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Control problems for traffic flow

Mauro Garavello

University of Milano Bicocca

OptHySYS

University of Trento, January 9-11, 2017

Mauro Garavello (University of Milano Bicocca) Control problems for traffic flow

The LWR traffic flow model (1955, 1956)

∂t ρ+ ∂x f(ρ) = 0

• ρ(t, x): density of cars at time t > 0 and at the position x• f(ρ): flux of cars

f(ρ) = ρv, where v is the average velocity

• v depends only on ρ in a decreasing wayv(ρ) = Vmax

(1− ρ

ρmax

), ρmax: maximum density, Vmax: maximum velocity

• f(0) = f(ρmax) = 0 is a strictly concave function

ρ ρ

v f

σ ρmaxρmax

Vmax



∂t ρ+ ∂x f(ρ) = 0

• ρ(t, x): density of cars at time t > 0 and at the position x

x

ρ

• f(ρ): flux of cars

f(ρ) = ρv, where v is the average velocity


(1− ρ

ρmax



ρ ρ

v f

σ ρmaxρmax

Vmax



∂t ρ+ ∂x f(ρ) = 0

• ρ(t, x): density of cars at time t > 0 and at the position x• f(ρ): flux of cars

f(ρ) = ρv, where v is the average velocity• v depends only on ρ in a decreasing way

v(ρ) = Vmax

(1− ρ

ρmax



ρ ρ

v f

σ ρmaxρmax

Vmax



∂t ρ+ ∂x f(ρ) = 0

• ρ(t, x): density of cars at time t > 0 and at the position x• f(ρ): flux of cars f(ρ) = ρv, where v is the average velocity


(1− ρ

ρmax



ρ ρ

v f

σ ρmaxρmax

Vmax



∂t ρ+ ∂x f(ρ) = 0

• ρ(t, x): density of cars at time t > 0 and at the position x• f(ρ): flux of cars f(ρ) = ρv, where v is the average velocity• v depends only on ρ in a decreasing way

v(ρ) = Vmax

(1− ρ

ρmax



ρ ρ

v f

σ ρmaxρmax

Vmax


Traffic flow on a network of roads

Conservation laws evolving on a directed graph (finitecollection of directed arcs and nodes)

Solutions at nodes

- node: m incoming arcs, n outgoing arcs- initial datum ρl in each arc Il, interval of R- This corresponds to m+ n IBV problems ∂t ρl+∂x f(ρl) = 0, x∈Il, l ∈ 1, · · · ,m+ n, t > 0

ρl(0, x) = ρl(x), x ∈ Il, l ∈ 1, · · · ,m+ nρl(t, 0) = ???, t > 0, l ∈ 1, · · · , n+m

Coupling transition condition at a node located at x = 0

Ψ(ρ1(t, 0), . . . , ρm+n(t, 0)) = 0



Conservation laws evolving on a directed graph (finitecollection of directed arcs and nodes)Solutions at nodes




Ψ(ρ1(t, 0), . . . , ρm+n(t, 0)) = 0




- node: m incoming arcs, n outgoing arcs

- initial datum ρl in each arc Il, interval of R- This corresponds to m+ n IBV problems ∂t ρl+∂x f(ρl) = 0, x∈Il, l ∈ 1, · · · ,m+ n, t > 0


I1

I2

I3

I4

I5

I6

I7

I8

I9


Ψ(ρ1(t, 0), . . . , ρm+n(t, 0)) = 0




- node: m incoming arcs, n outgoing arcs- initial datum ρl in each arc Il, interval of R

- This corresponds to m+ n IBV problems ∂t ρl+∂x f(ρl) = 0, x∈Il, l ∈ 1, · · · ,m+ n, t > 0ρl(0, x) = ρl(x), x ∈ Il, l ∈ 1, · · · ,m+ nρl(t, 0) = ???, t > 0, l ∈ 1, · · · , n+m

