Control problems for traffjc fmow Mauro Garavello University of - - PowerPoint PPT Presentation

Control problems for traffjc fmow

Mauro Garavello University of Milano Bicocca OptHySYS

University of Trento, January 9-11, 2017

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

The LWR traffjc fmow model (1955, 1956)

∂t ρ + ∂x f(ρ) = 0

: density of cars at time and at the position : fmux of cars , where is the average velocity depends only on in a decreasing way , : maximum density, : maximum velocity is a strictly concave function

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

The LWR traffjc fmow model (1955, 1956)

∂t ρ + ∂x f(ρ) = 0

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

x

ρ

: fmux of cars , where is the average velocity depends only on in a decreasing way , : maximum density, : maximum velocity is a strictly concave function

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

The LWR traffjc fmow 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): fmux of cars

, where is the average velocity depends only on in a decreasing way , : maximum density, : maximum velocity is a strictly concave function

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

The LWR traffjc fmow 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): fmux of cars f(ρ) = ρv, where v is the average velocity

depends only on in a decreasing way , : maximum density, : maximum velocity is a strictly concave function

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

The LWR traffjc fmow 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): fmux 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

) , ρmax: maximum density, Vmax: maximum velocity is a strictly concave function

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

The LWR traffjc fmow 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): fmux 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

) , ρmax: maximum density, Vmax: maximum velocity

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

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

The LWR traffjc fmow 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): fmux 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

) , ρmax: maximum density, Vmax: maximum velocity

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

ρ ρ v f σ ρmax ρmax Vmax

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Traffjc fmow on a network of roads

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

• node:

incoming arcs,

• utgoing arcs
• initial datum

in each arc , interval of

• This corresponds to

IBV problems

Coupling transition condition at a node located at

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Traffjc fmow on a network of roads

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

• node:

incoming arcs,

• utgoing arcs
• initial datum

in each arc , interval of

• This corresponds to

IBV problems

Coupling transition condition at a node located at

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Traffjc fmow on a network of roads

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

• node: m incoming arcs, n outgoing arcs
• initial datum

in each arc , interval of

• This corresponds to

IBV problems I1 I2 I3 I4 I5 I6 I7 I8 I9

Coupling transition condition at a node located at

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Traffjc fmow on a network of roads

Conservation laws evolving on a directed graph (fjnite collection 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

IBV problems

ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8 ρ9

Coupling transition condition at a node located at

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Traffjc fmow on a network of roads

Conservation laws evolving on a directed graph (fjnite collection 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 + n} ρl(t, 0) = ???, t > 0, l ∈ {1, · · · , n + m}

ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8 ρ9

Coupling transition condition at a node located at

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Traffjc fmow on a network of roads

Conservation laws evolving on a directed graph (fjnite collection 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 + n} ρl(t, 0) = ???, t > 0, l ∈ {1, · · · , n + m}

ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8 ρ9

Coupling transition condition at a node located at x = 0 Ψ(ρ1(t, 0), . . . , ρm+n(t, 0)) = 0

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

PDEs on networks

Vehicular traffjc (scalar and systems) Air traffjc management Gas pipelines (systems) Supply chains (scalar) Blood circulatory fmow & vascular stents (systems) Irrigation channels (systems) Biological networks (scalar) Telecommunication and data networks (scalar)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

PDEs on networks

Vehicular traffjc (scalar and systems) Air traffjc management Gas pipelines (systems) Supply chains (scalar) Blood circulatory fmow & vascular stents (systems) Irrigation channels (systems) Biological networks (scalar) Telecommunication and data networks (scalar)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

PDEs on networks

Vehicular traffjc (scalar and systems) Air traffjc management Gas pipelines (systems) Supply chains (scalar) Blood circulatory fmow & vascular stents (systems) Irrigation channels (systems) Biological networks (scalar) Telecommunication and data networks (scalar)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

PDEs on networks

Vehicular traffjc (scalar and systems) Air traffjc management Gas pipelines (systems) Supply chains (scalar) Blood circulatory fmow & vascular stents (systems) Irrigation channels (systems) Biological networks (scalar) Telecommunication and data networks (scalar)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

