  We investigated a cost-constrained static ergodic control problem of the
variance of measure-valued affine processes and its application in streamflow
management. The controlled system is a jump-driven mixed moving average process
that generates realistic subexponential autocorrelation functions, and the
static nature of the control originates from a realistic observability
assumption in the system. The Markovian lift was effectively used to discretize
the system into a finite-dimensional process, which is easier to analyze. The
resolution of the problem is based on backward Kolmogorov equations and a
quadratic solution ansatz. The control problem has a closed-form solution, and
the variance has both strict upper and lower bounds, indicating that the
variance cannot take an arbitrary value even when it is subject to a high
control cost. The correspondence between the discretized system based on the
Markovian lift and the original infinite-dimensional one is discussed. Then, a
convergent Markovian lift is presented to approximate the infinite-dimensional
system. Finally, the control problem was applied to real cases using available
data for a river reach. An extended problem subject to an additional constraint
on maintaining the flow variability was also analyzed without significantly
degrading the tractability of the proposed framework.
