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

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.