ρ1

ρ2

ρ3

ρ4

ρ5

ρ6

ρ7

ρ8

ρ9


Ψ(ρ1(t, 0), . . . , ρm+n(t, 0)) = 0






ρ1

ρ2

ρ3

ρ4

ρ5

ρ6

ρ7

ρ8

ρ9


Ψ(ρ1(t, 0), . . . , ρm+n(t, 0)) = 0


PDEs on networks

Vehicular traffic (scalar and systems)

Air traffic managementGas pipelines (systems)Supply chains (scalar)Blood circulatory flow & vascular stents (systems)Irrigation channels (systems)Biological networks (scalar)Telecommunication and data networks (scalar)


PDEs on networks

Vehicular traffic (scalar and systems)Air traffic management

Gas pipelines (systems)Supply chains (scalar)Blood circulatory flow & vascular stents (systems)Irrigation channels (systems)Biological networks (scalar)Telecommunication and data networks (scalar)


PDEs on networks

Vehicular traffic (scalar and systems)Air traffic managementGas pipelines (systems)

Supply chains (scalar)Blood circulatory flow & vascular stents (systems)Irrigation channels (systems)Biological networks (scalar)Telecommunication and data networks (scalar)


PDEs on networks

Vehicular traffic (scalar and systems)Air traffic managementGas pipelines (systems)Supply chains (scalar)

Blood circulatory flow & vascular stents (systems)Irrigation channels (systems)Biological networks (scalar)Telecommunication and data networks (scalar)


PDEs on networks

Vehicular traffic (scalar and systems)Air traffic managementGas pipelines (systems)Supply chains (scalar)Blood circulatory flow & vascular stents (systems)

Irrigation channels (systems)Biological networks (scalar)Telecommunication and data networks (scalar)


PDEs on networks

Vehicular traffic (scalar and systems)Air traffic managementGas pipelines (systems)Supply chains (scalar)Blood circulatory flow & vascular stents (systems)Irrigation channels (systems)

Biological networks (scalar)Telecommunication and data networks (scalar)


PDEs on networks

Vehicular traffic (scalar and systems)Air traffic managementGas pipelines (systems)Supply chains (scalar)Blood circulatory flow & vascular stents (systems)Irrigation channels (systems)Biological networks (scalar)

Telecommunication and data networks (scalar)


PDEs on networks

Vehicular traffic (scalar and systems)Air traffic managementGas pipelines (systems)Supply chains (scalar)Blood circulatory flow & vascular stents (systems)Irrigation channels (systems)Biological networks (scalar)Telecommunication and data networks (scalar)


PDEs on networks

Vehicular traffic (scalar and systems)Air traffic managementGas pipelines (systems)Supply chains (scalar)Blood circulatory flow & vascular stents (systems)Irrigation channels (systems)Biological networks (scalar)Telecommunication and data networks (scalar)

Achdou, Andreianov, Banda, Bastin, Bressan, Camilli, Canic, Chalons,Coclite, Colombo, Coron, Costeseque, D’Apice, Delle Monache,Donadello, Gasser, Goatin, Göttlich, Guerra, Gugat, Han, Herty, Holden,Imbert, Klar, Lattanzio, Lebacque, Leugering, Manzo, Marcellini, Marchi,Monneau, Moutari, Nguyen, Marigo, Piccoli, Rascle, Risebro, Rosini,Schleper, Shen, Tchou, Zidani, Ziegler...


Nodal conditions on road networks

Incoming roads : Il = (−∞, 0], Outgoing roads : Il = [0,+∞)

Conservation of the number of cars

m+n∑j=m+1

f(ρj(t, 0)) =

m∑i=1

f(ρi(t, 0)) for a.e. t > 0,

Distribution rules: aji (percentage of drivers coming fromi-th incoming road and turning into j-th outgoing road)

f(ρj(t, 0)) =m∑i=1

aji(t)f(ρi(t, 0)) for a.e. t > 0,

0 ≤ aji(t) ≤ 1 ∀ j, im+n∑

j=m+1

aji(t) = 1 ∀ i


Nodal conditions on road networks

Incoming roads : Il = (−∞, 0], Outgoing roads : Il = [0,+∞)

