The mathematical concept of multiplier robust control is applied to a dam operation problem, which is an urgent issue on river water environment, as a new industrial application of stochastic optimal control. The goal of the problem is to find a fit-for-purpose and environmentally sound operation policy of the flow discharge from a dam so that overgrowth of the harmful algae Cladophora glomerata Kützing in its downstream river is effectively suppressed. A minimal stochastic differential equation for the algae growth dynamics with uncertain growth rate is first presented. The performance index to be maximized by the operator of the dam while minimized by nature is formulated within the framework of differential games. The dynamic programming principle leads to a Hamilton-Jacobi-Bellman-Isaacs equation whose solution determines the worst-case optimal operation policy of the dam, ie, the policy that the operator wants to find. Application of the model to overgrowth suppression of Cladophora glomerata Kützing just downstream of a dam in a Japanese river is then carried out. Values of the model parameters are identified with which the model successfully reproduces the observed population dynamics. A series of numerical experiments are performed to find the most effective operation policy of the dam based on a relaxation of the current policy.
Excessive predation pressure from the waterfowl Phalacrocorax carbo (Great Cormorant) on Plecoglossus altivelis (Ayu) has recently been a severe problem of river environment in Japan. Local fishery cooperatives are currently suffering from economic difficulties due to decrease of the fish catch of P. altivelis. Local fishery cooperatives and municipalities have been enthusiastically trying to develop countermeasures that can effectively reduce the predation pressure; however, their effectiveness and efficiency have not been systematically quantified well. This aim can be achieved with the help of an appropriate mathematical model. In this paper, based on a pure death process, a practical stochastic control model for population dynamics of released P. altivelis in river environment under predation pressure from P. carbo, harvesting by human, and environmental fluctuations is proposed. Finding an optimal management strategy ultimately reduces to solving a 2D Hamilton–Jacobi–Bellman equation, which is performed with a finite element scheme. Its application to a Japanese river environment successfully computes the optimal management strategy that is consistent with the reality. Numerical sensitivity analysis of the presented mathematical model is also performed for comprehension of dependence of the optimal strategy on the model parameters.
A minimal stochastic generalization of a deterministic open-ended logistic growth model is proposed for efficiently describing the biological growth of individual organisms under natural environment. The model is a system of stochastic differential equations. Its unique solvability in a strong sense is proven, and the behaviour of the solution is analysed. The presented model is then applied to the migratory fish Plecoglossus altivelis altivelis (P. altivelis, Ayu) having a one-year life history based on the data sets collected in 2017 and 2018.
A stochastic control model for finding an ecologically sound, fit-for-purpose dam operation policy to suppress bloom of attached algae in its downstream is presented. A singular exactly solvable and a more realistic regular-singular cases are analysed in terms of a Hamilton-Jacobi-Bellman equation. Regularity and consistency of the value function are analysed and its classical verification theorem is established. Practical implications of the mathematical analysis results are discussed focusing on parameter dependence of the optimal controls. An asymptotic analysis with a numerical computation reveals solution behaviour of the Hamilton-Jacobi-Bellman equation near the origin, namely at the early stage of algae growth.
This short paper presents an optimal control model for cost‐effective and ecologically conscious management of released fish. Finding an optimal management strategy reduces to solving a Hamilton‐Jacobi‐Bellman equation having a solution by separation of variables. This paper analyzes the behavior of the solution and the associated optimal management strategy. The latter is verified with an actual strategy for Plecoglossus altivelis in Japan.
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