Analytical and numerical solutions to ergodic control problems arising in environmental management
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Environmental management should be based on a long-run sustainable viewpoint. A stochastic control problem optimizing a long-run objective function is an ergodic control problem whose resolution can be achieved by solving an associated Hamilton-Jacobi-Bellman (HJB) equation having an effective Hamiltonian. However, this key topic has not been well-studied in the context of environmental management. Focusing on non-smooth sediment storage management as an engineering problem, we formulate a new ergodic control problem under discrete observations: a natural assumption of realistic observations. We give an analytical solution to a simple problem and verify optimality and uniqueness of the corresponding HJB equation. To manage HJB equations in more general cases, we propose a fast sweep method resorting to neither pseudo-time integration nor vanishing discount. The optimal policy and the effective Hamiltonian are then computed simultaneously. A robust control counterpart where the dynamics involve uncertainties is also considered based on a multiplier robust formalism that harmonizes with the stochastic control approach. The resulting HJB equation has a stronger non-linearity but is managed using the presented numerical method. An extended problem constraining control policy is analyzed as well.
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