  Earth and soils are indispensable elements of river environment.
Dam-downstream environment and ecosystems have been severely affected by
reduced or even stopped sediment supply from the upstream. Replenishing earth
and soils from outside the river has been considered as an effective way to
mitigate this issue. However, its cost-effective implementation has not been
considered from a theoretical side. This paper presents a tractable new
stochastic control model to deal with this issue. The sediment dynamics in the
river environment follow non-smooth and continuous-time piecewise deterministic
dynamics. The model assumes that the observation of the sediment dynamics is
carried out only randomly and discretely, and that the sediment can be
replenished at each observation time with cost. This partial observation
assumption is consistent with the fact that continuously obtaining the
environmental information is difficult in applications. The performance index
to penalize the sediment depletion has a non-smooth term as well. We
demonstrate that these non-smoothness factors harmonize with a dynamic
programming principle, and derive the optimality equation in a degenerate
elliptic form governing the most cost-efficient sediment replenishment policy.
We analytically derive and verify an exact solution under a simplified
condition for a discounted case, an Ergodic case, and a complete information
case. A more realistic case is handled using a high-resolution finite
difference scheme. We then provide the optimal sediment replenishment policy
numerically.
