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

TL;DR

This paper investigates the long-term behavior of control strategies for a class of piecewise deterministic Markov processes inspired by temperate viruses, establishing conditions for the convergence of value functions without requiring dissipative dynamics.

Contribution

It introduces a novel analysis of asymptotic control for PDMPs related to viruses, showing the existence and coincidence of limits of discounted and average cost problems under broad conditions.

Findings

Uniform limits of discounted and average problems exist and coincide.

Limit value depends on initial configuration without dissipative assumptions.

Approximation of value functions using piecewise constant policies is demonstrated.

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.