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

We study optimal control problems in infinite horizon when the dynamics belong to a specific class of piecewise deterministic Markov processes constrained to star-shaped networks (inspired by traffic models). We adapt the results in [H. M. Soner. Optimal control with state-space constraint. II. 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. Finally, we prove that the same kind of linearization can be obtained by combining linearization for classical (unconstrained) problems and cost penalization. The latter method works for very general near-viable systems (possibly without further controllability) and discontinuous costs.