Inverse Problem for Non-Viscous Mean Field Control: Example From Traffic

This paper presents an inverse problem for mean field games where we find the mean field problem statement for which the given dynamics is the solution. We use distributed traffic as an example and derive the classic Lighthill Whitham Richards (LWR) model as a solution of the non-viscous mean field game. We also derive the same model by choosing a different problem where we use travel time, which is a distributed parameter, as the cost for the optimal control. We then study the stationary versions of these two problems and provide numerical solutions for the same.

[1]  S. Brendle,et al.  Calculus of Variations , 1927, Nature.

[2]  J. Douglas.  Solution of the Inverse Problem of the Calculus of Variations. , 1939, Proceedings of the National Academy of Sciences of the United States of America.

[3]  M J Lighthill,et al.  ON KINEMATIC WAVES.. , 1955 .

[4]  P. I. Richards.  Shock Waves on the Highway , 1956 .

[5]  R. B. Potts,et al.  Car-Following Theory of Steady-State Traffic Flow , 1959 .

[6]  D. Gazis,et al.  Nonlinear Follow-the-Leader Models of Traffic Flow , 1961 .

[7]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[8]  J. Nédélec,et al.  First order quasilinear equations with boundary conditions , 1979 .

[9]  E. Barron,et al.  The Pontryagin maximum principle from dynamic programming and viscosity solutions to first-order partial differential equations , 1986 .

[10]  P. Floch,et al.  Boundary conditions for nonlinear hyperbolic systems of conservation laws , 1988 .

[11]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[12]  I. Anderson,et al.  The Inverse Problem in the Calculus of Variations for Ordinary Differen-tial Equations , 1992 .

[13]  O. Krupkova.  On the inverse problem of the calculus of variations for ordinary differential equations , 1993 .

[14]  M. Ehrendorfer.  The Liouville Equation and Its Potential Usefulness for the Prediction of Forecast Skill. Part II: Applications , 1994 .

[15]  M. Ehrendorfer.  The Liouville Equation and Prediction of Forecast Skill , 1994 .

[16]  M. Ehrendorfer.  The Liouville Equation and Its Potential Usefulness for the Prediction of Forecast Skill. Part I: Theory , 1994 .

[17]  J. Lebacque.  THE GODUNOV SCHEME AND WHAT IT MEANS FOR FIRST ORDER TRAFFIC FLOW MODELS , 1996 .

[18]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[19]  S. Shreve,et al.  Stochastic differential equations , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[20]  Reka Albert,et al.  Mean-field theory for scale-free random networks , 1999 .

[21]  A. Bressan.  Hyperbolic systems of conservation laws : the one-dimensional Cauchy problem , 2000 .

[22]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[23]  P. Kachroo,et al.  Real-time travel time estimation using macroscopic traffic flow models , 2001, ITSC 2001. 2001 IEEE Intelligent Transportation Systems. Proceedings (Cat. No.01TH8585).

[24]  J. P. Lebacque.  Intersection Modeling, Application to Macroscopic Network Traffic Flow Models and Traffic Management , 2005 .

[25]  Mauro Garavello,et al.  Traffic Flow on Networks , 2006 .

[26]  Issam S. Strub,et al.  Weak formulation of boundary conditions for scalar conservation laws: an application to highway traffic modelling , 2006 .

[27]  P. Lions,et al.  Mean field games , 2007 .

[28]  Ian A. Hiskens,et al.  Decentralized charging control for large populations of plug-in electric vehicles , 2010, 49th IEEE Conference on Decision and Control (CDC).

[29]  Christian Dogbé,et al.  Modeling crowd dynamics by the mean-field limit approach , 2010, Math. Comput. Model..

[30]  Olivier Guéant,et al.  Mean Field Games and Applications , 2011 .

[31]  Marie-Therese Wolfram,et al.  On a mean field game approach modeling congestion and aversion in pedestrian crowds , 2011 .

[32]  Shaurya Agarwal.  Wavelets in Intelligent Transportation Systems: Data Compression and Incident Detection , 2012 .

[33]  Hamidou Tembine,et al.  A mean field game analysis of electric vehicles in the smart grid , 2012, 2012 Proceedings IEEE INFOCOM Workshops.

[34]  Claire J. Tomlin,et al.  On efficiency in mean field differential games , 2013, 2013 American Control Conference.

[35]  Pushkin Kachroo,et al.  Geospatial Framework for Integration of Transportation Data Using Voronoi Diagrams , 2013 .

[36]  Alexander Paz,et al.  Estimation of Performance Indices for the Planning of Sustainable Transportation Systems , 2013, Adv. Fuzzy Syst..

[37]  M. Burger,et al.  Mean field games with nonlinear mobilities in pedestrian dynamics , 2013, 1304.5201.

[38]  Pushkin Kachroo,et al.  Dynamic Traffic Assignment: A Survey of Mathematical Models and Techniques , 2013 .

[39]  Peter E. Caines,et al.  Nash, Social and Centralized Solutions to Consensus Problems via Mean Field Control Theory , 2013, IEEE Transactions on Automatic Control.

[40]  A. Bensoussan,et al.  Mean Field Games and Mean Field Type Control Theory , 2013 .

[41]  Pushkin Kachroo,et al.  Feedback-Coordinated Ramp Control of Consecutive On-Ramps Using Distributed Modeling and Godunov-Based Satisfiable Allocation , 2015, IEEE Transactions on Intelligent Transportation Systems.

[42]  Alexander Paz,et al.  Development of a Methodology to Evaluate Projects Using Dynamic Traffic Assignment Models , 2015 .

[43]  Alexander Paz,et al.  A land ferry system to alleviate increasing costs of maintaining the I-80 transportation corridor: An economic assessment , 2015 .

[44]  S. Shankar Sastry,et al.  Travel Time Dynamics for Intelligent Transportation Systems: Theory and Applications , 2016, IEEE Transactions on Intelligent Transportation Systems.

[45]  Pushkin Kachroo,et al.  Modeling and Estimation of the Vehicle-Miles Traveled Tax Rate Using Stochastic Differential Equations , 2016, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[46]  Pushkin Kachroo,et al.  Dynamic Modeling of Performance Indices for Planning of Sustainable Transportation Systems , 2016 .

[47]  Pushkin Kachroo,et al.  A Dynamic Network Modeling-Based Approach for Traffic Observability Problem , 2016, IEEE Transactions on Intelligent Transportation Systems.