  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.