PDEs on networks

Vehicular traffjc (scalar and systems) Air traffjc management Gas pipelines (systems) Supply chains (scalar) Blood circulatory fmow & vascular stents (systems) Irrigation channels (systems) Biological networks (scalar) Telecommunication and data networks (scalar)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

PDEs on networks

Vehicular traffjc (scalar and systems) Air traffjc management Gas pipelines (systems) Supply chains (scalar) Blood circulatory fmow & vascular stents (systems) Irrigation channels (systems) Biological networks (scalar) Telecommunication and data networks (scalar)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

PDEs on networks

Vehicular traffjc (scalar and systems) Air traffjc management Gas pipelines (systems) Supply chains (scalar) Blood circulatory fmow & vascular stents (systems) Irrigation channels (systems) Biological networks (scalar) Telecommunication and data networks (scalar)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

PDEs on networks

Vehicular traffjc (scalar and systems) Air traffjc management Gas pipelines (systems) Supply chains (scalar) Blood circulatory fmow & vascular stents (systems) Irrigation channels (systems) Biological networks (scalar) Telecommunication and data networks (scalar)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

PDEs on networks

Vehicular traffjc (scalar and systems) Air traffjc management Gas pipelines (systems) Supply chains (scalar) Blood circulatory fmow & 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...

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Nodal conditions on road networks

Incoming roads : Il = (−∞, 0], Outgoing roads : Il = [0, +∞) Conservation of the number of cars

for a.e.

Distribution rules: (percentage of drivers coming from

• th incoming road and turning into
• th outgoing road)

for a.e.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Nodal conditions on road networks

Incoming roads : Il = (−∞, 0], Outgoing roads : Il = [0, +∞) Conservation of the number of cars

for a.e.

Distribution rules: (percentage of drivers coming from

• th incoming road and turning into
• th outgoing road)

for a.e.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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: (percentage of drivers coming from

• th incoming road and turning into
• th outgoing road)

for a.e.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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 from i-th incoming road and turning into j-th outgoing road)

for a.e.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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 from i-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,

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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 from i-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, i

m+n

∑

j=m+1

aji(t) = 1 ∀ i

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Nodal conditions on road networks. . . continued

Remark: distribution rules are not suffjcient to select unique solution at the junction. Optimization criterion imposed on the traces of the solution [maximize the fmux through the junction] Priority rules: [percentage of time drivers from -th incoming road passing through the junction], (when and the total possible fmux on the incoming roads is larger than the maximal fmux that the outgoing roads can handle)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Nodal conditions on road networks. . . continued

Remark: distribution rules are not suffjcient to select unique solution at the junction.

• Optimization criterion imposed on the traces of the solution

[maximize the fmux through the junction] Priority rules: [percentage of time drivers from -th incoming road passing through the junction], (when and the total possible fmux on the incoming roads is larger than the maximal fmux that the outgoing roads can handle)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Nodal conditions on road networks. . . continued

Remark: distribution rules are not suffjcient to select unique solution at the junction.

• Optimization criterion imposed on the traces of the solution

[maximize the fmux through the junction] Priority rules: [percentage of time drivers from -th incoming road passing through the junction], (when and the total possible fmux on the incoming roads is larger than the maximal fmux that the outgoing roads can handle)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Nodal conditions on road networks. . . continued

Remark: distribution rules are not suffjcient to select unique solution at the junction.

• Optimization criterion imposed on the traces of the solution

[maximize the fmux through the junction] Priority rules: ci [percentage of time drivers from i-th incoming road passing through the junction], ∑m

i=1 ci(t) = 1

(when and the total possible fmux on the incoming roads is larger than the maximal fmux that the outgoing roads can handle)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Nodal conditions on road networks. . . continued

Remark: distribution rules are not suffjcient to select unique solution at the junction.

• Optimization criterion imposed on the traces of the solution

[maximize the fmux through the junction] Priority rules: ci [percentage of time drivers from i-th incoming road passing through the junction], ∑m

i=1 ci(t) = 1

(when m > n and the total possible fmux on the incoming roads is larger than the maximal fmux that the outgoing roads can handle)

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Riemann problem approach

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

Junction Riemann Solver (JRS): provide nodal conditions and a procedure for (uniquely) solving IBV (Riemann) problems at incoming and

• utgoing roads when initial

data are constants. Solutions are required to be self-similar as for classical Riemann problems. Construct front tracking approximate solutions with piecewise constant initial data on each road. A-priori BV bounds yield the convergence of front tracking approximations to the (unique?) solution of the Cauchy problem at the node with general initial data (having fjnite total variation).

