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

Journal Article · Sat Oct 15 04:00:00 EDT 2016 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S00245-015-9319-Z· OSTI ID:22617195

Goreac, Dan; Kobylanski, Magdalena; Martinez, Miguel ^[1]

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1. Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS (France)

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.

• OSTI ID: 22617195
• Journal Information: Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 2 Vol. 74; ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
BY-PRODUCTS
DYNAMIC PROGRAMMING
HAMILTON-JACOBI EQUATIONS
MAINTENANCE
MARKOV PROCESS
MATHEMATICAL SOLUTIONS
OPTIMAL CONTROL
OPTIMIZATION
PROBABILITY
PROGRAMMING
VISCOSITY