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

PDEs on networks Achdou, Andreianov, Banda, Bastin, Bressan, Camilli, Canic, Chalons, Mauro Garavello (University of Milano Bicocca) Schleper, Shen, Tchou, Zidani, Ziegler... Monneau, Moutari, Nguyen, Marigo, Piccoli, Rascle, Risebro, Rosini, Imbert, Klar, Lattanzio, Lebacque, Leugering, Manzo, Marcellini, Marchi, Donadello, Gasser, Goatin, Göttlich, Guerra, Gugat, Han, Herty, Holden, Coclite, Colombo, Coron, Costeseque, D’Apice, Delle Monache, Telecommunication and data networks (scalar) Vehicular traffjc (scalar and systems) Biological networks (scalar) Irrigation channels (systems) Blood circulatory fmow & vascular stents (systems) Supply chains (scalar) Gas pipelines (systems) Air traffjc management Control problems for traffjc fmow

Nodal conditions on road networks 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 Incoming roads : I l = ( −∞ , 0] , Outgoing roads : I l = [0 , + ∞ )

Nodal conditions on road networks 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 Incoming roads : I l = ( −∞ , 0] , Outgoing roads : I l = [0 , + ∞ )

Nodal conditions on road networks Distribution rules: Mauro Garavello (University of Milano Bicocca) for a.e. -th outgoing road) -th incoming road and turning into (percentage of drivers coming from Control problems for traffjc fmow Conservation of the number of cars Incoming roads : I l = ( −∞ , 0] , Outgoing roads : I l = [0 , + ∞ ) m + n m ∑ ∑ f ( ρ j ( t, 0)) = f ( ρ i ( t, 0)) for a.e. t > 0 , j = m +1 i =1

Nodal conditions on road networks for a.e. Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Conservation of the number of cars Incoming roads : I l = ( −∞ , 0] , Outgoing roads : I l = [0 , + ∞ ) m + n m ∑ ∑ f ( ρ j ( t, 0)) = f ( ρ i ( t, 0)) for a.e. t > 0 , j = m +1 i =1 Distribution rules: a ji (percentage of drivers coming from i -th incoming road and turning into j -th outgoing road)

Nodal conditions on road networks Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Conservation of the number of cars Incoming roads : I l = ( −∞ , 0] , Outgoing roads : I l = [0 , + ∞ ) m + n m ∑ ∑ f ( ρ j ( t, 0)) = f ( ρ i ( t, 0)) for a.e. t > 0 , j = m +1 i =1 Distribution rules: a ji (percentage of drivers coming from i -th incoming road and turning into j -th outgoing road) m ∑ f ( ρ j ( t, 0)) = a ji ( t ) f ( ρ i ( t, 0)) for a.e. t > 0 , i =1

Nodal conditions on road networks Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Conservation of the number of cars Incoming roads : I l = ( −∞ , 0] , Outgoing roads : I l = [0 , + ∞ ) m + n m ∑ ∑ f ( ρ j ( t, 0)) = f ( ρ i ( t, 0)) for a.e. t > 0 , j = m +1 i =1 Distribution rules: a ji (percentage of drivers coming from i -th incoming road and turning into j -th outgoing road) m ∑ f ( ρ j ( t, 0)) = a ji ( t ) f ( ρ i ( t, 0)) for a.e. t > 0 , i =1 m + n ∑ 0 ≤ a ji ( t ) ≤ 1 ∀ j, i a ji ( t ) = 1 ∀ i j = m +1

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

Control problems for traffjc fmow Priority rules: Mauro Garavello (University of Milano Bicocca) larger than the maximal fmux that the outgoing roads can handle) and the total possible fmux on the incoming roads is (when road passing through the junction], [percentage of time drivers from -th incoming [maximize the fmux through the junction] Remark: distribution rules are not suffjcient to select unique Optimization criterion imposed on the traces of the solution solution at the junction. Nodal conditions on road networks . . . continued � � � �

Control problems for traffjc fmow Priority rules: Mauro Garavello (University of Milano Bicocca) larger than the maximal fmux that the outgoing roads can handle) and the total possible fmux on the incoming roads is (when road passing through the junction], [percentage of time drivers from -th incoming [maximize the fmux through the junction] Remark: distribution rules are not suffjcient to select unique Optimization criterion imposed on the traces of the solution solution at the junction. Nodal conditions on road networks . . . continued � � � �

Control problems for traffjc fmow Remark: distribution rules are not suffjcient to select unique Mauro Garavello (University of Milano Bicocca) larger than the maximal fmux that the outgoing roads can handle) and the total possible fmux on the incoming roads is (when [maximize the fmux through the junction] solution at the junction. Optimization criterion imposed on the traces of the solution Nodal conditions on road networks . . . continued � � � � Priority rules: c i [percentage of time drivers from i -th incoming road passing through the junction], ∑ m i =1 c i ( t ) = 1

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] 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 � � � � Priority rules: c i [percentage of time drivers from i -th incoming road passing through the junction], ∑ m i =1 c i ( t ) = 1 (when m > n and the total possible fmux on the incoming roads is

problems at A-priori BV bounds yield the convergence of front tracking approximations to the (unique?) solution of the Cauchy Riemann problem approach Construct front tracking approximate solutions with piecewise Mauro Garavello (University of Milano Bicocca) total variation). problem at the node with general initial data (having fjnite constant initial data on each road. for classical Riemann problems. [Holden, Risebro (1995); Coclite, G., Piccoli (2005) - ] data are constants. Solutions are required to be self-similar as outgoing roads when initial incoming and IBV (Riemann) a procedure for (uniquely) solving Junction Riemann Solver (JRS): provide nodal conditions and Control problems for traffjc fmow