Conservation of the number of carsm+n∑

j=m+1

f(ρj(t, 0)) =

m∑i=1

f(ρi(t, 0)) for a.e. t > 0,

Distribution rules: aji (percentage of drivers coming fromi-th incoming road and turning into j-th outgoing road)

f(ρj(t, 0)) =m∑i=1

aji(t)f(ρi(t, 0)) for a.e. t > 0,

0 ≤ aji(t) ≤ 1 ∀ j, im+n∑

j=m+1

aji(t) = 1 ∀ i


Nodal conditions on road networks. . . continued

Remark: distribution rules are not sufficient to select uniquesolution at the junction.

wwwOptimization criterion imposed on the traces of the solution[maximize the flux through the junction]

Priority rules: ci [percentage of time drivers from i-th incomingroad passing through the junction],

∑mi=1 ci(t) = 1

(when m > n and the total possible flux on the incoming roads islarger than the maximal flux that the outgoing roads can handle)



Remark: distribution rules are not sufficient to select uniquesolution at the junction. www

Optimization criterion imposed on the traces of the solution[maximize the flux through the junction]


∑mi=1 ci(t) = 1




Remark: distribution rules are not sufficient to select uniquesolution at the junction. wwwOptimization criterion imposed on the traces of the solution[maximize the flux through the junction]


∑mi=1 ci(t) = 1



Riemann problem approach[Holden, Risebro (1995); Coclite, G., Piccoli (2005) - ]

Junction Riemann Solver (JRS): provide nodal conditions anda procedure for (uniquely) solving m+ n IBV (Riemann)problems at m incoming and n outgoing roads when initialdata are constants. Solutions are required to be self-similar asfor classical Riemann problems.

Construct front tracking approximate solutions with piecewiseconstant initial data on each road.

A-priori BV bounds yield the convergence of front trackingapproximations to the (unique?) solution of the Cauchyproblem at the node with general initial data (having finitetotal variation).


Junction Riemann Solver

A Riemann problem at a node is a Cauchy problem with constantinitial condition on each arc Il:

∂t ρl + ∂x f(ρl) = 0, x ∈ Il, l ∈ 1, . . . ,m+ n, t > 0

ρl(0, x) = ρl, x ∈ Il, l ∈ 1, . . . ,m+ n

Giving a solution is equivalent to giving its trace at the node:ρl(t, 0) = ρl t > 0, l ∈ 1, . . . ,m+ n

A Riemann solver at the node is a mapRS : [0, ρmax]

m+n → [0, ρmax]m+n,

(ρ1, . . . , ρm+n) 7→ (ρ1, . . . , ρm+n)

that associates to an (m+ n)-tuple of constant initial data an(m+ n)-tuple of constant boundary data which are the tracesat the node of the solution to the corresponding Riemann problem.


Junction Riemann Solver. . . continuedFor l ∈ 1, . . . ,m (incoming arcs) the classical Riemann problem

∂t ρl + ∂x f(ρl) = 0

ρl(0, x) =

ρl if x < 0

ρl if x > 0

is solved by waves with negative speeds.ρl ρl

x = 0

For l ∈ m+ 1, . . . ,m+ n (outgoing arcs) the classical Riemannproblem

∂t ρl + ∂x f(ρl) = 0

ρl(0, x) =

ρl if x < 0

ρl if x > 0

is solved by waves with positive speeds.ρl ρl

x = 0


Solution for the Junction Riemann Problem

Fix a distribution Markov matrix A ∈ M(n×m) (Drivers’ preferences)

Impose the linear constraints

A · (f(ρ1(t, 0)), . . . , f(ρm(t, 0)))T = (f(ρm+1(t, 0)), . . . , f(ρm+n(t, 0)))T

on the traces of the fluxes (flux distribution ↔ matrix A)Choose the self-similar solution which maximizes

∑nl=1 f(ρl(t, 0))

If needed (m > n), prescribe relative priority rules



Fix a distribution Markov matrix A ∈ M(n×m) (Drivers’ preferences)Impose the linear constraints


on the traces of the fluxes (flux distribution ↔ matrix A)

Choose the self-similar solution which maximizes∑n

l=1 f(ρl(t, 0))







∑nl=1 f(ρl(t, 0))







