River environmental restoration based on random observations of a non-smooth stochastic dynamical system

TL;DR

This paper develops a stochastic control model for river sediment replenishment, accounting for non-smooth dynamics and partial, discrete observations, providing analytical and numerical solutions for optimal environmental restoration strategies.

Contribution

It introduces a novel stochastic control framework with non-smooth dynamics and partial observations, offering analytical solutions and numerical methods for river sediment replenishment.

Findings

Analytical solutions for simplified cases including discounted, ergodic, and complete information scenarios.

Numerical schemes for realistic cases using high-resolution finite difference methods.

Optimal replenishment policies that minimize costs while maintaining sediment levels.

Abstract

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.