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Riemann problem approach

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

Junction Riemann Solver (JRS): provide nodal conditions and a procedure for (uniquely) solving m + n IBV (Riemann) problems at m incoming and n outgoing roads when initial data are constants. Solutions are required to be self-similar as for classical Riemann problems. Construct front tracking approximate solutions with piecewise constant initial data on each road. A-priori BV bounds yield the convergence of front tracking approximations to the (unique?) solution of the Cauchy problem at the node with general initial data (having fjnite total variation).

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Riemann problem approach

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

Junction Riemann Solver (JRS): provide nodal conditions and a procedure for (uniquely) solving m + n IBV (Riemann) problems at m incoming and n outgoing roads when initial data are constants. Solutions are required to be self-similar as for classical Riemann problems. Construct front tracking approximate solutions with piecewise constant initial data on each road. A-priori BV bounds yield the convergence of front tracking approximations to the (unique?) solution of the Cauchy problem at the node with general initial data (having fjnite total variation).

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Riemann problem approach

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

Junction Riemann Solver (JRS): provide nodal conditions and a procedure for (uniquely) solving m + n IBV (Riemann) problems at m incoming and n outgoing roads when initial data are constants. Solutions are required to be self-similar as for classical Riemann problems. Construct front tracking approximate solutions with piecewise constant initial data on each road. A-priori BV bounds yield the convergence of front tracking approximations to the (unique?) solution of the Cauchy problem at the node with general initial data (having fjnite total variation).

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Junction Riemann Solver

A Riemann problem at a node is a Cauchy problem with constant initial condition on each arc Il: Giving a solution is equivalent to giving its trace at the node: A Riemann solver at the node is a map that associates to an

• tuple of constant initial data an
• tuple of constant boundary data which are the traces

at the node of the solution to the corresponding Riemann problem.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Junction Riemann Solver

A Riemann problem at a node is a Cauchy problem with constant initial 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: A Riemann solver at the node is a map that associates to an

• tuple of constant initial data an
• tuple of constant boundary data which are the traces

at the node of the solution to the corresponding Riemann problem.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Junction Riemann Solver

A Riemann problem at a node is a Cauchy problem with constant initial 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 map that associates to an

• tuple of constant initial data an
• tuple of constant boundary data which are the traces

at the node of the solution to the corresponding Riemann problem.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Junction Riemann Solver

A Riemann problem at a node is a Cauchy problem with constant initial 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 map RS : [0, ρmax]m+n → [0, ρmax]m+n, (ρ1, . . . , ρm+n) → ( 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 traces at the node of the solution to the corresponding Riemann problem.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Junction Riemann Solver. . . continued

For 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 (outgoing arcs) the classical Riemann problem if if is solved by waves with positive speeds.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Junction Riemann Solver. . . continued

For 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 Riemann problem        ∂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

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Solution for the Junction Riemann Problem

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

• n the traces of the fmuxes (fmux distribution

matrix ) Choose the self-similar solution which maximizes If needed ( ), prescribe relative priority rules

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

• n the traces of the fmuxes (fmux distribution ↔ matrix A)

Choose the self-similar solution which maximizes If needed ( ), prescribe relative priority rules

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

• n the traces of the fmuxes (fmux distribution ↔ matrix A)

Choose the self-similar solution which maximizes ∑n

l=1 f(ρl(t, 0))

If needed ( ), prescribe relative priority rules

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

• n the traces of the fmuxes (fmux distribution ↔ matrix A)

Choose the self-similar solution which maximizes ∑n

l=1 f(ρl(t, 0))

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

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

• n the traces of the fmuxes (fmux distribution ↔ matrix A)

Choose the self-similar solution which maximizes ∑n

l=1 f(ρl(t, 0))

