Modeling stochastic operation of reservoir under ambiguity with an emphasis on river management

Yaegashi

2021

J.Math.Industry

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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.

Section: Empirical Numerical Approachmentioning

confidence: 99%

“…The HJBQVI (20) is computed with the two different numerical methods, which is the ODE-based method using (70) through (72) and a verified finite difference scheme having a formally first-order accuracy [44]. In addition, the statistical indices of the controlled state dynamics are calculated.…”

Section: Numerical Computationmentioning

confidence: 99%

Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions

Yaegashi

2021

J.Math.Industry

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“…The environmental and hydrological information of Hii River and Obara Dam constructed at mid‐stream of the river is documented in the literature 59,60 . From 2018, a local fishery cooperative and the local government are trying to suppress the algae bloom in the downstream of Obara Dam by wisely using sediment stored in a huge hydraulic structure placed further downstream where a large amount of sediment is collected at each flood (For its details, see Yoshioka et al 56 ).…”

Section: Applicationmentioning

confidence: 99%

“…A recursive formulation allows us to numerically handle the optimality equation. An uncertainty‐aversion (ambiguity‐aversion) problem 57 is also considered as an advanced problem, motivated by the fact that accurately identifying the environmental dynamics is often difficult 46,58,59 . Our methodology can handle cases both with and without model uncertainty.…”

Section: Introductionmentioning

confidence: 99%

A hybrid stochastic river environmental restoration modeling with discrete and costly observations

Optim Control Appl Methods

Tsujimura

Hamagami

et al. 2020

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Summary A recent river environmental restoration problem is approached from a standpoint of stochastic control of hybrid regime‐switching diffusion processes with discrete and costly observations. This setting harmonizes with real problems because continuously obtaining environmental and ecological information is difficult, and is often costly. The main problem is to decide when and how much of the sediment should be supplied into a river environment to effectively suppress bloom of benthic algae. The interventions are allowed only at observation times. Finding the optimal river restoration policy ultimately reduces to solving an optimality equation in an unconventional form due to the observation cost and discrete observations. We show its unique solvability and present a connection with degenerate parabolic partial differential equations, with which we can construct an effective algorithm for its approximation. An uncertainty‐averse optimization problem is also considered as an advanced problem. Coefficients and parameter values are identified from experimental and observation results to numerically compute the optimal restoration policy of an existing river environment with and without uncertainty.

“…1 Stochastic control is a branch of the optimal control specialized for problems with noise-driven dynamics. 2 Management problems not limited to but include those of energy and resources, 3,4 population, 5,6 environment, 7,8 finance and economics, 9,10 and planning 11,12 have been analyzed as stochastic control problems.…”

Section: Introductionmentioning

confidence: 99%

Stochastic impulse control of nonsmooth dynamics with partial observation and execution delay: Application to an environmental restoration problem

Optim Control Appl Methods

Yaegashi²

2021

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Nonsmooth dynamics driven by stochastic disturbance arise in a wide variety of engineering problems. Impulsive interventions are often employed to control stochastic systems; however, the modeling and analysis subject to execution delay have been less explored. In addition, continuously receiving information of the dynamics is not always possible. In this article, with an application to an environmental restoration problem, a continuous-time stochastic impulse control problem subject to execution delay under discrete and random observations is newly formulated and analyzed. The dynamics have a nonsmooth coefficient modulated by a Markov chain, and eventually attain an undesirable state like a depletion due to the nonsmoothness. The goal of the control problem is to find the most cost-efficient policy that can prevent the dynamics from attaining the undesirable state. We demonstrate that finding the optimal policy reduces to solving a nonstandard system of degenerate elliptic equations, the Hamilton-Jacobi-Bellman equation (HJBE), which is rigorously and analytically verified in a simplified case. The associated Fokker-Planck equation (FPE) for the controlled dynamics is derived and solved explicitly as well. The model is finally applied to numerical computation of a recent river environmental restoration problem. The HJBE and FPE are successfully computed, and the optimal policy and the probability density functions are numerically obtained.The impacts of execution delay are discussed to deeper analyze the model.