HJB and Fokker-Planck equations for river environmental management based on stochastic impulse control with discrete and random observation

TL;DR

This paper develops a stochastic impulse control framework using HJB and Fokker-Planck equations to optimize river environmental management, specifically sediment and algae control, with numerical solutions and analysis.

Contribution

It introduces a novel two-variable stochastic control model for river management based on jump SDEs, incorporating discrete and random observations, and analyzes associated HJB and FP equations.

Findings

Optimal threshold-type control identified

Numerical schemes effectively solve complex equations

Probability density functions analyzed for control strategies

Abstract

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.