Generalized Lax-Hopf Formulas for Cournot Maps and Hamilton-Jacobi-McKendrick Equations

Aubin, Jean‐Pierre; Chen, Luxi

doi:10.1007/978-3-319-06917-3_1

Cited by 1 publication

(1 citation statement)

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“…Road traffic information exists at least since the milestones have been used by Roman Empire for deducing the travel durations between two positions 18 . At the time, only the position was indicated, the celerity being not yet an issue !…”

Section: Motivation : Road Trafficmentioning

confidence: 99%

Transports Regulators of Networks with Junctions Detected by Durations Functions

Aubin¹

2013

Preprint

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This study advocates a mathematical framework of "transport relations" on a network. They single out a subset of "traffic states" described by time, duration, position and other traffic attributes (called "monads" for short). Duration evolutions are non negative, decreasing toward 0 for incoming durations, increasing from 0 for outgoing durations, allowing the detection of "junction states" defined as traffic states with "zero duration". A "junction relation" (cross-roads, synapses, clearing houses, etc.) is a subset of the transport relation made of junction states. The objective is to construct a "transport regulator" associating with traffic states a set of "celerities" that mobiles circulating in the network can use as velocities. In other word, a network is regarded as a "provider of velocity information" to the mobiles for travelling from one departure state to an arrival state across a junction relation (a kind of geodesic problem). This investigation assumes that a system governs the evolution of monads in function of time, duration and position using celerities as controls and provide the transport regulator, a feedback from transport states to celerities. The proposed mathematical framework can acclimate road or aerial networks, endocrine (hormonal) or synaptic (neurotransmitters) networks, financial or economic networks, which motivated this investigation. This framework could probably accommodate computer and even social networks. This investigation is restricted to junctions between two routes. An extension to junctions between a set or prejunction routes and postjunction is under way, as well as the development of specific examples.

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Section: Motivation : Road Trafficmentioning

confidence: 99%