A-priori BV bounds yield the convergence of front tracking approximations to the (unique?) solution of the Cauchy Riemann problem approach Construct front tracking approximate solutions with piecewise Mauro Garavello (University of Milano Bicocca) total variation). problem at the node with general initial data (having fjnite constant initial data on each road. for classical Riemann problems. [Holden, Risebro (1995); Coclite, G., Piccoli (2005) - ] data are constants. Solutions are required to be self-similar as IBV (Riemann) a procedure for (uniquely) solving Junction Riemann Solver (JRS): provide nodal conditions and Control problems for traffjc fmow m + n problems at m incoming and n outgoing roads when initial

A-priori BV bounds yield the convergence of front tracking approximations to the (unique?) solution of the Cauchy Riemann problem approach Construct front tracking approximate solutions with piecewise Mauro Garavello (University of Milano Bicocca) total variation). problem at the node with general initial data (having fjnite constant initial data on each road. for classical Riemann problems. [Holden, Risebro (1995); Coclite, G., Piccoli (2005) - ] data are constants. Solutions are required to be self-similar as IBV (Riemann) a procedure for (uniquely) solving Junction Riemann Solver (JRS): provide nodal conditions and Control problems for traffjc fmow m + n problems at m incoming and n outgoing roads when initial

Riemann problem approach [Holden, Risebro (1995); Coclite, G., Piccoli (2005) - ] Mauro Garavello (University of Milano Bicocca) total variation). A-priori BV bounds yield the convergence of front tracking constant initial data on each road. Construct front tracking approximate solutions with piecewise for classical Riemann problems. data are constants. Solutions are required to be self-similar as IBV (Riemann) a procedure for (uniquely) solving Junction Riemann Solver (JRS): provide nodal conditions and Control problems for traffjc fmow m + n problems at m incoming and n outgoing roads when initial approximations to the (unique?) solution of the Cauchy problem at the node with general initial data (having fjnite

-tuple of constant boundary data which are the traces Junction Riemann Solver A Riemann problem at a node is a Cauchy problem with constant 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 at the node of the solution to the corresponding Riemann problem. Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow initial condition on each arc I l :

-tuple of constant boundary data which are the traces Junction Riemann Solver A Riemann problem at a node is a Cauchy problem with constant 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 at the node of the solution to the corresponding Riemann problem. Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow initial condition on each arc I l : { ∂ t ρ l + ∂ x f ( ρ l ) = 0 , x ∈ I l , l ∈ { 1 , . . . , m + n } , t > 0 ρ l (0 , x ) = ρ l , x ∈ I l , l ∈ { 1 , . . . , m + n }

-tuple of constant boundary data which are the traces Junction Riemann Solver A Riemann problem at a node is a Cauchy problem with constant Mauro Garavello (University of Milano Bicocca) at the node of the solution to the corresponding Riemann problem. -tuple of constant initial data an that associates to an A Riemann solver at the node is a map Control problems for traffjc fmow initial condition on each arc I l : { ∂ t ρ l + ∂ x f ( ρ l ) = 0 , x ∈ I l , l ∈ { 1 , . . . , m + n } , t > 0 ρ l (0 , x ) = ρ l , x ∈ I l , l ∈ { 1 , . . . , m + n } Giving a solution is equivalent to giving its trace at the node: ρ l ( t, 0) = � ρ l t > 0 , l ∈ { 1 , . . . , m + n }

Junction Riemann Solver A Riemann problem at a node is a Cauchy problem with constant Mauro Garavello (University of Milano Bicocca) at the node of the solution to the corresponding Riemann problem. A Riemann solver at the node is a map Control problems for traffjc fmow initial condition on each arc I l : { ∂ t ρ l + ∂ x f ( ρ l ) = 0 , x ∈ I l , l ∈ { 1 , . . . , m + n } , t > 0 ρ l (0 , x ) = ρ l , x ∈ I l , l ∈ { 1 , . . . , m + n } Giving a solution is equivalent to giving its trace at the node: ρ l ( t, 0) = � ρ l t > 0 , l ∈ { 1 , . . . , m + n } RS : [0 , ρ max ] m + n → [0 , ρ max ] m + n , ( ρ 1 , . . . , ρ m + n ) �→ ( � ρ 1 , . . . , � ρ m + n ) that associates to an ( m + n ) -tuple of constant initial data an ( m + n ) -tuple of constant boundary data which are the traces

Control problems for traffjc fmow For Mauro Garavello (University of Milano Bicocca) is solved by waves with positive speeds. if if problem (outgoing arcs) the classical Riemann Junction Riemann Solver . . . continued For l ∈ { 1 , . . . , m } (incoming arcs) the classical Riemann problem   ∂ t ρ l + ∂ x f ( ρ l ) = 0   { ρ l if x < 0   ρ l (0 , x ) =  ρ l � ρ l � ρ l if x > 0 is solved by waves with negative speeds. x = 0