∑nl=1 f(ρl(t, 0))


Algorithm

• Ω =(γ1, · · · , γn) ∈

∏ni=1 Ωi : A · (γ1, · · · , γn)T ∈

∏n+mj=n+1 Ωj

• Maximize E = γ1 + · · ·+ γn on Ω

• Find the corresponding densities

• Select the densities that satisfy the priority rules, if needed


Solution for the Junction Riemann ProblemFix a distribution Markov matrix A ∈ M(n×m) (Drivers’ preferences)Impose the linear constraints



∑nl=1 f(ρl(t, 0))


γ1

γ2

γ1,max

γ2,max





∑nl=1 f(ρl(t, 0))


γ1

γ2

γ1,max

γ2,max

α1,3γ1 + α2,3γ2 = γ3,max





∑nl=1 f(ρl(t, 0))


γ1

γ2

γ1,max

γ2,max

α1,3γ1 + α2,3γ2 = γ3,max

α1,4γ1 + α2,4γ2 = γ4,max





∑nl=1 f(ρl(t, 0))


γ1

γ2

γ1,max

γ2,max

α1,3γ1 + α2,3γ2 = γ3,max

α1,4γ1 + α2,4γ2 = γ4,max

Ω





∑nl=1 f(ρl(t, 0))


γ1

γ2

γ1,max

γ2,max

α1,3γ1 + α2,3γ2 = γ3,max

α1,4γ1 + α2,4γ2 = γ4,max

Ω

Level curve of γ1 + γ2


Matrix A as a control function

Theorem: existence of solution (G., Piccoli)Assume that the traffic distribution matrix A = A(t) is time dependentand BV (control function). Fix initial data ρl,0 ∈

(BV ∩ L1

)(Il;R).

Then, for every T > 0, there exists a solution (ρ1, . . . , ρn+m) for theCauchy problem

∂∂tρl +

∂∂xf(ρl) = 0

ρl(0, x) = ρl,0(x)l = 1, . . . , n+m

such that

RSA(t)(ρ1(t, 0), . . . , ρn+m(t, 0)) = (ρ1(t, 0), . . . , ρn+m(t, 0))

for a.e. t ∈ [0, T ].


A different approach: a control theoretic point of view

Fix T > 0. Find the solution on [0, T ] to the Cauchy problem∂t ρl + ∂x f(ρl) = 0, x ∈ Il, l ∈ 1, . . . , n+m, t > 0

ρl(0, x) = ρl(x), x ∈ Il, l ∈ 1, . . . , n+m,

that satisfies the flux constraint


and maximizes an integral cost∫ T

0

m∑l=1

f (ρl(t, 0)) dt

The maximization is global on [0, T ] among all admissiblesolutions (fulfilling linear flux constraints) and not pointwisein time


A different approach: a control theoretic point of view

Fix T > 0. Find the solution on [0, T ] to the Cauchy problem∂t ρl + ∂x f(ρl) = 0, x ∈ Il, l ∈ 1, . . . , n+m, t > 0

ρl(0, x) = ρl(x), x ∈ Il, l ∈ 1, . . . , n+m,

that satisfies the flux constraint


and maximizes an integral cost∫ T

0J (f (ρ1(t, 0)) , . . . , f (ρm(t, 0))) dt (J : Rm → R).

The maximization is global on [0, T ] among all admissiblesolutions (fulfilling linear flux constraints) and not pointwisein time


New features of optimal solutions

In general there is no Junction Riemann Solver compatiblewith an optimal solution.

Optimal solutions with constant initial densities are in generalnot self-similar.

No uniqueness of optimal solutions without further conditions(control theoretic perspective not a modelling one).


