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Applied Mathematics & Optimization

, Volume 74, Issue 2, pp 375–421 | Cite as

A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton–Jacobi Integrodifferential Systems on Networks

  • Dan GoreacEmail author
  • Magdalena Kobylanski
  • Miguel Martinez
Article
  • 144 Downloads

Abstract

We study optimal control problems in infinite horizon whxen the dynamics belong to a specific class of piecewise deterministic Markov processes constrained to star-shaped networks (corresponding to a toy traffic model). We adapt the results in Soner (SIAM J Control Optim 24(6):1110–1122, 1986) to prove the regularity of the value function and the dynamic programming principle. Extending the networks and Krylov’s “shaking the coefficients” method, we prove that the value function can be seen as the solution to a linearized optimization problem set on a convenient set of probability measures. The approach relies entirely on viscosity arguments. As a by-product, the dual formulation guarantees that the value function is the pointwise supremum over regular subsolutions of the associated Hamilton–Jacobi integrodifferential system. This ensures that the value function satisfies Perron’s preconization for the (unique) candidate to viscosity solution.

Keywords

Piecewise deterministic Markov process Viscosity solutions Network constraints Linear programming Traffic and maintenance 

Mathematics Subject Classification

49L25 93E20 60J25 49L20 

Notes

Acknowledgments

The authors would like to thank the anonymous referees for constructive remarks allowing to improve the manuscript. The work of the first author has been partially supported by the French National Research Agency project PIECE, Number ANR-12-JS01-0006.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Dan Goreac
    • 1
    Email author
  • Magdalena Kobylanski
    • 1
  • Miguel Martinez
    • 1
  1. 1.Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRSMarne-la-ValléeFrance

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