Mathematical Analysis on a Conforming Finite Element Scheme for Advection-Dispersion-Decay Equations on Connected Graphs

Appl Stoch Models Bus & Ind

Yano

2019

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The operation of a stand‐alone photovoltaic (PV) system ultimately aims for the optimization of its energy storage. We present a mathematical model for cost‐effective control of a stand‐alone system based on a PV panel equipped with an angle adjustment device. The model is based on viscosity solutions to partial differential equations, which serve as a new and mathematically rigorous tool for modeling, analyzing, and controlling PV systems. We formulate a stochastic optimal switching problem of the panel angle, which is here a binary variable to be dynamically controlled under stochastic weather condition. The stochasticity comes from cloud cover dynamics, which is modeled with a nonlinear stochastic differential equation. In finding the optimal control policy of the panel angle, switching the angle is subject to impulsive cost and reduces to solving a system of Hamilton‐Jacobi‐Bellman quasi‐variational inequalities (HJBQVIs). We show that the stochastic differential equation is well posed and that the HJBQVIs admit a unique viscosity solution. In addition, a finite‐difference scheme is proposed for the numerical discretization of HJBQVIs. A demonstrative computational example of the HJBQVIs, with emphasis on a stand‐alone experimental system, is finally presented with practical implications for its cost‐effective operation.

normalΔ t = N_{t}^{- 1} T

.…”

Section: Numerical Computationmentioning

confidence: 99%

An optimal switching approach toward cost‐effective control of a stand‐alone photovoltaic panel system under stochastic environment

Appl Stoch Models Bus & Ind

Yano

2019

Self Cite

“…The reduced HJBI(13) is a degenerate parabolic partial differential equation whose form is suited to numerically solve with the 1-D Petrov-Galerkin finite element scheme [71]. The scheme can handle degenerate parabolic equations in a stable manner, and has been successfully applied to degenerate parabolic partial differential equations such as Hamilton-Jacobi-Bellman equations arising in ecological and fisheries problems [67,68,73,76].…”

Section: Methodsmentioning

confidence: 99%

Stochastic differential game for management of non-renewable fishery resource under model ambiguity

Journal of Biological Dynamics

Yaegashi

2018

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A new bio-economic model for managing population of nonrenewable 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. ARTICLE HISTORY

“…The mathematical analysis results are validated through numerical computation of the HJB equation (10). The finite element scheme with the help of a penalization technique suitable for solving the degenerate elliptic boundary value problems such as variational inequalities is employed [57]. The detail of the scheme and its variant, and their application examples are not described here, but found in literatures [58,53,55,56,59].…”

Section: Proposition 44: the Equationmentioning

confidence: 99%

Singular stochastic control model for algae growth management in dam downstream

Journal of Biological Dynamics

Yaegashi

2018

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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.