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A countermeasure against sediment depletion in rivers is sand replenishment. The cost-efficient sizing of its capacity is critical for sustainable environmental management. However, only temporally discrete replenishment is possible, and sediment transport is intermittent and uncertain. In this paper, we approach this new design problem using a robust ergodic control. First, a stochastic differential equation driven by a (tempered) stable subordinator describes the sediment storage dynamics. Sand replenishment is considered an impulse control, wherein observation/replenishment chances arrive at Poisson jump times. The subordinator is ambiguous since the decision-maker does not precisely know its Lévy measure. Using a dynamic risk measure that balances replenishment cost, disutility triggered by sediment depletion, and penalization of ambiguity, we derived the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation to determine the optimal storage capacity. We obtained that the HJBI equation has an analytical solution when a stable subordinator is used. Furthermore, we prove it admits a unique continuous viscosity solution and that it is optimal and applied the proposed model to real data.

Optimal sizing of the sediment replenishment capacity based on robust ergodic control of subordinator-driven dynamics