Admissible flux traces

Given:T,M > 0,

initial data ρl : Il → [0, ρmax], l ∈ 1, . . . ,m+ n,



Given:T,M > 0,initial data ρl : Il → [0, ρmax], l ∈ 1, . . . ,m+ n,




Consider:


Admissible flux tracesGiven:

T,M > 0,initial data ρl : Il → [0, ρmax], l ∈ 1, . . . ,m+ n,

Consider:

For l ∈ 1, . . . ,m, Il = (−∞, 0] (incoming arcs)∂t ρl + ∂x f(ρl) = 0 x < 0, t > 0

ρl(0, x) = ρl(x) x < 0

ρl(t, 0) = ρl(t) t > 0

Fl = FTl (ρl)

.=f(ρl(·, 0)) | ρl sol on [0, T ]× Il, ρl(t) ∈ [0, ρmax]

FMl = FT,M

l (ρl).=f(ρl(·, 0)) ∈ Fl | TV(ρl(·, 0)) ≤ M


Admissible flux tracesGiven:


Consider:

For l ∈ m+ 1, . . . ,m+ n, Il = [0,+∞) (outgoing arcs)∂t ρl + ∂x f(ρl) = 0 x > 0, t > 0

ρl(0, x) = ρl(x) x > 0

ρl(t, 0) = ρl(t) t > 0

Fl = FTl (ρl)

.=f(ρl(·, 0)) | ρl sol on [0, T ]× Il, ρl(t) ∈ [0, ρmax]

FMl = FT,M

l (ρl).=f(ρl(·, 0)) ∈ Fl | TV(ρl(·, 0)) ≤ M




Consider:

Admissible flux traces compatible with ρ.= (ρ1, . . . , ρm+n)

GM (ρ)=

(g1, . . . , gm)∈

m∏i=1

FMi (ρi) :

m∑i=1

αjigi∈Fj(ρj), j=m+1, . . . ,m+n


Admissible flux traces - inflow controls


Consider:


GM (ρ)=

(g1, . . . , gm)∈

m∏i=1

FMi (ρi) :

m∑i=1


Treat the elements of GM (ρ) as admissible inflow controls at thejunction


Admissible flux traces - inflow controls


Consider:


GM (ρ)=

(g1, . . . , gm)∈

m∏i=1

FMi (ρi) :

m∑i=1



- Implementation of inflow controls at junctions: traffic lights,ramp metering control


Admissible flux traces - inflow controlsGiven:


Consider:


GM (ρ)=

(g1, . . . , gm)∈

m∏i=1

FMi (ρi) :

m∑i=1



- Implementation of inflow controls at junctions: traffic lights,ramp metering control- Goals: decrease traffic congestion, diminish fuel consumption,improve driver safety, improve performance of the traffic system


Existence of optimal solutions

Theorem (Ancona, Cesaroni, Coclite, G.)Let J : Rm → R be continuous. For every M > 0, and for anyinitial data ρ :

∏m+nl=1 Il → [0, ρmax]

m+n, there exists g ∈ GM (ρ)s.t. ∫ T

0J (g(t)) dt = sup

g∈GM (ρ)

∫ T

0J (g(t)) dt

Ex: J (g1, . . . , gm) =∑

i gi, J (g1, . . . , gm) =∏

i gi,

Proof:

Uniform BV bounds and Helly’s Compactness Theorem=⇒ convergence of subsequence of flux tracesDivergence Theorem on [−max |f ′(ρ)| · T, 0]× [0, T ]=⇒ convergence of flux traces of solutions to flux traceof limit solution






0J (g(t)) dt = sup

g∈GM (ρ)

∫ T

0J (g(t)) dt

Ex: J (g1, . . . , gm) =∑

i gi, J (g1, . . . , gm) =∏

i gi,

Proof:Uniform BV bounds and Helly’s Compactness Theorem=⇒ convergence of subsequence of flux traces

Divergence Theorem on [−max |f ′(ρ)| · T, 0]× [0, T ]=⇒ convergence of flux traces of solutions to flux traceof limit solution






0J (g(t)) dt = sup

g∈GM (ρ)

∫ T

0J (g(t)) dt

Ex: J (g1, . . . , gm) =∑

i gi, J (g1, . . . , gm) =∏

i gi,

Proof:Uniform BV bounds and Helly’s Compactness Theorem=⇒ convergence of subsequence of flux tracesDivergence Theorem on [−max |f ′(ρ)| · T, 0]× [0, T ]=⇒ convergence of flux traces of solutions to flux traceof limit solution