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

• Ω =

{ (γ1, · · · , γn) ∈ ∏n

i=1 Ωi : A · (γ1, · · · , γn)T ∈ ∏n+m j=n+1 Ωj

}

• Maximize E = γ1 + · · · + γn on Ω
• Find the corresponding densities
• Select the densities that satisfy the priority rules, if needed

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

• n the traces of the fmuxes (fmux distribution ↔ matrix A)

Choose the self-similar solution which maximizes ∑n

l=1 f(ρl(t, 0))

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

γ1 γ2

γ1,max γ2,max Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

• n the traces of the fmuxes (fmux distribution ↔ matrix A)

Choose the self-similar solution which maximizes ∑n

l=1 f(ρl(t, 0))

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

γ1 γ2

γ1,max γ2,max Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

• n the traces of the fmuxes (fmux distribution ↔ matrix A)

Choose the self-similar solution which maximizes ∑n

l=1 f(ρl(t, 0))

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

γ1 γ2

γ1,max γ2,max α1,3γ1 + α2,3γ2 = γ3,max Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

• n the traces of the fmuxes (fmux distribution ↔ matrix A)

Choose the self-similar solution which maximizes ∑n

l=1 f(ρl(t, 0))

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

γ1 γ2

γ1,max γ2,max α1,3γ1 + α2,3γ2 = γ3,max α1,4γ1 + α2,4γ2 = γ4,max Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

• n the traces of the fmuxes (fmux distribution ↔ matrix A)

Choose the self-similar solution which maximizes ∑n

l=1 f(ρl(t, 0))

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

γ1 γ2

γ1,max γ2,max α1,3γ1 + α2,3γ2 = γ3,max α1,4γ1 + α2,4γ2 = γ4,max

Ω

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

• n the traces of the fmuxes (fmux distribution ↔ matrix A)

Choose the self-similar solution which maximizes ∑n

l=1 f(ρl(t, 0))

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

γ1 γ2

γ1,max γ2,max α1,3γ1 + α2,3γ2 = γ3,max α1,4γ1 + α2,4γ2 = γ4,max

Ω Level curve of γ1 + γ2

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Matrix A as a control function

Theorem: existence of solution (G., Piccoli)

Assume that the traffjc distribution matrix A = A(t) is time dependent and 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 the Cauchy 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].

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

A difgerent 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 satisfjes the fmux constraint

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

and maximizes an integral cost ∫ T

m

∑

l=1

f (ρl(t, 0)) dt The maximization is global on among all admissible solutions (fulfjlling linear fmux constraints) and not pointwise in time

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

A difgerent 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 satisfjes the fmux constraint

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

and maximizes an integral cost ∫ T J (f (ρ1(t, 0)) , . . . , f (ρm(t, 0))) dt (J : Rm → R). The maximization is global on among all admissible solutions (fulfjlling linear fmux constraints) and not pointwise in time

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

A difgerent 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 satisfjes the fmux constraint

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

and maximizes an integral cost ∫ T J (f (ρ1(t, 0)) , . . . , f (ρm(t, 0))) dt (J : Rm → R). The maximization is global on [0, T] among all admissible solutions (fulfjlling linear fmux constraints) and not pointwise in time

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

New features of optimal solutions

In general there is no Junction Riemann Solver compatible with an optimal solution. Optimal solutions with constant initial densities are in general not self-similar. No uniqueness of optimal solutions without further conditions (control theoretic perspective not a modelling one).

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

New features of optimal solutions

In general there is no Junction Riemann Solver compatible with an optimal solution. Optimal solutions with constant initial densities are in general not self-similar. No uniqueness of optimal solutions without further conditions (control theoretic perspective not a modelling one).

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

New features of optimal solutions

In general there is no Junction Riemann Solver compatible with an optimal solution. Optimal solutions with constant initial densities are in general not self-similar. No uniqueness of optimal solutions without further conditions (control theoretic perspective not a modelling one).

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Admissible fmux traces

Given: T, M > 0, initial data , ,

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Admissible fmux traces

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

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Admissible fmux traces

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

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Admissible fmux traces

Given: 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 = FT

l (ρl) .

= { f(ρl(·, 0)) | ρl sol on [0, T] × Il, ˜ ρl(t) ∈ [0, ρmax] } FM

l

= FT,M

l

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

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Admissible fmux traces

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

l (ρl) .

