  The problem of controlling the mean and the variance of a species of interest
in a simple gene expression is addressed. It is shown that the protein mean
level can be globally and robustly tracked to any desired value using a simple
PI controller that satisfies certain sufficient conditions. Controlling both
the mean and variance however requires an additional control input, e.g. the
mRNA degradation rate, and local robust tracking of mean and variance is proved
to be achievable using multivariable PI control, provided that the reference
point satisfies necessary conditions imposed by the system. Even more
importantly, it is shown that there exist PI controllers that locally, robustly
and simultaneously stabilize all the equilibrium points inside the admissible
region. The results are then extended to the mean control of a gene expression
with protein dimerization. It is shown that the moment closure problem can be
circumvented without invoking any moment closure technique. Local stabilization
and convergence of the average dimer population to any desired reference value
is ensured using a pure integral control law. Explicit bounds on the controller
gain are provided and shown to be valid for any reference value. As a
byproduct, an explicit upper-bound of the variance of the monomer species,
acting on the system as unknown input due to the moment openness, is obtained.
The results are illustrated by simulation.
