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

TL;DR

This paper develops a mathematical framework for optimal control of piecewise deterministic Markov processes on networks, proving the value function's regularity and its characterization as a viscosity solution to Hamilton-Jacobi integrodifferential systems.

Contribution

It extends existing control theory to network-constrained Markov processes, proving regularity and duality results using viscosity solutions and linearization techniques.

Findings

Value function is the supremum over regular subsolutions.

Proved the value function satisfies Perron's preconization.

Applicable to systems with discontinuous costs and limited controllability.

Abstract

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.