= { f(ρl(·, 0)) | ρl sol on [0, T] × Il, ˜ ρl(t) ∈ [0, ρmax] } FM

l

= FT,M

l

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

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Admissible fmux traces

Given: T, M > 0, initial data ρl : Il → [0, ρmax], l ∈ {1, . . . , m + n}, Consider: Admissible fmux traces compatible with ρ . = (ρ1, . . . , ρm+n)

GM(ρ)= { (g1, . . . , gm)∈

m

∏

i=1

FM

i (ρi) : m

∑

i=1

αjigi ∈Fj(ρj), j =m+1, . . . , m+n }

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Admissible fmux traces - infmow controls

Given: T, M > 0, initial data ρl : Il → [0, ρmax], l ∈ {1, . . . , m + n}, Consider: Admissible fmux traces compatible with ρ . = (ρ1, . . . , ρm+n)

GM(ρ)= { (g1, . . . , gm)∈

m

∏

i=1

FM

i (ρi) : m

∑

i=1

αjigi ∈Fj(ρj), j =m+1, . . . , m+n } Treat the elements of GM(ρ) as admissible infmow controls at the junction

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Admissible fmux traces - infmow controls

Given: T, M > 0, initial data ρl : Il → [0, ρmax], l ∈ {1, . . . , m + n}, Consider: Admissible fmux traces compatible with ρ . = (ρ1, . . . , ρm+n)

GM(ρ)= { (g1, . . . , gm)∈

m

∏

i=1

FM

i (ρi) : m

∑

i=1

αjigi ∈Fj(ρj), j =m+1, . . . , m+n } Treat the elements of GM(ρ) as admissible infmow controls at the junction

• Implementation of infmow controls at junctions: traffjc lights,

ramp metering control

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Admissible fmux traces - infmow controls

Given: T, M > 0, initial data ρl : Il → [0, ρmax], l ∈ {1, . . . , m + n}, Consider: Admissible fmux traces compatible with ρ . = (ρ1, . . . , ρm+n)

GM(ρ)= { (g1, . . . , gm)∈

m

∏

i=1

FM

i (ρi) : m

∑

i=1

αjigi ∈Fj(ρj), j =m+1, . . . , m+n } Treat the elements of GM(ρ) as admissible infmow controls at the junction

• Implementation of infmow controls at junctions: traffjc lights,

ramp metering control

• Goals: decrease traffjc congestion, diminish fuel consumption,

improve driver safety, improve performance of the traffjc system

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Existence of optimal solutions

Theorem (Ancona, Cesaroni, Coclite, G.) Let J : Rm → R be continuous. For every M > 0, and for any initial data ρ : ∏m+n

l=1 Il → [0, ρmax]m+n, there exists

g ∈ GM(ρ) s.t. ∫ T J ( g(t)) dt = sup

g∈GM(ρ)

∫ T J (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 fmux traces Divergence Theorem on max = convergence of fmux traces of solutions to fmux trace

• f limit solution

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Existence of optimal solutions

Theorem (Ancona, Cesaroni, Coclite, G.) Let J : Rm → R be continuous. For every M > 0, and for any initial data ρ : ∏m+n

l=1 Il → [0, ρmax]m+n, there exists

g ∈ GM(ρ) s.t. ∫ T J ( g(t)) dt = sup

g∈GM(ρ)

∫ T J (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 fmux traces Divergence Theorem on max = convergence of fmux traces of solutions to fmux trace

• f limit solution

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Existence of optimal solutions

Theorem (Ancona, Cesaroni, Coclite, G.) Let J : Rm → R be continuous. For every M > 0, and for any initial data ρ : ∏m+n

l=1 Il → [0, ρmax]m+n, there exists

g ∈ GM(ρ) s.t. ∫ T J ( g(t)) dt = sup

g∈GM(ρ)

∫ T J (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 fmux traces Divergence Theorem on [− max |f′(ρ)| · T, 0] × [0, T] = ⇒ convergence of fmux traces of solutions to fmux trace

• f limit solution

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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 = FM

1

∩ F2 sup

g∈GM

∫ T J (g(t)) dt (max)M

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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 = FM

