Mathematical and computational approaches for stochastic control of river environment and ecology: from fisheries viewpoint

TL;DR

This paper develops a stochastic control framework for optimizing river ecology and fisheries management, demonstrating computational methods on various environmental models and a high-dimensional dam-reservoir control problem.

Contribution

It introduces a modern stochastic control approach with computational techniques for complex environmental systems, including high-dimensional problems, advancing modeling and analysis in environmental engineering.

Findings

Finite difference schemes effectively compute optimal controls.

Models cover diverse ecological phenomena with unified mathematical approach.

Semi-Lagrangian discretization enables high-dimensional control problem solving.

Abstract

We present a modern stochastic control framework for dynamic optimization of river environment and ecology. We focus on a fisheries problem in Japan, and show several examples of simplified optimal control problems of stochastic differential equations modeling fishery resource dynamics, reservoir water balance dynamics, benthic algae dynamics, and sediment storage dynamics. These problems concern different phenomena with each other, but they all reduce to solving degenerate parabolic or elliptic equations. Optimal controls and value functions of these problems are computed using finite difference schemes. Finally, we present a higher-dimensional problem of controlling a dam-reservoir system using a semi-Lagrangian discretization on sparse grids. Our contribution shows the state-of-art of modeling, analysis, and computation of stochastic control in environmental engineering and science, and related research areas.