Regular and exploratory resource extraction models considering sustainability

We formulate an optimal control problem of resource extraction, where a decision maker with sustainability concern dynamically controls the extraction rate. We assume harvesting to increase profit and incur a risk of resource depletion and aim to resolve sustainability concerns. The optimality equation of the control problem is the Hamilton–Jacobi–Bellman (HJB) equation with an unbounded Hamiltonian. A regularization technique to bound the Hamiltonian is proposed to prove the existence of a unique viscosity solution to both the modified and original HJB equations. We also investigate a relaxed control case, an exploratory control counterpart of our mathematical model, with the control variable belonging to a set of probability measures. Convergent, fully implicit finite difference methods to compute the viscosity solutions to the HJB equations are presented as well. These numerical methods exploit the characteristic direction of the Hamiltonians to avoid using any matrix inversions. Finally, a demonstrative application example of the proposed model to a fishery management problem is presented.