Optimal control of Markov jump processes : Asymptotic analysis, algorithms and applications to the modelling of chemical reaction systems

TL;DR

This paper analyzes the optimal control of Markov jump processes, especially in chemical systems, providing asymptotic analysis, convergence results, and algorithms to manage complexity in large systems.

Contribution

It introduces a hybrid control policy algorithm for large-scale systems and proves convergence of control value functions as species copy numbers grow.

Findings

Convergence of control value functions as copy numbers increase

Development of a hybrid control policy algorithm

Numerical validation on chemical reaction models

Abstract

Markov jump processes are widely used to model natural and engineered processes. In the context of biological or chemical applications one typically refers to the chemical master equation (CME), which models the evolution of the probability mass of any copy-number combination of the interacting particles. When many interacting particles ("species") are considered, the complexity of the CME quickly increases, making direct numerical simulations impossible. This is even more problematic when one aims at controlling the Markov jump processes defined by the CME. In this work, we study both open loop and feedback optimal control problems of the Markov jump processes in the case that the controls can only be switched at fixed control stages. Based on Kurtz's limit theorems, we prove the convergence of the respective control value functions of the underlying Markov decision problem as the copy numbers of the species go to infinity. In the case of the optimal control problem on a finite time-horizon, we propose a hybrid control policy algorithm to overcome the difficulties due to the curse of dimensionality when the copy number of the involved species is large. Two numerical examples demonstrate the suitability of both the analysis and the proposed algorithms.