1

∩ F2 sup

g∈GM

∫ T J (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] } , FM

l

. = { f(ρl(·, 0)) ∈ Fl | TV(ρl(·, 0)) ≤ M } , l = 1, 2.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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 = FM

1

∩ F2 sup

g∈GM

∫ T J (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.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Case n = m = 1: comparison with entropy solution of (CP) Consider J (g) = g Theorem (Ancona, Cesaroni, Coclite, G.) For every , let be the entropy admissible solution to (CP). Then, solves the maximization problem , i.e. sup for every TV . The proof relies on Hopf-Lax formula for explicit representation of viscosity solutions to IBV for Hamilton-Jacobi equation entropy weak sol’n of is viscosity sol’n of

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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 f (ρe(t, 0)) dt = sup

g∈GM(ρe(0,·))

∫ T g(t)dt, for every M ≥ TV(ρe(·, 0)). The proof relies on Hopf-Lax formula for explicit representation of viscosity solutions to IBV for Hamilton-Jacobi equation entropy weak sol’n of is viscosity sol’n of

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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 f (ρe(t, 0)) dt = sup

g∈GM(ρe(0,·))

∫ T g(t)dt, for every M ≥ TV(ρe(·, 0)). The proof relies on Hopf-Lax formula for explicit representation of viscosity 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

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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) = 1

4

{ ∂t ρ2 + ∂x (ρ2(1 − ρ2)) = 0 ρ2(0, x) = 1

4

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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) = 1

4

{ ∂t ρ2 + ∂x (ρ2(1 − ρ2)) = 0 ρ2(0, x) = 1

4

The functions ρ1(t, x) ≡ 1 4 ρ2(t, x) ≡ 1 4 provide an optimal solution for any T, M > 0, i.e. 3 16T = ∫ T f (ρ1(t, 0)) dt = sup

g∈GM

∫ T g(t)dt

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Case n = m = 1: non entropic optimal solutions

The functions

1 3

8 3 1 4 7 8 5 8 1 4 1 4 3 8 1 8

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

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Case n = m = 1: non entropic optimal solutions

The functions

1 3

8 3 1 4 7 8 5 8 1 4 1 4 3 8 1 8

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

∫ T f (ρ(t, 0)) dt = f ( 7 8 ) · 1 + f (5 8 ) · 5 3 + f ( 1 4 ) · ( T − 8 3 )

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Case n = m = 1: non entropic optimal solutions

The functions

1 3

8 3 1 4 7 8 5 8 1 4 1 4 3 8 1 8

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

∫ T f (ρ(t, 0)) dt = 7 64 + 15 64 · 5 3 + 3 16 · ( T − 8 3 )

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Case n = m = 1: non entropic optimal solutions

The functions

1 3

8 3 1 4 7 8 5 8 1 4 1 4 3 8 1 8

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

∫ T f (ρ(t, 0)) dt = 7 64 + 15 64 · 5 3 + 3 16 · ( T − 8 3 ) = 3 16 T

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Optimization w.r.t. drivers’ preference parameters

Given: T, M > 0 initial data , a Markov matrix valued map Admissible fmux traces compatible with and satisfying constraints given by A:

a.e. TV

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Optimization w.r.t. drivers’ preference parameters

Given: T, M > 0 initial data ρl : Il → [0, ρmax], l ∈ {1, . . . , m + n} a Markov matrix valued map Admissible fmux traces compatible with and satisfying constraints given by A:

a.e. TV

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Optimization w.r.t. drivers’ preference parameters

Given: T, M > 0 initial data ρl : Il → [0, ρmax], l ∈ {1, . . . , m + n} a Markov matrix valued map t → A(t), t ∈ [0, T] Admissible fmux traces compatible with and satisfying constraints given by A:

a.e. TV

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Optimization w.r.t. drivers’ preference parameters

Given: T, M > 0 initial data ρl : Il → [0, ρmax], l ∈ {1, . . . , m + n} a Markov matrix valued map t → A(t), t ∈ [0, T] Consider: Admissible fmux traces compatible with and satisfying constraints given by A:

a.e. TV

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Optimization w.r.t. drivers’ preference parameters

