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1、Workshop on “Irrigation Channels and Related Problems” Variation of permeability parameters in Barcelona networks,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.,Description of dy
2、namics on a single road,a,b,Dynamics on roads,L,Length of the road: L; Congested part of length: l; Free part: L l,l,L l,Slope at low densities : V0 ; Slope at high densities: c.,a,b,Dynamics on roads,L,l,L l,Incoming flow,Outgoing flow,T: safe time.,Maximal flux,a,b,The permeability parameter,L,If
3、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
4、. The permeability can control traffic flows!,a,b,Traffic jams modelled by a DDE,L,The number of delayed vehicles can be expressed by the following DDE (Delayed Differential Equation):,Road networks,Barcelona networks,A Barcelona network is seen as a finite collection of roads (arcs), meeting at som
5、e junctions (nodes). Every road has not a linear shape.,Assumptions,Helbing model for Barcelona networks,Dynamics on roads are solved by the Helbing model.,For every road an initial data. Boundary data for roads with infinite endpoints. For inner roads of the network, solving dynamics at nodes is fu
6、ndamental!,Boundary data! The arrival flow,Boundary data! The arrival flow,Junctions,It is necessary solving dynamics at road junctions,Riemann Solver (RS),A RS for the node (i, j) is a map that allows to obtain a solution for the 4 tuple,Rule A Distribution of Traffic,Some coefficients are introduc
7、ed 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 Matrixsuch that,Some coefficients are introduced in order to describe the preferences of drive
8、rs. 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.,Rule A Distribution of Traffic,Rule B Max
9、imization 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.,Dynamics at a node,Assumption: one lane. Solution for the junction:,(A),P,(B),Dynamics at a node,Three possibl
10、e cases for RS at (i, j).,Assumption: presence of queues on roads.,RS1,RS2,RS3,Formulation of an optimal control problem,Optimization and control for Barcelona networks,Dynamics in form of a control system: the state is the number of delayed vehicles, the control is the permeability.,Presence of del
11、ayed permeabilities.,Extra variable.,U = set of controls; R = set of roads.,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.,Empty queues: the nesting phe
12、nomenon,Nesting equation!,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:,A hybrid approach,To describe the whole dynamics at (i, j), we define the logic variables,as follows:,For , the definition is simi
13、lar.,A complete hybrid dynamic for the node (i, j) can be described by the following equation:,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) determ
14、ine 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).,Dynamics of needle variations,Needle variation and variational equations,Let t be a Lebesgue point for . For , we can
15、define a family of controls in this way;,Variational equations,For g* to be optimal, we require that:,The tangent vector v satisfies the following equations:,For t t, while:,Continuous dynamic:,Discrete dynamics of needle variations,Consider a time interval 0, T and a Lebesgue point .,Notice that: .,Discrete dynamics of needle variations,Some preliminary numerical results,Preliminary simulations,Simulations,Jump of g implies jumps of O,Period of g wave: 15.,