Control problems for traffjc fmow is solved by waves with positive speeds. Mauro Garavello (University of Milano Bicocca) problem Junction Riemann Solver . . . continued For l ∈ { 1 , . . . , m } (incoming arcs) the classical Riemann problem   ∂ t ρ l + ∂ x f ( ρ l ) = 0   { ρ l if x < 0   ρ l (0 , x ) =  ρ l � ρ l ρ l � if x > 0 is solved by waves with negative speeds. x = 0 For l ∈ { m + 1 , . . . , m + n } (outgoing arcs) the classical Riemann   ∂ t ρ l + ∂ x f ( ρ l ) = 0   { � ρ l if x < 0   ρ l (0 , x ) =  ρ l � ρ l ρ l if x > 0 x = 0

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

Solution for the Junction Riemann Problem (Drivers’ preferences) Impose the linear constraints Choose the self-similar solution which maximizes If needed ( ), prescribe relative priority rules Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Fix a distribution Markov matrix A ∈ M ( n × m ) A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T on the traces of the fmuxes (fmux distribution ↔ matrix A )

Solution for the Junction Riemann Problem (Drivers’ preferences) Impose the linear constraints If needed ( ), prescribe relative priority rules Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Fix a distribution Markov matrix A ∈ M ( n × m ) A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T on the traces of the fmuxes (fmux distribution ↔ matrix A ) Choose the self-similar solution which maximizes ∑ n l =1 f ( ρ l ( t, 0))

Solution for the Junction Riemann Problem (Drivers’ preferences) Impose the linear constraints Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Fix a distribution Markov matrix A ∈ M ( n × m ) A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T on the traces of the fmuxes (fmux distribution ↔ matrix A ) Choose the self-similar solution which maximizes ∑ n l =1 f ( ρ l ( t, 0)) If needed ( m > n ), prescribe relative priority rules

Solution for the Junction Riemann Problem Algorithm Mauro Garavello (University of Milano Bicocca) Select the densities that satisfy the priority rules, if needed Find the corresponding densities Control problems for traffjc fmow (Drivers’ preferences) Impose the linear constraints Fix a distribution Markov matrix A ∈ M ( n × m ) A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T on the traces of the fmuxes (fmux distribution ↔ matrix A ) Choose the self-similar solution which maximizes ∑ n l =1 f ( ρ l ( t, 0)) If needed ( m > n ), prescribe relative priority rules { } ( γ 1 , · · · , γ n ) ∈ ∏ n i =1 Ω i : A · ( γ 1 , · · · , γ n ) T ∈ ∏ n + m • Ω = j = n +1 Ω j • Maximize E = γ 1 + · · · + γ n on Ω • •

Solution for the Junction Riemann Problem (Drivers’ preferences) Impose the linear constraints Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Fix a distribution Markov matrix A ∈ M ( n × m ) A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T on the traces of the fmuxes (fmux distribution ↔ matrix A ) Choose the self-similar solution which maximizes ∑ n l =1 f ( ρ l ( t, 0)) If needed ( m > n ), prescribe relative priority rules γ 2 γ 2 ,max γ 1 γ 1 ,max

Solution for the Junction Riemann Problem (Drivers’ preferences) Impose the linear constraints Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Fix a distribution Markov matrix A ∈ M ( n × m ) A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T on the traces of the fmuxes (fmux distribution ↔ matrix A ) Choose the self-similar solution which maximizes ∑ n l =1 f ( ρ l ( t, 0)) If needed ( m > n ), prescribe relative priority rules γ 2 γ 2 ,max γ 1 γ 1 ,max

Solution for the Junction Riemann Problem (Drivers’ preferences) Impose the linear constraints Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Fix a distribution Markov matrix A ∈ M ( n × m ) A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T on the traces of the fmuxes (fmux distribution ↔ matrix A ) Choose the self-similar solution which maximizes ∑ n l =1 f ( ρ l ( t, 0)) If needed ( m > n ), prescribe relative priority rules γ 2 γ 2 ,max α 1 , 3 γ 1 + α 2 , 3 γ 2 = γ 3 ,max γ 1 γ 1 ,max

Solution for the Junction Riemann Problem Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow (Drivers’ preferences) Impose the linear constraints Fix a distribution Markov matrix A ∈ M ( n × m ) A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T on the traces of the fmuxes (fmux distribution ↔ matrix A ) Choose the self-similar solution which maximizes ∑ n l =1 f ( ρ l ( t, 0)) If needed ( m > n ), prescribe relative priority rules γ 2 α 1 , 4 γ 1 + α 2 , 4 γ 2 = γ 4 ,max γ 2 ,max α 1 , 3 γ 1 + α 2 , 3 γ 2 = γ 3 ,max γ 1 γ 1 ,max

Solution for the Junction Riemann Problem Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Impose the linear constraints (Drivers’ preferences) Fix a distribution Markov matrix A ∈ M ( n × m ) A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T on the traces of the fmuxes (fmux distribution ↔ matrix A ) Choose the self-similar solution which maximizes ∑ n l =1 f ( ρ l ( t, 0)) If needed ( m > n ), prescribe relative priority rules γ 2 α 1 , 4 γ 1 + α 2 , 4 γ 2 = γ 4 ,max γ 2 ,max Ω α 1 , 3 γ 1 + α 2 , 3 γ 2 = γ 3 ,max γ 1 γ 1 ,max

Solution for the Junction Riemann Problem Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow (Drivers’ preferences) Impose the linear constraints Fix a distribution Markov matrix A ∈ M ( n × m ) A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T on the traces of the fmuxes (fmux distribution ↔ matrix A ) Choose the self-similar solution which maximizes ∑ n l =1 f ( ρ l ( t, 0)) If needed ( m > n ), prescribe relative priority rules γ 2 α 1 , 4 γ 1 + α 2 , 4 γ 2 = γ 4 ,max Level curve of γ 1 + γ 2 γ 2 ,max Ω α 1 , 3 γ 1 + α 2 , 3 γ 2 = γ 3 ,max γ 1 γ 1 ,max

Control problems for traffjc fmow Cauchy problem Mauro Garavello (University of Milano Bicocca) such that Theorem: existence of solution (G., Piccoli) Matrix A as a control function Assume that the traffjc distribution matrix A = A ( t ) is time dependent ( BV ∩ L 1 ) and BV (control function). Fix initial data ρ l, 0 ∈ ( I l ; R ) . Then, for every T > 0 , there exists a solution ( ρ 1 , . . . , ρ n + m ) for the { ∂ ∂ ∂t ρ l + ∂x f ( ρ l ) = 0 l = 1 , . . . , n + m ρ l (0 , x ) = ρ l, 0 ( x ) RS A ( t ) ( ρ 1 ( t, 0) , . . . , ρ n + m ( t, 0)) = ( ρ 1 ( t, 0) , . . . , ρ n + m ( t, 0)) for a.e. t ∈ [0 , T ] .

The maximization is global on A difgerent approach: a control theoretic point of view Mauro Garavello (University of Milano Bicocca) in time solutions (fulfjlling linear fmux constraints) and not pointwise among all admissible Control problems for traffjc fmow that satisfjes the fmux constraint Fix T > 0 . Find the solution on [0 , T ] to the Cauchy problem { ∂ t ρ l + ∂ x f ( ρ l ) = 0 , x ∈ I l , l ∈ { 1 , . . . , n + m } , t > 0 ρ l (0 , x ) = ρ l ( x ) , x ∈ I l , l ∈ { 1 , . . . , n + m } , A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T and maximizes an integral cost ∫ T m ∑ f ( ρ l ( t, 0)) dt 0 l =1

The maximization is global on A difgerent approach: a control theoretic point of view Mauro Garavello (University of Milano Bicocca) in time solutions (fulfjlling linear fmux constraints) and not pointwise among all admissible Control problems for traffjc fmow that satisfjes the fmux constraint Fix T > 0 . Find the solution on [0 , T ] to the Cauchy problem { ∂ t ρ l + ∂ x f ( ρ l ) = 0 , x ∈ I l , l ∈ { 1 , . . . , n + m } , t > 0 ρ l (0 , x ) = ρ l ( x ) , x ∈ I l , l ∈ { 1 , . . . , n + m } , A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T and maximizes an integral cost ∫ T ( J : R m → R ) . J ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) dt 0

A difgerent approach: a control theoretic point of view that satisfjes the fmux constraint Mauro Garavello (University of Milano Bicocca) in time solutions (fulfjlling linear fmux constraints) and not pointwise Control problems for traffjc fmow Fix T > 0 . Find the solution on [0 , T ] to the Cauchy problem { ∂ t ρ l + ∂ x f ( ρ l ) = 0 , x ∈ I l , l ∈ { 1 , . . . , n + m } , t > 0 ρ l (0 , x ) = ρ l ( x ) , x ∈ I l , l ∈ { 1 , . . . , n + m } , A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T and maximizes an integral cost ∫ T ( J : R m → R ) . J ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) dt 0 The maximization is global on [0 , T ] among all admissible

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: initial data , , Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow T, M > 0 ,

Admissible fmux traces Given: Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow T, M > 0 , initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } ,

Admissible fmux traces Given: Consider: Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow T, M > 0 , initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } ,

Admissible fmux traces Given: Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Consider: T, M > 0 , initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } , For l ∈ { 1 , . . . , m } , I l = ( −∞ , 0] (incoming arcs)   ∂ t ρ l + ∂ x f ( ρ l ) = 0 x < 0 , t > 0  ρ l (0 , x ) = ¯ ρ l ( x ) x < 0   ρ l ( t, 0) = ˜ ρ l ( t ) t > 0 { } l ( ρ l ) . F l = F T = f ( ρ l ( · , 0)) | ρ l sol on [0 , T ] × I l , ˜ ρ l ( t ) ∈ [0 , ρ max ] { } ( ρ l ) . = F T,M F M = f ( ρ l ( · , 0)) ∈ F l | TV ( ρ l ( · , 0)) ≤ M l l

Admissible fmux traces Given: Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Consider: T, M > 0 , initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } , For l ∈ { m + 1 , . . . , m + n } , I l = [0 , + ∞ ) (outgoing arcs)   ∂ t ρ l + ∂ x f ( ρ l ) = 0 x > 0 , t > 0  ρ l (0 , x ) = ¯ ρ l ( x ) x > 0   ρ l ( t, 0) = ˜ ρ l ( t ) t > 0 { } l ( ρ l ) . F l = F T = f ( ρ l ( · , 0)) | ρ l sol on [0 , T ] × I l , ˜ ρ l ( t ) ∈ [0 , ρ max ] { } ( ρ l ) . = F T,M F M = f ( ρ l ( · , 0)) ∈ F l | TV ( ρ l ( · , 0)) ≤ M l l

Admissible fmux traces Given: Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Consider: T, M > 0 , initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } , Admissible fmux traces compatible with ρ . = ( ρ 1 , . . . , ρ m + n ) { } m m ∏ ∑ G M ( ρ )= F M ( g 1 , . . . , g m ) ∈ i ( ρ i ) : α ji g i ∈F j ( ρ j ) , j = m +1 , . . . , m + n i =1 i =1

Admissible fmux traces - infmow controls Given: Mauro Garavello (University of Milano Bicocca) junction Control problems for traffjc fmow Consider: T, M > 0 , initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } , Admissible fmux traces compatible with ρ . = ( ρ 1 , . . . , ρ m + n ) { } ∏ m ∑ m G M ( ρ )= F M ( g 1 , . . . , g m ) ∈ i ( ρ i ) : α ji g i ∈F j ( ρ j ) , j = m +1 , . . . , m + n i =1 i =1 Treat the elements of G M ( ρ ) as admissible infmow controls at the

Admissible fmux traces - infmow controls Given: Mauro Garavello (University of Milano Bicocca) ramp metering control - Implementation of infmow controls at junctions: traffjc lights, junction Control problems for traffjc fmow Consider: T, M > 0 , initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } , Admissible fmux traces compatible with ρ . = ( ρ 1 , . . . , ρ m + n ) { } ∏ m ∑ m G M ( ρ )= F M ( g 1 , . . . , g m ) ∈ i ( ρ i ) : α ji g i ∈F j ( ρ j ) , j = m +1 , . . . , m + n i =1 i =1 Treat the elements of G M ( ρ ) as admissible infmow controls at the

Admissible fmux traces - infmow controls Given: Mauro Garavello (University of Milano Bicocca) improve driver safety, improve performance of the traffjc system - Goals: decrease traffjc congestion, diminish fuel consumption, ramp metering control junction Control problems for traffjc fmow Consider: T, M > 0 , initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } , Admissible fmux traces compatible with ρ . = ( ρ 1 , . . . , ρ m + n ) { } m m ∏ ∑ G M ( ρ )= F M ( g 1 , . . . , g m ) ∈ i ( ρ i ) : α ji g i ∈F j ( ρ j ) , j = m +1 , . . . , m + n i =1 i =1 Treat the elements of G M ( ρ ) as admissible infmow controls at the - Implementation of infmow controls at junctions: traffjc lights,

Existence of optimal solutions Theorem (Ancona, Cesaroni, Coclite, G.) Mauro Garavello (University of Milano Bicocca) of limit solution convergence of fmux traces of solutions to fmux trace = max Divergence Theorem on convergence of subsequence of fmux traces = Uniform BV bounds and Helly’s Compactness Theorem Proof: Ex: Control problems for traffjc fmow sup s.t. Let J : R m → R be continuous. For every M > 0 , and for any initial data ρ : ∏ m + n l =1 I l → [0 , ρ max ] m + n , there exists � g ∈ G M ( ρ ) ∫ T ∫ T J ( � g ( t )) dt = J ( g ( t )) dt 0 0 g ∈G M ( ρ ) J ( g 1 , . . . , g m ) = ∑ J ( g 1 , . . . , g m ) = ∏ i g i , i g i ,

Existence of optimal solutions Theorem (Ancona, Cesaroni, Coclite, G.) Mauro Garavello (University of Milano Bicocca) of limit solution convergence of fmux traces of solutions to fmux trace = max Divergence Theorem on = Uniform BV bounds and Helly’s Compactness Theorem Proof: Ex: Control problems for traffjc fmow sup s.t. Let J : R m → R be continuous. For every M > 0 , and for any initial data ρ : ∏ m + n l =1 I l → [0 , ρ max ] m + n , there exists � g ∈ G M ( ρ ) ∫ T ∫ T J ( � g ( t )) dt = J ( g ( t )) dt 0 0 g ∈G M ( ρ ) J ( g 1 , . . . , g m ) = ∑ J ( g 1 , . . . , g m ) = ∏ i g i , i g i , ⇒ convergence of subsequence of fmux traces

Existence of optimal solutions sup Mauro Garavello (University of Milano Bicocca) of limit solution = = Uniform BV bounds and Helly’s Compactness Theorem Proof: Ex: Theorem (Ancona, Cesaroni, Coclite, G.) Control problems for traffjc fmow s.t. Let J : R m → R be continuous. For every M > 0 , and for any initial data ρ : ∏ m + n l =1 I l → [0 , ρ max ] m + n , there exists � g ∈ G M ( ρ ) ∫ T ∫ T J ( � g ( t )) dt = J ( g ( t )) dt 0 0 g ∈G M ( ρ ) J ( g 1 , . . . , g m ) = ∑ J ( g 1 , . . . , g m ) = ∏ i g i , i g i , ⇒ 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

Control problems for traffjc fmow Incoming road Outgoing road Mauro Garavello (University of Milano Bicocca) sup Case n = m = 1 : comparison with entropy solution of (CP) { { ∂ t ρ 1 + ∂ x f ( ρ 1 )=0 x< 0 , t> 0 , ∂ t ρ 2 + ∂ x f ( ρ 2 )=0 x> 0 , t> 0 ρ 1 (0 , x ) = ¯ ρ 1 ( x ) x < 0 , ρ 2 (0 , x ) = ¯ ρ 2 ( x ) x > 0 , G M = F M G = F 1 ∩ F 2 ∩ F 2 1 ∫ T J ( g ( t )) dt ( max ) M g ∈G M 0

Control problems for traffjc fmow sup Outgoing road Mauro Garavello (University of Milano Bicocca) Incoming road Case n = m = 1 : comparison with entropy solution of (CP) { { ∂ t ρ 1 + ∂ x f ( ρ 1 )=0 x< 0 , t> 0 , ∂ t ρ 2 + ∂ x f ( ρ 2 )=0 x> 0 , t> 0 ρ 1 (0 , x ) = ¯ ρ 1 ( x ) x < 0 , ρ 2 (0 , x ) = ¯ ρ 2 ( x ) x > 0 , G M = F M G = F 1 ∩ F 2 ∩ F 2 1 ∫ T J ( g ( t )) dt ( max ) M 0 g ∈G M { } F 1 . = f ( ρ 1 ( · , 0)) | ρ 1 sol on [0 , T ] × ( −∞ , 0] , ˜ ρ 1 ( t ) ∈ [0 , ρ max ] , { } F 2 . = f ( ρ 2 ( · , 0)) | ρ 2 sol on [0 , T ] × [0 , + ∞ ) , ˜ ρ 2 ( t ) ∈ [0 , ρ max ] , { } . F M = f ( ρ l ( · , 0)) ∈ F l | TV ( ρ l ( · , 0)) ≤ M , l = 1 , 2 . l

Control problems for traffjc fmow sup Outgoing road Mauro Garavello (University of Milano Bicocca) = (CP) Incoming road Case n = m = 1 : comparison with entropy solution of (CP) { { ∂ t ρ 1 + ∂ x f ( ρ 1 )=0 x< 0 , t> 0 , ∂ t ρ 2 + ∂ x f ( ρ 2 )=0 x> 0 , t> 0 ρ 1 (0 , x ) = ¯ ρ 1 ( x ) x < 0 , ρ 2 (0 , x ) = ¯ ρ 2 ( x ) x > 0 , G M = F M G = F 1 ∩ F 2 ∩ F 2 1 ∫ T J ( g ( t )) dt ( max ) M 0 g ∈G M   ∂ t ρ + ∂ x f ( ρ ) = 0  { ρ 1 ( x ) if x < 0 ¯  ρ (0 , x ) =  ρ 2 ( x ) if x > 0 ¯ } ρ e : [0 , T ] × R → R entropy admissible sol. to (CP) ρ e ( · , 0) ∈ G M . ⇒ TV ( ρ e ( · , 0)) ≤ M

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

Control problems for traffjc fmow sup Theorem (Ancona, Cesaroni, Coclite, G.) Mauro Garavello (University of Milano Bicocca) is viscosity sol’n of entropy weak sol’n of viscosity solutions to IBV for Hamilton-Jacobi equation The proof relies on Hopf-Lax formula for explicit representation of Case n = m = 1 : comparison with entropy solution of (CP) Consider J ( g ) = g For every T > 0 , let ρ e be the entropy admissible solution to (CP). Then, ρ e ( · , 0) solves the maximization problem ( max ) M , i.e. ∫ T ∫ T f ( ρ e ( t, 0)) dt = g ( t ) dt, 0 0 g ∈G M ( ρ e (0 , · )) for every M ≥ TV ( ρ e ( · , 0)) .

Control problems for traffjc fmow sup Theorem (Ancona, Cesaroni, Coclite, G.) Mauro Garavello (University of Milano Bicocca) viscosity solutions to IBV for Hamilton-Jacobi equation Case n = m = 1 : comparison with entropy solution of (CP) Consider J ( g ) = g For every T > 0 , let ρ e be the entropy admissible solution to (CP). Then, ρ e ( · , 0) solves the maximization problem ( max ) M , i.e. ∫ T ∫ T f ( ρ e ( t, 0)) dt = g ( t ) dt, 0 0 g ∈G M ( ρ e (0 , · )) for every M ≥ TV ( ρ e ( · , 0)) . The proof relies on Hopf-Lax formula for explicit representation of ρ ( t, x ) entropy weak sol’n of ∂ t ρ + ∂ x f ( ρ ) = 0 ⇓ ∫ x v ( t, x ) . = −∞ ρ ( t, z ) dz is viscosity sol’n of ∂ t v + f ( ∂ x v ) = 0

Incoming road Outgoing road Mauro Garavello (University of Milano Bicocca) Control problems for traffjc fmow Case n = m = 1 : non entropic optimal solutions f ( ρ ) = ρ (1 − ρ ) , ρ max = 1 { { ∂ t ρ 1 + ∂ x ( ρ 1 (1 − ρ 1 )) = 0 ∂ t ρ 2 + ∂ x ( ρ 2 (1 − ρ 2 )) = 0 ρ 1 (0 , x ) = 1 ρ 2 (0 , x ) = 1 4 4

Control problems for traffjc fmow The functions Outgoing road Mauro Garavello (University of Milano Bicocca) Incoming road Case n = m = 1 : non entropic optimal solutions f ( ρ ) = ρ (1 − ρ ) , ρ max = 1 { { ∂ t ρ 1 + ∂ x ( ρ 1 (1 − ρ 1 )) = 0 ∂ t ρ 2 + ∂ x ( ρ 2 (1 − ρ 2 )) = 0 ρ 1 (0 , x ) = 1 ρ 2 (0 , x ) = 1 4 4 ρ 1 ( t, x ) ≡ 1 ρ 2 ( t, x ) ≡ 1 4 4 provide an optimal solution for any T, M > 0 , i.e. ∫ T ∫ T 3 16 T = f ( ρ 1 ( t, 0)) dt = sup g ( t ) dt 0 0 g ∈G M

Control problems for traffjc fmow The functions Mauro Garavello (University of Milano Bicocca) Case n = m = 1 : non entropic optimal solutions 3 1 4 8 3 5 3 8 8 1 7 1 8 8 1 1 4 0 4 provide another optimal solution for any T > 8/3 .

Control problems for traffjc fmow The functions Mauro Garavello (University of Milano Bicocca) Case n = m = 1 : non entropic optimal solutions 3 1 4 8 3 5 3 8 8 1 7 1 8 8 1 1 4 0 4 provide another optimal solution for any T > 8/3 . ( 7 ) ( 5 ) ( 1 ) ( ) ∫ T · 5 T − 8 f ( ρ ( t, 0)) dt = f · 1 + f 3 + f · 8 8 4 3 0

Control problems for traffjc fmow The functions Mauro Garavello (University of Milano Bicocca) Case n = m = 1 : non entropic optimal solutions 3 1 4 8 3 5 3 8 8 1 7 1 8 8 1 1 4 0 4 provide another optimal solution for any T > 8/3 . ( ) ∫ T f ( ρ ( t, 0)) dt = 7 64 + 15 64 · 5 3 + 3 T − 8 16 · 3 0

Control problems for traffjc fmow The functions Mauro Garavello (University of Milano Bicocca) Case n = m = 1 : non entropic optimal solutions 3 1 4 8 3 5 3 8 8 1 7 1 8 8 1 1 4 0 4 provide another optimal solution for any T > 8/3 . ( ) ∫ T f ( ρ ( t, 0)) dt = 7 64 + 15 64 · 5 3 + 3 T − 8 = 3 16 · 16 T 3 0

Optimization w.r.t. drivers’ preference parameters Given: 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 T, M > 0

Optimization w.r.t. drivers’ preference parameters Given: 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 T, M > 0 initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n }

Optimization w.r.t. drivers’ preference parameters Given: 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 T, M > 0 initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } a Markov matrix valued map t �→ A ( t ) , t ∈ [0 , T ]

Optimization w.r.t. drivers’ preference parameters Given: 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 T, M > 0 initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } a Markov matrix valued map t �→ A ( t ) , t ∈ [0 , T ]

Optimization w.r.t. drivers’ preference parameters and satisfying constraints given by A: Mauro Garavello (University of Milano Bicocca) Given: Control problems for traffjc fmow Consider: T, M > 0 initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } a Markov matrix valued map t �→ A ( t ) , t ∈ [0 , T ] Admissible fmux traces compatible with ρ . = ( ρ 1 , . . . , ρ m + n ) { ∏ m ∑ m G M F M A ( ρ ) = ( g 1 , . . . , g m ) ∈ i ( ρ i ) : a ji ( t ) g i ( t ) ∈ F j ( ρ j ) , } i =1 i =1 for a.e. t ∈ [0 , T ] , j = m +1 , . . . , m + n, TV ( a ij ) ≤ M ∀ i, j

Optimization w.r.t. drivers’ preference parameters and satisfying constraints given by A: Mauro Garavello (University of Milano Bicocca) controls at the junction Given: Control problems for traffjc fmow Consider: T, M > 0 initial data ρ l : I l → [0 , ρ max ] , l ∈ { 1 , . . . , m + n } a Markov matrix valued map t �→ A ( t ) , t ∈ [0 , T ] Admissible fmux traces compatible with ρ . = ( ρ 1 , . . . , ρ m + n ) { ∏ m ∑ m G M F M A ( ρ ) = ( g 1 , . . . , g m ) ∈ i ( ρ i ) : a ji ( t ) g i ( t ) ∈ F j ( ρ j ) , } i =1 i =1 for a.e. t ∈ [0 , T ] , j = m +1 , . . . , m + n, TV ( a ij ) ≤ M ∀ i, j Treat ( A, g ) , with A an n × m Markov matrix valued map, and g ∈ G M A ( ρ ) , as admissible pair of infmow and driver preference

Existence of optimal solutions w.r.t. drivers’ preference sup Mauro Garavello (University of Milano Bicocca) . fjxed distributional matrix Same arguments as for the optimal control problems with a Proof: Theorem (Ancona, Cesaroni, Coclite, G.) Control problems for traffjc fmow Let J : R m → R be continuous. For every M > 0 , and for any initial data ρ : ∏ m + n l =1 I l → [0 , ρ max ] m + n , there exists an n × m Markov matrix valued map � g ∈ G M A , and � A ( ρ ) , s.t. � ∫ T ∫ T J ( � g ( t )) dt = J ( g ( t )) dt. 0 0 A ∈M M n × m , g ∈G M A ( ρ ) ( M M n × m : set of n × m Markov matrix valued maps A ( · ) = ( a ij ( · )) ij with TV { a ij } ≤ M , for all i, j )

Existence of optimal solutions w.r.t. drivers’ preference Theorem (Ancona, Cesaroni, Coclite, G.) Mauro Garavello (University of Milano Bicocca) Same arguments as for the optimal control problems with a Proof: sup Control problems for traffjc fmow Let J : R m → R be continuous. For every M > 0 , and for any initial data ρ : ∏ m + n l =1 I l → [0 , ρ max ] m + n , there exists an n × m Markov matrix valued map � g ∈ G M A , and � A ( ρ ) , s.t. � ∫ T ∫ T J ( � g ( t )) dt = J ( g ( t )) dt. 0 0 A ∈M M n × m , g ∈G M A ( ρ ) ( M M n × m : set of n × m Markov matrix valued maps A ( · ) = ( a ij ( · )) ij with TV { a ij } ≤ M , for all i, j ) fjxed distributional matrix A .

Additional optimization criterium sup Mauro Garavello (University of Milano Bicocca) min Then minimizes Control problems for traffjc fmow with fmux constraints Fix T, M > 0 . In connection with the nodal problem { ∂ t ρ l + ∂ x f ( ρ l ) = 0 , x ∈ I l , l ∈ { 1 , . . . , m + n } , t > 0 , ρ l (0 , x ) = ρ l ( x ) , x ∈ I l , l ∈ { 1 , . . . , m + n } , A · ( f ( ρ 1 ( t, 0)) , . . . , f ( ρ m ( t, 0))) T = ( f ( ρ m +1 ( t, 0)) , . . . , f ( ρ m + n ( t, 0))) T , let G M ( ρ ) be the set of admissible fmux traces as above, and denote by D M ( ρ ) the set of optimal solutions of ∫ T J ( g ( t )) dt. ( max ) M 0 g ∈G M ( ρ ) m ∑ TV [0 ,T ] g i . ( min ) M g ∈D M ( ρ ) i =1

Additional optimization criterium: existence of solution and Mauro Garavello (University of Milano Bicocca) -topology and the lower semicontinuity of the total variation the with respect to The proof relies on the compactness of the set oscillation the smallest possible of maximize the total fmux through the junction keeping the Notice: solutions of the min-max problems fulfjlls the requirement min Theorem (Ancona, Cesaroni, Coclite, G.) Control problems for traffjc fmow sup s.t. Let J : R n → R be continuous. For every M > 0 , and for any initial data ρ : ∏ m + n l =1 I l → [0 , ρ max ] m + n , there exists � g ∈ G M ( ρ ) ∫ T ∫ T J ( � g ( t )) dt = J ( g ( t )) dt 0 0 g ∈G M ( ρ ) m m ∑ ∑ TV [0 ,T ] � g i = TV [0 ,T ] g i . g ∈D M ( ρ ) i =1 i =1

Additional optimization criterium: existence of solution and Mauro Garavello (University of Milano Bicocca) -topology and the lower semicontinuity of the total variation the with respect to The proof relies on the compactness of the set oscillation the smallest possible of maximize the total fmux through the junction keeping the Notice: solutions of the min-max problems fulfjlls the requirement min Theorem (Ancona, Cesaroni, Coclite, G.) Control problems for traffjc fmow sup s.t. Let J : R n → R be continuous. For every M > 0 , and for any initial data ρ : ∏ m + n l =1 I l → [0 , ρ max ] m + n , there exists � g ∈ G M ( ρ ) ∫ T ∫ T J ( � g ( t )) dt = J ( g ( t )) dt 0 0 g ∈G M ( ρ ) m m ∑ ∑ TV [0 ,T ] � g i = TV [0 ,T ] g i . g ∈D M ( ρ ) i =1 i =1

Additional optimization criterium: existence of solution sup Mauro Garavello (University of Milano Bicocca) oscillation the smallest possible of maximize the total fmux through the junction keeping the Notice: solutions of the min-max problems fulfjlls the requirement min Theorem (Ancona, Cesaroni, Coclite, G.) and Control problems for traffjc fmow s.t. Let J : R n → R be continuous. For every M > 0 , and for any initial data ρ : ∏ m + n l =1 I l → [0 , ρ max ] m + n , there exists � g ∈ G M ( ρ ) ∫ T ∫ T J ( � g ( t )) dt = J ( g ( t )) dt 0 0 g ∈G M ( ρ ) m m ∑ ∑ TV [0 ,T ] � g i = TV [0 ,T ] g i . g ∈D M ( ρ ) i =1 i =1 The proof relies on the compactness of the set G M with respect to the L 1 -topology and the lower semicontinuity of the total variation

Equivalent variational formulation Let Mauro Garavello (University of Milano Bicocca) TV min TV sup such that and subsequence , there exists a be continuous. For every Theorem (Ancona, Cesaroni, Coclite, G.) Control problems for traffjc fmow sup For every δ > 0 consider (∫ T ) m ∑ J ( g ( t )) dt − δ TV [0 ,T ] g i 0 ( g,A ) ∈G M (¯ ρ ) i =1

Equivalent variational formulation Theorem (Ancona, Cesaroni, Coclite, G.) Mauro Garavello (University of Milano Bicocca) min sup Control problems for traffjc fmow sup For every δ > 0 consider (∫ T ) m ∑ J ( g ( t )) dt − δ TV [0 ,T ] g i 0 ( g,A ) ∈G M (¯ ρ ) i =1 Let J : R n → R be continuous. For every δ ν → 0 , there exists a ( ) g, � ∈ G M (¯ subsequence δ ν k and � A ρ ) such that ( ) ( ) g δ νk , � g, � � A δ νk → � A ∫ T ∫ T J ( � g ( t )) dt = J ( g ( t )) dt 0 0 g ∈G M ( ρ ) ∑ m ∑ m TV [0 ,T ] � g i = TV [0 ,T ] g i . g ∈D M ( ρ ) i =1 i =1