Case n = m = 1: comparison with entropy solution of (CP)

Incoming road Outgoing road

∂t ρ1 + ∂x f(ρ1)=0 x<0, t>0,

ρ1(0, x) = ρ1(x) x < 0,

∂t ρ2 + ∂x f(ρ2)=0 x>0, t>0

ρ2(0, x) = ρ2(x) x > 0,

G = F1 ∩ F2 GM = FM1 ∩ F2

supg∈GM

∫ T

0J (g(t)) dt (max)M


Case n = m = 1: comparison with entropy solution of (CP)Incoming road Outgoing road

∂t ρ1 + ∂x f(ρ1)=0 x<0, t>0,

ρ1(0, x) = ρ1(x) x < 0,

∂t ρ2 + ∂x f(ρ2)=0 x>0, t>0

ρ2(0, x) = ρ2(x) x > 0,

G = F1 ∩ F2 GM = FM1 ∩ F2

supg∈GM

∫ T

0J (g(t)) dt (max)M

F1.=f(ρ1(·, 0)) | ρ1 sol on [0, T ]× (−∞, 0], ρ1(t) ∈ [0, ρmax]

,

F2.=f(ρ2(·, 0)) | ρ2 sol on [0, T ]× [0,+∞), ρ2(t) ∈ [0, ρmax]

,

FMl

.=f(ρl(·, 0)) ∈ Fl | TV(ρl(·, 0)) ≤ M

, l = 1, 2.


Case n = m = 1: comparison with entropy solution of (CP)Incoming road Outgoing road

∂t ρ1 + ∂x f(ρ1)=0 x<0, t>0,

ρ1(0, x) = ρ1(x) x < 0,

∂t ρ2 + ∂x f(ρ2)=0 x>0, t>0

ρ2(0, x) = ρ2(x) x > 0,

G = F1 ∩ F2 GM = FM1 ∩ F2

supg∈GM

∫ T

0J (g(t)) dt (max)M

∂t ρ+ ∂x f(ρ) = 0

ρ(0, x) =

ρ1(x) if x < 0

ρ2(x) if x > 0

(CP)

ρe : [0, T ]× R → R entropy admissible sol. to (CP)TV(ρe(·, 0)) ≤ M

=⇒ ρe(·, 0) ∈ GM .


Case n = m = 1: comparison with entropy solution of (CP)

Consider J (g) = g

Theorem (Ancona, Cesaroni, Coclite, G.)For every T > 0, let ρe be the entropy admissible solution to (CP).Then, ρe(·, 0) solves the maximization problem (max)M , i.e.∫ T

0f (ρe(t, 0)) dt = sup

g∈GM (ρe(0,·))

∫ T

0g(t)dt,

for every M ≥ TV(ρe(·, 0)).

The proof relies on Hopf-Lax formula for explicit representation ofviscosity solutions to IBV for Hamilton-Jacobi equation

ρ(t, x) entropy weak sol’n of ∂t ρ+ ∂x f(ρ) = 0⇓

v(t, x).=∫ x−∞ ρ(t, z)dz is viscosity sol’n of ∂t v + f(∂x v) = 0


Case n = m = 1: non entropic optimal solutions

Incoming road Outgoing road

f(ρ) = ρ(1− ρ), ρmax = 1

∂t ρ1 + ∂x (ρ1(1− ρ1)) = 0

ρ1(0, x) =14

∂t ρ2 + ∂x (ρ2(1− ρ2)) = 0

ρ2(0, x) =14


Case n = m = 1: non entropic optimal solutionsIncoming road Outgoing road

f(ρ) = ρ(1− ρ), ρmax = 1

∂t ρ1 + ∂x (ρ1(1− ρ1)) = 0

ρ1(0, x) =14

∂t ρ2 + ∂x (ρ2(1− ρ2)) = 0

ρ2(0, x) =14

The functionsρ1(t, x) ≡

1

4ρ2(t, x) ≡

1

4

provide an optimal solution for any T,M > 0, i.e.

