  We formulate a new two-variable river environmental restoration problem based
on jump stochastic differential equations (SDEs) governing the sediment storage
and nuisance benthic algae population dynamics in a dam-downstream river.
Controlling the dynamics is carried out through impulsive sediment
replenishment with discrete and random observation/intervention to avoid
sediment depletion and thick algae growth. We consider a cost-efficient
management problem of the SDEs to achieve the objectives whose resolution
reduces to solving a Hamilton-Jacobi-Bellman (HJB) equation. We also consider a
Fokker-Planck (FP) equation governing the probability density function of the
controlled dynamics. The HJB equation has a discontinuous solution, while the
FP equation has a Dirac's delta along boundaries. We show that the value
function, the optimized objective function, is governed by the HJB equation in
the simplified case and further that a threshold-type control is optimal. We
demonstrate that simple numerical schemes can handle these equations. Finally,
we numerically analyze the optimal controls and the resulting probability
density functions.
