Traffic Specifications
Many other requirements, besides the minimal demand of staying on the network, require other variables, called specifications \(z \in \mathbb {R}^{|z|}\) (acceleration, congestion, vehicles ahead, etc.), must be added to time, duration and position. Conse
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Traffic Specifications
Many other requirements, besides the minimal demand of staying on the network, require other variables, called specifications1 z ∈ R|z| (velocity, acceleration, congestion, vehicles ahead, etc.), must be added to time, duration and position. Consequently, we introduce the specific traffic variable described by (t, o, p, z), replacing the simpler traffic variable (t, o, p). Examples of specifications reviewed are duration with variable fluidities, accelerations as well as positions and velocities of a fleet of vehicles. Other ones (accumulation of vehicles on an evolving road segment, for instance) will be introduced together with indicators in Chap. 6, p.145, dealing with micro-meso-macro systems. The purpose of this chapter is thus to study the specific traffic relations, subsets of specific traffic states (t, o, p, z) generating traffic evolutions ( p(·), z(·)) satisfying other specific viability requirements and departure conditions. We thus shall answer the same questions than the ones of Chap. 2, p. 13, by taking them into account. The only added difficulty is the addition of other cumbersome notations, but the approach based on viability theory is exactly the same. In particular, we carve in the celerity regulator depending only on time, duration and position a sub-celerity regulator providing velocities depending also on the specifications and their celerities by using the Viability Theorem. This assumes that the drivers have access to the latter two variables.
1 We suggest the word specification as a “neutral” common abstract concept for avoiding confusion by choosing other terms such as attributes (used in the investigations of [89, Costesèque and Lebacque], [92, Costesèque, Lebacque and Monneau], [167, 168, Lebacque], for instance, among other authors), criteria, positions, incomes, events, monads (from the Greek “monos”, unique, introduced by Gottfried Leibniz in philosophy, and borrowed in mathematical category theory and abstract programming), characteristics (used to define “characteristic” systems in conservation laws, Hamilton–Jacobi partial differential equations and viability theory). They may induce a meaning adequate in one field but not in other ones and thus, bring muddiness. This allows us to place under the same abstract umbrella many examples motivated by various fields.
© Springer-Verlag Berlin Heidelberg 2017 J.-P. Aubin and A. Désilles, Traffic Networks as Information Systems, Mathematical Engineering, DOI 10.1007/978-3-642-54771-3_5
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5 Traffic Specifications
5.1 Micro-Meso-Macro Cascade of Traffic Evolutionary Systems Among the various examples of specifications, we single out scalar specifications, called indicators, which enable us to select among viable traffic evolutions intertemporally optimal viable traffic evolutions. This leads us to a classification of traffic systems according to the nature of the specifications. Selection Procedures of Celerities Introducing specifications thus allows us to shrink the celerity regulator by retaining only
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