Asymptotic Control for a Class of Piecewise Deterministic Markov Processes Associated to Temperate Viruses

Abstract

We aim at characterizing the asymptotic behavior of value functions in the control of piece-wise deterministic Markov processes (PDMP) of switch type under nonexpansive assumptions. For a particular class of processes inspired by temperate viruses, we show that uniform limits of discounted problems as the discount decreases to zero and time-averaged problems as the time horizon increases to infinity exist and coincide. The arguments allow the limit value to depend on initial configuration of the system and do not require dissipative properties on the dynamics. The approach strongly relies on viscosity techniques, linear programming arguments and coupling via random measures associated to PDMP. As an intermediate step in our approach, we present the approximation of discounted value functions when using piecewise constant (in time) open-loop policies.

arXiv commentIn this revised version, statements of the main results are gathered in Section 3. Proofs of the main results (Theorem 4 and Theorem 7) make the object of separate sections (Section 5, resp. Section 6). The biological example makes the object of Section 4. Notations are gathered in Subsection 2.1. This is the final version to be published in SICON

References in corpus (6)

• Noise-based switches and amplifiers for gene expression
• Quantitative ergodicity for some switched dynamical systems
• Interacting multi-class transmissions in large stochastic networks
• Ergodic BSDEs and related PDEs with Neumann boundary conditions
• Viability, Invariance and Reachability for Controlled Piecewise Deterministic Markov Processes Associated to Gene Networks
• A uniform Tauberian theorem in optimal control