3

16T =

∫ T

0f (ρ1(t, 0)) dt = sup

g∈GM

∫ T

0g(t)dt



The functions

0

1

3

83

14

78

58

14

14

38

18

provide another optimal solution for any T > 8/3.

∫ T

0f (ρ(t, 0)) dt =



The functions

0

1

3

83

14

78

58

14

14

38

18

provide another optimal solution for any T > 8/3.∫ T

0f (ρ(t, 0)) dt = f

(7

8

)· 1 + f

(5

8

)·5

3+ f

(1

4

)·(T −

8

3

)



The functions

0

1

3

83

14

78

58

14

14

38

18


0f (ρ(t, 0)) dt =

7

64+

15

64·5

3+

3

16·(T −

8

3

)



The functions

0

1

3

83

14

78

58

14

14

38

18


0f (ρ(t, 0)) dt =

7

64+

15

64·5

3+

3

16·(T −

8

3

)=

3

16T


Optimization w.r.t. drivers’ preference parameters

Given:T,M > 0

initial data ρl : Il → [0, ρmax], l ∈ 1, . . . ,m+ na Markov matrix valued map t 7→ A(t), t ∈ [0, T ]


and satisfying constraints given by A:

GM

A(ρ) =

(g1, . . . , gm)∈

m∏i=1

FMi (ρi) :

m∑i=1

aji(t)gi(t) ∈ Fj(ρj),

for a.e. t∈ [0, T ], j=m+1, . . . ,m+n, TV(aij) ≤ M ∀i, j



Given:T,M > 0

initial data ρl : Il → [0, ρmax], l ∈ 1, . . . ,m+ n

a Markov matrix valued map t 7→ A(t), t ∈ [0, T ]



GM

A(ρ) =

(g1, . . . , gm)∈

m∏i=1

FMi (ρi) :

m∑i=1





Given:T,M > 0




GM

A(ρ) =

(g1, . . . , gm)∈

m∏i=1

FMi (ρi) :

m∑i=1





Given:T,M > 0


Consider:



GM

A(ρ) =

(g1, . . . , gm)∈

m∏i=1

FMi (ρi) :

m∑i=1




Optimization w.r.t. drivers’ preference parametersGiven:

T,M > 0


Consider:



GM

A(ρ) =

(g1, . . . , gm)∈

m∏i=1

FMi (ρi) :

m∑i=1



Treat (A, g), with A an n×m Markov matrix valued map, andg ∈ GM

A (ρ), as admissible pair of inflow and driver preferencecontrols at the junction


Existence of optimal solutions w.r.t. drivers’ preference



m+n, there exists an n×m

Markov matrix valued map A, and g ∈ GMA

(ρ), s.t.∫ T

0J (g(t)) dt = sup

A∈MMn×m, g∈GM

A (ρ)

∫ T

0J (g(t)) dt.

(MMn×m : set of n×m Markov matrix valued maps

A(·) = (aij(·))ij with TVaij ≤ M , for all i, j)

Proof:

Same arguments as for the optimal control problems with afixed distributional matrix A.


Existence of optimal solutions w.r.t. drivers’ preference



m+n, there exists an n×m

Markov matrix valued map A, and g ∈ GMA

(ρ), s.t.∫ T

0J (g(t)) dt = sup

A∈MMn×m, g∈GM

A (ρ)

∫ T

0J (g(t)) dt.

(MMn×m : set of n×m Markov matrix valued maps

A(·) = (aij(·))ij with TVaij ≤ M , for all i, j)

Proof:Same arguments as for the optimal control problems with afixed distributional matrix A.


Additional optimization criterium

Fix T,M > 0. In connection with the nodal problem∂t ρl + ∂x f(ρl) = 0, x ∈ Il, l ∈ 1, . . . ,m+ n, t > 0,

ρl(0, x) = ρl(x), x ∈ Il, l ∈ 1, . . . ,m+ n,

with flux constraints

A·(f(ρ1(t, 0)), . . . , f(ρm(t, 0)))T = (f(ρm+1(t, 0)), . . . , f(ρm+n(t, 0)))T ,

let GM (ρ) be the set of admissible flux traces as above, anddenote by DM (ρ) the set of optimal solutions of

supg∈GM (ρ)

