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A measure theoretic approach to traffic flow optimisation on networks

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Miscellaneous topics - Partial differential equations Existence theories

Published online by Cambridge University Press:  16 October 2018

SIMONE CACACE ,
FABIO CAMILLI ,
RAUL DE MAIO  and
ANDREA TOSIN Open the ORCID record for ANDREA TOSIN [Opens in a new window]
SIMONE CACACE
Affiliation:
Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, L. go S. Leonardo Murialdo 1, 00146 Roma, Italy email: cacace@mat.uniroma3.it
FABIO CAMILLI
Affiliation:
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, “Sapienza” Università di Roma, Via Scarpa 16, 00161 Roma, Italy emails: camilli@sbai.uniroma1.it; raul.demaio@sbai.uniroma1.it
RAUL DE MAIO
Affiliation:
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, “Sapienza” Università di Roma, Via Scarpa 16, 00161 Roma, Italy emails: camilli@sbai.uniroma1.it; raul.demaio@sbai.uniroma1.it
ANDREA TOSIN*
Affiliation:
Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy email: andrea.tosin@polito.it
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Abstract

We consider a class of optimal control problems for measure-valued nonlinear transport equations describing traffic flow problems on networks. The objective is to minimise/maximise macroscopic quantities, such as traffic volume or average speed, controlling few agents, e.g. smart traffic lights and automated cars. The measure theoretic approach allows to study in a same setting local and non-local drivers interactions and to consider the control variables as additional measures interacting with the drivers distribution. We also propose a gradient descent adjoint-based optimisation method, obtained by deriving first-order optimality conditions for the control problem, and we provide some numerical experiments in the case of smart traffic lights for a 2–1 junction.

Keywords

Networktransport equationmeasure-valued solutionstransmission conditionsoptimisation

MSC classification

Primary: 49J20: Optimal control problems involving partial differential equations
Secondary: 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces) 65M99: None of the above, but in this section
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 6: Applied Optimal Transport , December 2019 , pp. 1187 - 1209
Copyright
© Cambridge University Press 2018 

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References

Albi, G., Bongini, M., Cristiani, E. & Kalise, D. (2016) Invisible control of self-organising agents leaving unknown environments. SIAM J. Appl. Math. 76, 1683–1710.CrossRefGoogle Scholar
Ambrosio, L., Gigli, N. & Savaré, G. (2008) Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics, ETH Zürich, Birkhäuser Verlag, Basel.Google Scholar
Bongini, M. & Buttazzo, G. (2017) Optimal control problems in transport dynamics. Math. Models Methods Appl. Sci. 27, 427–451.CrossRefGoogle Scholar
Bressan, A., Čanić, S., Garavello, M., Herty, M. & Piccoli, B. (2014) Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1(1), 47–111.CrossRefGoogle Scholar
Camilli, F., De Maio, R. & Tosin, A. (2017) Transport of measures on networks. Netw. Heterog. Media 12, 191–215.CrossRefGoogle Scholar
Camilli, F., De Maio, R. & Tosin, A. (2018) Measure-valued solutions to transport equations on networks with nonlocal velocity. J. Diff. Equ. 254, 7213–7241.CrossRefGoogle Scholar
Cañizo, J. A., Carrillo, J. A. & Rosado, J. (2011) A well-posedness theory in measures for some kinetic models of collective motion. Math. Models Methods Appl. Sci. 21(3), 515–539.CrossRefGoogle Scholar
Cascetta, E. (2009) Transportation Systems Analysis, Springer, Heidelberg.CrossRefGoogle Scholar
Cavagnari, G., Marigonda, A., Nguyen, K. T. & Priuli, F. S. (2017) Generalized control systems in the space of probability measures. Set-Valued Var. Anal. 26(3), 1–29.Google Scholar
Cavagnari, G., Marigonda, A. & Piccoli, B. (2018) Averaged time-optimal control problem in the space of positive Borel measures. ESAIM COCV 24(2), 721–740.Google Scholar
Colombo, R. M., Garavello, M. & Lécureux-Mercier, M. (2012) A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci. 22(4), 1150023.CrossRefGoogle Scholar
Crippa, G. & Lécureux-Mercier, M. (2013) Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow. Nonlin. Diff. Equ. Appl. 20(3), 523–537.CrossRefGoogle Scholar
Cristiani, E., Piccoli, B. & Tosin, A. (2014) Multiscale Modeling of Pedestrian Dynamics. MS&A: Modeling, Simulation and Applications, Vol. 12, Springer, Cham.CrossRefGoogle Scholar
Cucker, F. & Smale, S. (2007) Emergent behaviour in flocks. IEEE Trans. Auto. Con. 52, 852–862.CrossRefGoogle Scholar
Garavello, M., Han, K. & Piccoli, B. (2016) Models for Vehicular Traffic on Networks. AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences, Springfield, MO.Google Scholar
Gentile, G. & Meschini, L. (2011) Using dynamic assignment models for real-time traffic forecast on large urban networks. In: 2nd International Conference on Models and Technologies for Intelligent Transportation Systems, Leuven, Belgium.Google Scholar
Goatin, P. & Scialanga, S. (2016) Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Netw. Heterog. Media 11(1), 107–121.CrossRefGoogle Scholar
Göttlich, S. & Ziegler, U. (2014) Traffic light control: a case study. Dis. Con. Dyn. Syst. 7, 483–501.Google Scholar
Trozzi, V., Kaparias, I., Bell, M. G. H. & Gentile, G. (2013) A dynamic route choice model for public transport networks with boarding queues. Trans. Plan. Tech. 36, 44–61.CrossRefGoogle Scholar
Wang, R., Li, Y. & Work, D. B. (2017) Comparing traffic state estimators for mixed human and automated traffic flows. Transp. Res. Part C Emerg. Technol. 78, 95–110.CrossRefGoogle Scholar