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

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.

Authors:
; ;  [1]
  1. Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS (France)
Publication Date:
OSTI Identifier:
22617195
Resource Type:
Journal Article
Journal Name:
Applied Mathematics and Optimization
Additional Journal Information:
Journal Volume: 74; Journal Issue: 2; Other Information: Copyright (c) 2016 Springer Science+Business Media New York; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0095-4616
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; BY-PRODUCTS; DYNAMIC PROGRAMMING; HAMILTON-JACOBI EQUATIONS; MAINTENANCE; MARKOV PROCESS; MATHEMATICAL SOLUTIONS; OPTIMAL CONTROL; OPTIMIZATION; PROBABILITY; PROGRAMMING; VISCOSITY

Citation Formats

Goreac, Dan, Kobylanski, Magdalena, and Martinez, Miguel. A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton–Jacobi Integrodifferential Systems on Networks. United States: N. p., 2016. Web. doi:10.1007/S00245-015-9319-Z.
Goreac, Dan, Kobylanski, Magdalena, & Martinez, Miguel. A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton–Jacobi Integrodifferential Systems on Networks. United States. https://doi.org/10.1007/S00245-015-9319-Z
Goreac, Dan, Kobylanski, Magdalena, and Martinez, Miguel. 2016. "A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton–Jacobi Integrodifferential Systems on Networks". United States. https://doi.org/10.1007/S00245-015-9319-Z.
@article{osti_22617195,
title = {A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton–Jacobi Integrodifferential Systems on Networks},
author = {Goreac, Dan and Kobylanski, Magdalena and Martinez, Miguel},
abstractNote = {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.},
doi = {10.1007/S00245-015-9319-Z},
url = {https://www.osti.gov/biblio/22617195}, journal = {Applied Mathematics and Optimization},
issn = {0095-4616},
number = 2,
volume = 74,
place = {United States},
year = {Sat Oct 15 00:00:00 EDT 2016},
month = {Sat Oct 15 00:00:00 EDT 2016}
}