Time-varying Riemann solvers for conservation laws on networks

Garavello, Mauro; Piccoli, Benedetto

doi:10.1016/j.jde.2008.12.017

Cited by 11 publications

(4 citation statements)

References 24 publications

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“…This book reports the pertinent literature in the field. The analysis of networks shows interesting mathematical problems such as Riemann solvers for conservation laws [81] and challenging applications [80]. Particularly important for engineering applications is the analysis of optimization and control problems [89], [77], [114].…”

Section: Critical Analysismentioning

confidence: 99%

On the Modeling of Traffic and Crowds: A Survey of Models, Speculations, and Perspectives

Bellomo¹,

Dogbe²

2011

SIAM Rev.

383

284

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This paper presents a review and critical analysis of the mathematical literature concerning the modeling of vehicular traffic and crowd phenomena. The survey of models deals with the representation scales and the mathematical frameworks that are used for the modeling approach. The paper also considers the challenging objective of modeling complex systems consisting of large systems of individuals interacting in a nonlinear manner, where one of the modeling difficulties is the fact that these systems are difficult to model at a global level when based only on the description of the dynamics of individual elements. The review is concluded with a critical analysis focused on research perspectives that consider the development of a unified modeling strategy.

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Section: Critical Analysismentioning

confidence: 99%

On the Modeling of Traffic and Crowds: A Survey of Models, Speculations, and Perspectives

Bellomo¹,

Dogbe²

2011

SIAM Rev.

383

284

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show abstract

“…• The case treated in this paper refers to fixed traffic distribution coefficient. In reality such coefficients may vary during the day and for this case an existence theory is already available, see [9]. • Besides redirecting traffic, a stronger measure is the closure of roads.…”

Section: Discussionmentioning

confidence: 99%

“…The latter may well change depending on the hour of the day, thus rendering the matrix A dependent on time. The case of time-varying coefficients was treated in [9]; however, here we focus on the simpler case of fixed coefficients.…”

Section: Road Networkmentioning

confidence: 99%

Optimal distribution of traffic flows in emergency cases

2012

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The aim of this work is to present a technique for the optimisation of emergency vehicles travel times on assigned paths when critical situations, such as car accidents, occur. Using a fluid-dynamic model for the description of car density evolution, the attention is focused on a decentralised approach reducing to simple junctions with two incoming roads and two outgoing ones (junctions of 2 × 2 type). We assume the redirection of cars at junctions is possible and choose a cost functional that describes the asymptotic average velocity of emergency vehicles. Fixing an incoming road and an outgoing road for the emergency vehicle, we determine the local distribution coefficients that maximise such functional at a single junction. Then we use the local optimal coefficients at each node of the network. The overall traffic evolution is studied via simulations, both for simple junctions or cascade networks, evaluating global performances when optimal parameters on the network are used.

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“…In these contexts, a crucial role is played by the design, analysis and numerical implementation of controls acting at the nodes of the network. The investigation of optimal control properties of time varying parameters corresponding to junction distribution coefficients have been considered for example in [21,22,23,42,46,50,51,69] while inflow controls have been analyzed in [28,36,45,65].…”

Section: Introductionmentioning

confidence: 99%

On the Optimization of Conservation Law Models at a Junction with Inflow and Flow Distribution Controls

Ancona¹,

Cesaroni²,

Coclite³

et al. 2018

SIAM J. Control Optim.

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The paper proposes a general framework to analyze control problems for conservation law models on a network. Namely we consider a general class of junction distribution controls and inflow controls and we establish the compactness in L 1 of a class of flux-traces of solutions. We then derive the existence of solutions for two optimization problems: (I) the maximization of an integral functional depending on the flux-traces of solutions evaluated at points of the incoming and outgoing edges; (II) the minimization of the total variation of the optimal solutions of problem (I). Finally we provide an equivalent variational formulation of the min-max problem (II) and we discuss some numerical simulations for a junction with two incoming and two outgoing edges.

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Time-varying Riemann solvers for conservation laws on networks

Cited by 11 publications

References 24 publications

On the Modeling of Traffic and Crowds: A Survey of Models, Speculations, and Perspectives

On the Modeling of Traffic and Crowds: A Survey of Models, Speculations, and Perspectives

Optimal distribution of traffic flows in emergency cases

On the Optimization of Conservation Law Models at a Junction with Inflow and Flow Distribution Controls

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