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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.

A series of hydraulic experiments were conducted for quantitative evaluation on the detachment effect of benthic algae on uneven riverbeds caused by sediment loading. We focused on the shielding part created by the exposure height of gravel and the width between pieces of gravel. The experimental results showed that the shielding was affected by both the exposure height of gravel and the width between pieces of gravel, with the exposure height of gravel having the larger effect. In addition, it was suggested that the movement of the sediment was restricted by an uneven riverbed. In order to take into account the effect of the shielding, a shielding coefficient was introduced to examine the effective tractive force and effective amount of work. There was a positive correlation between effective tractive force and the effective amount of work with the detachment characteristic value. Thus, it is possible to estimate the detachment characteristic value in uneven riverbed using the effective tractive force and effective amount of work.

Abstract Hamilton–Jacobi–Bellman equation (HJBE) and backward stochastic differential equation (BSDE) are the two faces of stochastic control. We explore their equivalence focusing on a system of self-exciting and affine stochastic differential equations (SDEs) arising in streamflow dynamics. Our SDE is a finite-dimensional Markovian embedding of an infinite-dimensional jump-driven process called the superposition of continuous-state branching processes (a supCBI process). We formulate new ergodic control problems to evaluate the worst-case streamflow discharge in the long run and derive their HJBEs and ergodic BSDEs. The constant ambiguity aversion classically used in assessing model ambiguity must be modified in our case so that the optimality equations become well-posed. With a suitable modification of the ambiguity-aversion coefficient depending on the distributed reversion speed, we demonstrate that the solutions to the optimality equations are equivalent to each other in the sense that they lead to the same result. Finally, we apply the proposed framework to the computation of realistic cases with an existing record of discharge through a numerical Markovian embedding.

Abstract A stochastic impulse control problem with imperfect controllability of interventions is formulated with an emphasis on applications to ecological and environmental management problems. The imperfectness comes from uncertainties with respect to the magnitude of interventions. Our model is based on a dynamic programming formalism to impulsively control a 1-D diffusion process of a geometric Brownian type. The imperfectness leads to a non-local operator different from the many conventional ones, and evokes a slightly different optimal intervention policy. We give viscosity characterizations of the Hamilton–Jacobi–Bellman Quasi-Variational Inequality (HJBQVI) governing the value function focusing on its numerical computation. Uniqueness and verification results of the HJBQVI are presented and a candidate exact solution is constructed. The HJBQVI is solved with the two different numerical methods, an ordinary differential equation (ODE) based method and a finite difference scheme, demonstrating their consistency. Furthermore, the resulting controlled dynamics are extensively analyzed focusing on a bird population management case from a statistical standpoint.

We propose a simple but non-trivial mathematical model of a dam–reservoir system balancing between the downstream environmental management and hydropower production. We consider a stochastic control problem of the system to find its balanced operation policy. We point out that finding the optimal policy reduces to solving a nonlinear degenerate elliptic equation. This equation is exactly solvable in a reduced case without hydropower production, while it requires numerical computation in more complicated cases. A demonstrative application example with a high-resolution finite difference scheme is finally presented.

The development of a multidimensional phosphorus dynamic model for lake–floodplain systems is still limited due to its complex hydrodynamics and environmental processes, which constrains our understanding and environmental management on eutrophication. Therefore, we developed a horizontal two-dimensional hydrodynamics-based phosphorus model and applied it to understand the phosphorus dynamics in the Tonle Sap Lake, a large tropical lake–floodplain system in Cambodia. This phosphorus dynamic model is based on a hydrodynamic model (local inertial model) and integrates the major phosphorus processes in the lake and the phosphorus inputs from tributaries and floating villages. The phosphorus dynamic model was applied to the period from September 1999 to December 2003 for elucidating the total phosphorus balance and the spatiotemporal distribution of its concentration. The model reproduced the seasonality of total phosphorus concentration with its root mean square error of 12 and 16 µg/L for the calibration and validation periods, respectively. During 1999–2003, the lake was in a mesotrophic to eutrophic state, whereas the average eutrophic area expanded from 8.0% in October to 45.3% in May, when the inundated area was reduced, due to a relatively high internal loading and the high concentration of tributary inputs. In addition, the model estimated that most of the external total phosphorus loading was settled down and the internal processes play an important role in phosphorus dynamics in the Tonle Sap Lake. As presented, the model analysis integrating hydrodynamics and internal processes substantially helps us understand the spatiotemporal dynamics of phosphorus in the lake–floodplain systems.

A logistic type stochastic control model for cost-effective single-species population management subject to an ambiguous jump intensity is presented based on the modern multiplier-robust formulation. The Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation for finding the optimal control is then derived. Mathematical analysis of the HJBI equation from the viewpoint of viscosity solutions is carried out with an emphasis on the non-linear and non-local term, which is a key term arising due to the jump ambiguity. We show that this term can be efficiently handled in the framework of viscosity solutions by utilizing its monotonicity property. A numerical scheme to discretize the HJBI equation is presented as well. Our model is finally applied to management of algae population in river environment. Optimal management policies ranging from the short-term to long-term viewpoints are numerically investigated.

Abstract We formulate a stochastic impulse control model for animal population management and a candidate of exact solutions to a Hamilton–Jacobi–Bellman quasi-variational inequality. This model has a qualitatively different functional form of the performance index from the existing monotone ones. So far, optimality and unique solvability of the Hamilton–Jacobi–Bellman quasi-variational inequality has not been investigated, which are thus addressed in this paper. We present a candidate of exact solutions to the Hamilton–Jacobi–Bellman quasi-variational inequality and prove its optimality and unique solvability within a certain class of solutions in a viscosity sense. We also present and examine a dynamical system-based numerical method for computing coefficients in the exact solutions.

