Communications in Mathematical Sciences

Volume 16 (2018)

Number 2

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

Pages: 293 – 331

DOI: http://dx.doi.org/10.4310/CMS.2018.v16.n2.a1

Authors

Wei Zhang (Institute of Mathematics, Freie Universität Berlin, Germany)

Carsten Hartmann (Institute of Mathematics, Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus, Germany)

Max von Kleist (Institute of Mathematics, Freie Universität Berlin, Germany)

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.

Keywords

Markov jump process, optimal control problem, large number limit, feedback control policy, hybrid control policy

2010 Mathematics Subject Classification

60J27, 93E20

Full Text (PDF format)

Received 15 September 2016

Accepted 9 August 2017

Published 14 May 2018