Given: T, M > 0 initial data ρl : Il → [0, ρmax], l ∈ {1, . . . , m + n} a Markov matrix valued map t → A(t), t ∈ [0, T] Consider: Admissible fmux traces compatible with ρ . = (ρ1, . . . , ρm+n) and satisfying constraints given by A:

GM

A (ρ) =

{ (g1, . . . , gm)∈

m

∏

i=1

FM

i (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 }

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Optimization w.r.t. drivers’ preference parameters

Given: T, M > 0 initial data ρl : Il → [0, ρmax], l ∈ {1, . . . , m + n} a Markov matrix valued map t → A(t), t ∈ [0, T] Consider: Admissible fmux traces compatible with ρ . = (ρ1, . . . , ρm+n) and satisfying constraints given by A:

GM

A (ρ) =

{ (g1, . . . , gm)∈

m

∏

i=1

FM

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

A (ρ), as admissible pair of infmow and driver preference

controls at the junction

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

Theorem (Ancona, Cesaroni, Coclite, G.) Let J : Rm → R be continuous. For every M > 0, and for any initial data ρ : ∏m+n

l=1 Il → [0, ρmax]m+n, there exists an n×m

Markov matrix valued map A, and g ∈ GM

• A (ρ), s.t.

∫ T J ( g(t)) dt = sup

A∈MM

n×m, g∈GM A (ρ)

∫ T J (g(t)) dt. (MM

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

A(·) = (aij(·))ij with TV{aij} ≤ M, for all i, j) Proof: Same arguments as for the optimal control problems with a fjxed distributional matrix .

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

Theorem (Ancona, Cesaroni, Coclite, G.) Let J : Rm → R be continuous. For every M > 0, and for any initial data ρ : ∏m+n

l=1 Il → [0, ρmax]m+n, there exists an n×m

Markov matrix valued map A, and g ∈ GM

• A (ρ), s.t.

∫ T J ( g(t)) dt = sup

A∈MM

n×m, g∈GM A (ρ)

∫ T J (g(t)) dt. (MM

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

A(·) = (aij(·))ij with TV{aij} ≤ M, for all i, j) Proof: Same arguments as for the optimal control problems with a fjxed distributional matrix A.

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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 fmux 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 fmux traces as above, and denote by DM(ρ) the set of optimal solutions of sup

g∈GM(ρ)

∫ T J (g(t)) dt. (max)M Then minimizes min

g∈DM(ρ) m

∑

i=1

TV[0,T] gi . (min)M

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Additional optimization criterium: existence of solution

Theorem (Ancona, Cesaroni, Coclite, G.) Let J : Rn → R be continuous. For every M > 0, and for any initial data ρ : ∏m+n

l=1 Il → [0, ρmax]m+n, there exists

g ∈ GM(ρ) s.t. ∫ T J ( g(t)) dt = sup

g∈GM(ρ)

∫ T J (g(t)) dt and

m

∑

i=1

TV[0,T] gi = min

g∈DM(ρ) m

∑

i=1

TV[0,T] gi . Notice: solutions of the min-max problems fulfjlls the requirement

• f maximize the total fmux through the junction keeping the
• scillation the smallest possible

The proof relies on the compactness of the set with respect to the

• topology and the lower semicontinuity of the total variation

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Additional optimization criterium: existence of solution

Theorem (Ancona, Cesaroni, Coclite, G.) Let J : Rn → R be continuous. For every M > 0, and for any initial data ρ : ∏m+n

l=1 Il → [0, ρmax]m+n, there exists

g ∈ GM(ρ) s.t. ∫ T J ( g(t)) dt = sup

g∈GM(ρ)

∫ T J (g(t)) dt and

m

∑

i=1

TV[0,T] gi = min

g∈DM(ρ) m

∑

i=1

TV[0,T] gi . Notice: solutions of the min-max problems fulfjlls the requirement

• f maximize the total fmux through the junction keeping the
• scillation the smallest possible

The proof relies on the compactness of the set with respect to the

• topology and the lower semicontinuity of the total variation

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Additional optimization criterium: existence of solution