This paper proposes a simple mathematical model forupstream fish migration along rivers. The model describes the fish migration along a river based on a mixed optimal control approach having swimming velocity, school size, and stopping time of migration as control variables. The optimization problem reduces to a variational inequality. Its explicit “viscosity” solution is presented with the dependence of the fish migration on river environment. To prove uniqueness of the solution to the variational inequality requires a constructive argument not based on the conventional theorems. A novel finite difference scheme for solving the variational inequality is also proposed with its convergence results. An application example of the model discusses the upstream migration of Plecoglossus altivelis (Ayu) in Japan, which evaluates the dependence of the fish migration on the habitat quality and provides recommendations for managing river environment. This is an interdisciplinary research between environmental and mathematical fields.

An optimal switching control formalism combined with the stochastic dynamic programming is, for the first time, applied to modelling life cycle of migrating population dynamics with non-overlapping generations. The migration behaviour between habitats is efficiently described as impulsive switching based on stochastic differential equations, which is a new standpoint for modelling the biological phenomenon. The population dynamics is assumed to occur so that the reproductive success is maximized under an expectation. Finding the optimal migration strategy ultimately reduces to solving an optimality equation of the quasi-variational type. We show an effective linkage between our optimality equation and the basic reproduction number. Our model is applied to numerical computation of optimal migration strategy and basic reproduction number of an amphidromous fish Plecoglossus altivelis altivelis in Japan as a target species.

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 simplified stochastic control model for optimization of logistic dynamics with the control-dependent carrying capacity, which is motivated by a recent algae population management problem in the river environment, is presented. Solving the optimization problem reduces to finding a solution to a non-local first-order differential equation called tt–Jacobi–Bellman (HJB) equation. It is shown that the HJB equation has a unique viscosity solution and that the solution can be approximated with a finite difference scheme. Asymptotic estimates of the solution and the optimal control are derived and compared with numerical solutions. Finally, parameter dependence of the optimal control is examined numerically with implications to river environmental management.

Abstract A new kind of multiple stochastic optimal stopping problem is formulated and its associated recursive variational inequalities are derived. We show that these variational inequalities can be solved exactly in a cascading manner. The relevance of the present problem in analyzing animal migration, which is an ecologically important problem, is also briefly discussed.

A new bio-economic model for managing population of non-renewable inland fishery resource in uncertain environment is presented. Population dynamics of the resource is described with stochastic differential equations (SDEs) having ambiguous growth and mortality rates. The performance index to be maximized by the manager of the resource while minimized by nature is presented in the context of differential game theory. The dynamic programming principle leads to a Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation that governs the optimal resource management strategy under the ambiguity. The main contribution of this paper is a series of theoretical analysis on the reduced HJBI equation for non-renewable fishery resources in a broad sense, indicating that the ambiguity critically affects the resulting optimal controls. Practical implications of the theoretical analysis results are also presented focusing on artificially hatched Plecoglossus altivelis (Ayu), an important inland fishery resource in Japan.

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.

Abstract Comprehension of moisture dynamics in fibrous sheets is indispensable in a wide variety of industrial areas. This paper proposes a practical mathematical model, which is referred to as the 2-D extended porous medium equation (PME), to physically describe moisture dynamics in fibrous sheets under evaporative environment. A numerical method, which is referred to as the 2-D dual-finite volume method (DFVM), to approximate its solutions in a stable manner is also presented so that the moisture dynamics is reasonably simulated. The 2-D DFVM, which can optionally be equipped with isotone numerical fluxes, is examined with test cases to show its satisfactory accuracy and versatility. The parameters and coefficients involved in the mathematical model for a non-woven fibrous sheet are identified with laboratory experiments. Numerical simulation of moisture dynamics in the horizontally or vertically placed sheet is performed as a demonstrative application example of the present model and the numerical method.

Abstract A stochastic optimization model for management of aquacultured fish for sale and ecological education is established. Population dynamics of the fish is described with a system of stochastic differential equations assuming that they are stochastically harvested after an opening time: the variable to be optimized. A remarkable difference between the present and conventional models for the aquaculture is that the former considers the harvesting for ecological education, which is a key for current inland fishery especially in Japan. Finding the cost-effective optimal opening time effectively reduces to solving an algebraic equation. Mathematical analysis on the optimal opening time clearly reveals its unique existence and qualitative behavior, such as its dependence on the model parameters, which have practical implications. A demonstrative application example of the model is also presented focusing on an aquaculture of Plecoglossus altivelis in Japan.

An increasing population and limited arable land area endanger sufficient and variegated food supplies worldwide. Greenhouse cultivation enables highly intensive plant production and thereby enables the production of abundant fresh vegetables and fruits. The salient benefits of greenhouse cultivation are supported by ingenious management of crop environments, assisted by fossil fuel and grid electricity supplies. To reduce dependence on traditional energy resources, various studies have investigated exploitation of renewable energies for greenhouse environment management. Among them, solar photovoltaic (PV) technologies are anticipated to feed electrical energy to greenhouse appliances for microclimate control. This study proposes a venetian-blind-type shading system consisting of semi-transparent PV modules as blind blades based on micro-spherical solar cell technology to achieve greenhouse shading and electricity production concurrently. In response to the solar irradiance level, the PV blind inclination was altered automatically using a direct current (DC) motor driven by electrical energy generated by the PV blind itself. The PV blind was operated continuously during a five-month test period without outage. Moreover, the PV blind generated surplus electrical energy of 2125 kJ for blind system operations during the test period. The annual surplus energy calculated under the present experimental condition was 7.8 kWh m−2 year−1, suggesting that application of the PV blind to a greenhouse roof enables sunlight level control and electrical appliance operations in the greenhouse with a diminished fuel and grid electricity supply, particularly in high-insolation regions.