) φ(t, 0) dt ≥ 0 for any φ ∈ C^∞_c (]0, +∞[ × R; R), for any k ∈ ]0, V₀], where

 h i+

ARZ with constraint

Assume to have at x = 0 a constraint and consider the problem Y~t+ F (~Y )x = ~0, Y (0, x) = ~~ Y₀, f (~Y (t, 0^±)) ≤ F . (cARZ)

The Cauchy problem

We propose a finite volumes numerical scheme for (cARZ) and we show that it can correctly locate contact discontinuities and take the constraint into account.

Chalons, Goatin, “Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling”, Communications in Mathematical Sciences, 2007

44 / 53

ARZ with constraint

Assume to have at x = 0 a constraint and consider the problem Y~t+ F (~Y )x = ~0, Y (0, x) = ~~ Y₀, f (~Y (t, 0^±)) ≤ F . (cARZ)

The Cauchy problem

The convergence of the wave-front tracking is proved in

B.Andreianov, C.Donadello, M.D.Rosini, A second order model for vehicular traffics with local point constraints on the flow, to appear on Mathematical Models and Methods in Applied Sciences

ARZ with constraint

Assume to have at x = 0 a constraint and consider the problem Y~t+ F (~Y )x = ~0, Y (0, x) = ~~ Y₀, f (~Y (t, 0^±)) ≤ F . (cARZ)

The Cauchy problem

The constraint F is essentially constant.

f

ρ F

ˇ

ρ ˙ρ ρˆ

˙f w

v w

˜ w

ˇ v ˆ v ˜v

ρ 7→ [w − p(ρ)] ρ v 7→ [v + p(F /v )]

44 / 53

Section 7

Two phase model (M.Benyahia, GSSI L’Aquila)

Free flow (LWR)









u = (ρ, v ) ∈ Ωf, ρ_t+ [ρ v ]_x = 0, v = vf(ρ),

Congested flow (ARZ)









u = (ρ, v ) ∈ Ωc, ρ_t+ [ρ v ]_x = 0, [ρ w ]_t+ [ρ w v ]_x = 0.

(PT)

ρ qmax

ρ v V_max Vf

Vc Ω^′f

Ωc Ω^′′f

Ωc Ω^′′f

Ω^′f

v w

Wc W_min Wmax

Vf Vc

P. Goatin, “The Aw-Rascle vehicular traffic flow with phase transitions”, Mathematical and Computer Modeling, 2006

46 / 53

Definition (Benyahia, Rosini work in progress)

Fix an initial distribution ¯u in BV(R; Ω). Let u be a function in L^∞(]0, +∞[; BV(R; Ω)) ∩ C⁰(R₊; L¹_loc(R; Ω)) satisfying the initial condition u(0, x) = ¯u(x) for a.e. x ∈ R. We say that u is an entropy solution to the two phase model (PT) if:

1 It is an entropy solution of LWR in u⁻¹(Ωf).

2 It is an entropy solution of ARZ in u⁻¹(Ωc).

3 If u performs a phase transition from u− to u+, then the speed of propagation of the phase transition satisfies the Rankine-Hugoniot conditions and (u₋, u₊) ∈ G_e.

Theorem (Benyahia, Rosini work in progress)

For any initial distribution ¯u in BV(R; Ω), the approximate solution for the Cauchy problem of (PT) with initial datum ¯u converges (up to a subsequence) in L¹_loc(R₊× R; Ω) to a function u and for all t, s ≥ 0 we have that

TV(u(t)) ≤ TV(¯u), ku(t)k_L^∞_(R;Ω)≤ C , ku(t) − u(s)k_L1(R;Ω) ≤ L |t − s|,

where L = TV(¯u) max{V_max, p⁻¹(W_max) p^′(p⁻¹(W_max))} and C = max{|Wmax|, |Wmin|} + Vf.

Moreover, ifV_max = Vf and u performs a finite number of phase transitions, then u is an entropy solution.

48 / 53

Let R : Ω × Ω → L^∞(R; Ω) be the Riemann solver associated to the constrained(PT).

Properties of R

R is neither L¹_loc-continuous, nor consistent.

R introduces non-classical shocks that do not “maximize” the flux through the constraint.

The study of the constrained Cauchy problem for (PT) will be very interesting. . .

THANK YOU

FOR YOUR

ATTENTION

Shocks

If ~W_ℓ ∈ W, ~W_r ∈ W₀^c with w_ℓ= w_r =w and v_r < v_ℓ, then we let S[ ~W_ℓ, ~Wr](x/t) =

( W~_ℓ x <σt,

W~r x >σt, σ = fr − fℓ

ρr − ρℓ

.

The solution is then f

fℓ

f_r

vℓ

vr

w

w