Theorem (Ancona, Cesaroni, Coclite, G.) Let J : Rn → R be continuous. For every M > 0, and for any initial data ρ : ∏m+n

l=1 Il → [0, ρmax]m+n, there exists

g ∈ GM(ρ) s.t. ∫ T J ( g(t)) dt = sup

g∈GM(ρ)

∫ T J (g(t)) dt and

m

∑

i=1

TV[0,T] gi = min

g∈DM(ρ) m

∑

i=1

TV[0,T] gi . Notice: solutions of the min-max problems fulfjlls the requirement

• f maximize the total fmux through the junction keeping the
• scillation the smallest possible

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

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Equivalent variational formulation

For every δ > 0 consider sup

(g,A)∈GM(¯ ρ)

(∫ T J (g(t)) dt − δ

m

∑

i=1

TV[0,T]gi ) Theorem (Ancona, Cesaroni, Coclite, G.) Let be continuous. For every , there exists a subsequence and such that sup TV min TV

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Equivalent variational formulation

For every δ > 0 consider sup

(g,A)∈GM(¯ ρ)

(∫ T J (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 a subsequence δνk and (

• g,

A ) ∈ GM(¯ ρ) such that (

• gδνk,

Aδνk ) → (

• g,

A ) ∫ T J ( g(t)) dt = sup

g∈GM(ρ)

∫ T J (g(t)) dt

m

∑

i=1

TV[0,T] gi = min

g∈DM(ρ) m

∑

i=1

TV[0,T] gi .

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

Consider J (g) = g Theorem (Ancona, Cesaroni, Coclite, G.) For every , let be the entropy admissible solution to (CP) and assume that , are monotone. Then, solves the min-max problem , i.e. TV min TV for every TV .

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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)) = min

g∈DM(ρe(0,·)) TV[0,T] g

for every M ≥ TV(ρe(·, 0)).

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

Notice: If the entropy admissible solution to (CP) does not satisfy the monotonicity requirement, then it is not solution of the min-max problem

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

Notice: If the entropy admissible solution to (CP) does not satisfy the monotonicity requirement, then it is not solution of the min-max problem Initial data (a2 < a1 < σ < a3) ρ1(x) =    a1 if x < ¯ x1 a2 if ¯ x1 < x < ¯ x2 a3 if x > ¯ x2 ρ2(x) = a3 t T x ¯ x1 ¯ x2 a1 a2 a3

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

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

Notice: If the entropy admissible solution to (CP) does not satisfy the monotonicity requirement, then it is not solution of the min-max problem Initial data (a2 < a1 < σ < a3) ρ1(x) =    a1 if x < ¯ x1 a2 if ¯ x1 < x < ¯ x2 a3 if x > ¯ x2 ρ2(x) = a3 t T x ¯ x1 ¯ x2 a1 a2 a3 t T x ¯ x1 ¯ x2 a1 a2 a3 b c a3

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Prospective research directions

Analyze optimal solutions for 1 × 2, 2 × 1 and 2 × 2 junctions Analyze optimal solutions involving infmow junctions controls at two (or more) nodes Consider cost functionals depending on the whole value of the solution on along the incoming roads (and not just on the infmow on at the junction) Consider feedback type optimal controls at junctions

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Prospective research directions

Analyze optimal solutions for 1 × 2, 2 × 1 and 2 × 2 junctions Analyze optimal solutions involving infmow junctions controls at two (or more) nodes Consider cost functionals depending on the whole value of the solution on along the incoming roads (and not just on the infmow on at the junction) Consider feedback type optimal controls at junctions

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Prospective research directions

Analyze optimal solutions for 1 × 2, 2 × 1 and 2 × 2 junctions Analyze optimal solutions involving infmow junctions controls at two (or more) nodes Consider cost functionals depending on the whole value of the solution on [0, T] along the incoming roads (and not just on the infmow on [0, T] at the junction) Consider feedback type optimal controls at junctions

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Prospective research directions

Analyze optimal solutions for 1 × 2, 2 × 1 and 2 × 2 junctions Analyze optimal solutions involving infmow junctions controls at two (or more) nodes Consider cost functionals depending on the whole value of the solution on [0, T] along the incoming roads (and not just on the infmow on [0, T] at the junction) Consider feedback type optimal controls at junctions

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow

Thank you for your attention!

Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow