Workshop on “Irrigation Channels and Related Problems” Variation of permeability parameters

1. University of Salerno Department of Information Engineering and Applied Mathematics October, 2°, 2008 Workshop on “Irrigation Channels and Related Problems” Variation of permeability parameters in Barcelona networks Luigi Rarità Joint work with: Ciro D’Apice, Dirk Helbing, Benedetto Piccoli.

2. Organization of the presentation • A model for car traffic on a single road. • Dynamics at nodes. • Formulation of an optimal control problem. • Simulations of queues on roads.

3. Description of dynamics on a single road

4. Dynamics on roads a b L l L – l Length of the road: L; Congested part of length: l; Free part: L – l Slope at low densities : V0 ; Slope at high densities: c.

5. Dynamics on roads a b L l L – l Incoming flow Maximal flux Outgoing flow T: safe time.

6. The permeability parameter a b L If the permeability parameter is zero, traffic is stopped (outgoing flow equal to zero). If the permeability parameter is one, traffic can flow and the outflow can depend either on queues on the road or the arrival flow. We can study some situation of traffic when the permeability is among zero and one. The permeability can control traffic flows!!

7. Traffic jams modelled by a DDE a b L The number of delayed vehicles can be expressed by the following DDE (Delayed Differential Equation):

8. Road networks

9. Barcelona networks A Barcelona network is seen as a finite collection of roads (arcs), meeting at some junctions (nodes). Every road has not a linear shape. Assumptions

10. Helbing modelfor Barcelona networks Dynamics on roads are solved by the Helbing model.

11. Junctions It is necessary solving dynamics at road junctions Boundary data!! The arrival flow… Boundary data!! The arrival flow… For every road an initial data. Boundary data for roads with infinite endpoints. For inner roads of the network, solving dynamics at nodes is fundamental!

12. Riemann Solver (RS) A RS for the node (i, j) is a map that allows to obtain a solution for the 4 – tuple

13. Rule ADistribution of Traffic • Some coefficients are introduced in order to describe the preferences of drivers. Such coefficients indicate the distribution of traffic from incoming to outgoing roads. For this reason, it is necessary to define a traffic Distribution Matrix such that

14. Rule ADistribution of Traffic • Some coefficients are introduced in order to describe the preferences of drivers. Such coefficients indicate the distribution of traffic from incoming to outgoing roads. For this reason, it is necessary to define a traffic Distribution Matrix is the percentage by which cars arrive from the incoming road i and take the outgoing road j.

15. Rule BMaximization of the flux • Assuming that (A) holds, drivers choose destination so as to obtain the maximization of the flux. No one can stop in front of the traffic junction without crossing it.

16. Dynamics at a node • Assumption: one lane. • Solution for the junction: (A) P (B)

17. Dynamics at a node Assumption: presence of queues on roads. Three possible cases for RS at (i, j). RS1 RS2 RS3

18. Formulation of an optimal control problem

19. Optimization and controlfor Barcelona networks Dynamics in form of a control system: the state is the number of delayed vehicles, the control is the permeability. Presence of delayed permeabilities. Extra variable. U = set of controls; R = set of roads.

20. Not empty queues A non linear control system, with delayed controls, given by permeabilities. In this case, RS for the node (i, j) depends only on controls (permeabilities) and not on the state.

21. Empty queues: the nesting phenomenon Nesting equation!!

22. A hybrid approach The evolution of y and do not depend only on dynamics at (i, j). depends on and . is described by RS at (i, j) by: depend on:

23. A hybrid approach To describe the whole dynamics at (i, j), we define the logic variables as follows: For , the definition is similar. A complete hybrid dynamic for the node (i, j) can be described by the following equation:

24. A hybrid approach The dynamic of a control parameter g (or a distribution coefficient a or b) influences the dynamic (which is of continuous type) of the couple (A, O) through RS. Dynamics of (A, O) determine a continuous dynamic of both (A, O) through RS and . The dynamic of implies a continuous dynamic of y and a discrete dynamic, through the logic variable e, of the couple (A, O).

25. Dynamics of needle variations

26. Needle variation and variational equations Let t be a Lebesgue point for . For , we can define a family of controls in this way;

27. Variational equations For g* to be optimal, we require that: The tangent vector v satisfies the following equations: Continuous dynamic: For t < t, while:

28. Discrete dynamics of needle variations Consider a time interval [0, T] and a Lebesgue point . Notice that: .

29. Discrete dynamics of needle variations

30. Some preliminary numerical results

31. Preliminary simulations

32. Simulations Jump of g implies jumps of O Period of g wave: 15.

33. Simulations O influences dynamics at the node (i, j). Hence, we have variations of A.

34. Queues Problems of saturation!!! Congested roads!

35. Queues

36. Some references Rarità L., D’Apice C., Piccoli B., Helbing D., Control of urban network flows through variation of permeability parameters, Preprint D.I.I.M.A. D.Helbing, J. Siegmeier and S. Lammer, Self-organized network flows, NHM, 2, 2007, no. 2, 193 – 210.. D. Helbing, S. Lammer and J.-P. Lebacque, Self-organized control of irregular or perturbed network traffic, in C. Deissenberg and R. F. Hartl (eds.), Optimal Control and Dynamic Games, Springer, Dordrecht, 2005, pp. 239 – 274.