∫ T

0J (g(t)) dt. (max)M

Then minimizesmin

g∈DM (ρ)

m∑i=1

TV[0,T ]

gi . (min)M


Additional optimization criterium: existence of solution

Theorem (Ancona, Cesaroni, Coclite, G.)Let J : Rn → R be continuous. For every M > 0, and for anyinitial data ρ :



0J (g(t)) dt = sup

g∈GM (ρ)

∫ T

0J (g(t)) dt

andm∑i=1

TV[0,T ]

gi = ming∈DM (ρ)

m∑i=1

TV[0,T ]

gi .

Notice: solutions of the min-max problems fulfills the requirementof maximize the total flux through the junction keeping theoscillation the smallest possibleThe proof relies on the compactness of the set GM with respect tothe L1-topology and the lower semicontinuity of the total variation






0J (g(t)) dt = sup

g∈GM (ρ)

∫ T

0J (g(t)) dt

andm∑i=1

TV[0,T ]

gi = ming∈DM (ρ)

m∑i=1

TV[0,T ]

gi .

Notice: solutions of the min-max problems fulfills the requirementof maximize the total flux through the junction keeping theoscillation the smallest possible

The proof relies on the compactness of the set GM with respect tothe L1-topology and the lower semicontinuity of the total variation






0J (g(t)) dt = sup

g∈GM (ρ)

∫ T

0J (g(t)) dt

andm∑i=1

TV[0,T ]

gi = ming∈DM (ρ)

m∑i=1

TV[0,T ]

gi .

Notice: solutions of the min-max problems fulfills the requirementof maximize the total flux through the junction keeping theoscillation the smallest possibleThe proof relies on the compactness of the set GM with respect tothe L1-topology and the lower semicontinuity of the total variation


Equivalent variational formulationFor every δ > 0 consider

sup(g,A)∈GM (ρ)

(∫ T

0J (g(t)) dt− δ

m∑i=1

TV[0,T ]gi

)

Theorem (Ancona, Cesaroni, Coclite, G.)Let J : Rn → R be continuous. For every δν → 0, there exists asubsequence δνk and

(g, A

)∈ GM (ρ) such that(

gδνk , Aδνk

)→(g, A

)∫ T

0J (g(t)) dt = sup

g∈GM (ρ)

∫ T

0J (g(t)) dt

m∑i=1

TV[0,T ]

gi = ming∈DM (ρ)

m∑i=1

TV[0,T ]

gi .


Case n = m = 1: comparison with entropy solution of CP

Consider J (g) = g

Theorem (Ancona, Cesaroni, Coclite, G.)For every T > 0, let ρe be the entropy admissible solution to (CP)and assume that ρe(0, ·), f(ρe(0, ·)) are monotone. Then, ρe(·, 0)solves the min-max problem (min)M , i.e.

TV[0,T ]

(ρe(·, 0)) = ming∈DM (ρe(0,·))

TV[0,T ]

g

for every M ≥ TV(ρe(·, 0)).


Case n = m = 1: comparison with entropy solution of CPNotice: If the entropy admissible solution to (CP) does

not satisfy the monotonicity requirement, thenit is not solution of the min-max problem




Initial data (a2 < a1 < σ < a3)

ρ1(x) =

a1 if x < x1a2 if x1 < x < x2a3 if x > x2

ρ2(x) = a3

t

T

xx1 x2

a1

a2 a3




Initial data (a2 < a1 < σ < a3)

ρ1(x) =

a1 if x < x1a2 if x1 < x < x2a3 if x > x2

ρ2(x) = a3

t

T

xx1 x2

a1

a2 a3t

T

xx1x2

a1

a2a3

b

c

a3


Prospective research directions

Analyze optimal solutions for 1× 2, 2× 1 and 2× 2 junctions

Analyze optimal solutions involving inflow junctions controlsat two (or more) nodes

Consider cost functionals depending on the whole value of thesolution on [0, T ] along the incoming roads (and not just onthe inflow on [0, T ] at the junction)

Consider feedback type optimal controls at junctions


Thank you for your attention!


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