Analytical and numerical solutions to ergodic control problems arising in environmental management

TL;DR

This paper addresses ergodic control problems in environmental management, specifically sediment storage, by analyzing and numerically solving associated HJB equations with novel methods and considering uncertainties.

Contribution

It introduces a new numerical fast-sweep method for solving ergodic HJB equations with non-smoothness and discontinuities, applied to environmental management.

Findings

Developed a fast-sweep numerical method for ergodic HJB equations.

Proved optimality and comparison results for non-smooth HJB solutions.

Analyzed convergence rates and extended to robust control with uncertainties.

Abstract

Environmental management optimizing a long-run objective is an ergodic control problem whose resolution can be achieved by solving an associated non-local Hamilton-Jacobi-Bellman (HJB) equation having an effective Hamiltonian. Focusing on sediment storage management as a modern engineering problem, we formulate, analyze, and compute a new ergodic control problem under discrete observations: a simple but non-trivial mathematical problem. We give optimality and comparison results of the corresponding HJB equation having unique non-smoothness and discontinuity. To numerically compute HJB equations, we propose a new fast-sweep method resorting to neither pseudo-time integration nor vanishing discount. The optimal policy and the effective Hamiltonian are then computed simultaneously. Convergence rate of numerical solutions is computationally analyzed. An advanced robust control counterpart where the dynamics involve uncertainties is also numerically considered.