A finite difference scheme for variational inequalities arising in stochastic control problems with several singular control variables

Yoshioka, Hidekazu; Yaegashi, Yuta

doi:10.1016/j.matcom.2018.06.013

Cited by 4 publications

(6 citation statements)

References 78 publications

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“…However, for larger a, there exists a subdomain near x = 1 such that q * = q min is optimal and q * is discontinuous at the left boundary of this sub-domain. This type of discontinuous optimal control has not been found in similar stochastic control models with constant K [63,65]. This sudden decrease of q * is considered to be due to the functional form K = aq + b of the carrying capacity that depends on the control q.…”

Section: Parameter Dependence Of the Optimal Controlmentioning

confidence: 87%

“…Therefore, g is Lipschitz continuous in the compact set D × Q. The function g models the cost of the management and the disutility caused by the existence of the population [58,[63][64][65] in a general manner. On the function g, we assume that it is increasing with respect to the second argument, meaning that the disutility caused by the population is larger for larger population.…”

Section: Performance Index To Be Minimizedmentioning

confidence: 99%

“…Formally setting a = 0 in (21) recovers the conventional constant carrying capacity K = b. In this paper, the control q appearing in the stochastic control problem is identified with the flow speed V. For the coefficient g in the performance index φ, following the models on management of harmful species population [63][64][65][66], set…”

Section: Specific Model For the Algae Population Controlmentioning

confidence: 99%

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A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

Yoshioka

2019

Journal of Biological Dynamics

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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.

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Section: Parameter Dependence Of the Optimal Controlmentioning

confidence: 87%

Section: Performance Index To Be Minimizedmentioning

confidence: 99%

Section: Specific Model For the Algae Population Controlmentioning

confidence: 99%

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A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

Yoshioka

2019

Journal of Biological Dynamics

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“…Policy iteration algorithms have been used for solving nonlinear equations resulting from HJB and related equations. 53,54 The policy iteration algorithm here is explained as follows. First, the discretized Equation (20) is rewritten as…”

Section: Policy Iteration and Numerical Optimal Controlmentioning

confidence: 99%

“…Powerful verification arguments are therefore not applicable. To overcome this issue, a finite‐difference scheme based on exponential discretization is applied to their numerical resolution. For deterministic systems, convergence results of several finite‐difference schemes to discretize HJBQVIs are available …”

Section: Introductionmentioning

confidence: 99%

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

Yoshioka

Yano

2019

Appl Stoch Models Bus & Ind

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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.

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A primitive model for stochastic regular‐impulse population control and its application to ecological problems

et al. 2019

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A primitive mathematical model for population control in natural environment is formulated under a unified regular‐impulse stochastic control formalism. This kind of mathematical models, although they are candidates of the models of population control, seem not to be well studied. In this paper, the impulse control uses deteriorating items and is scheduled only at predetermined times as in many management problems. Finding the optimal control reduces to solving a system of recursive Hamilton‐Jacobi‐Bellman equations. Mathematical analysis on the system provides a viscosity characterization of its solution. The model is applied to management problems of the predatory bird and harmful benthic alga, both of which are concerns in recent inland fisheries. We show that the optimal regular controls for the two problems have qualitatively different behavior with each other, which is due to different interactions between the regular and the impulse controls.

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A finite difference scheme for variational inequalities arising in stochastic control problems with several singular control variables

Cited by 4 publications

References 78 publications

A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

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

A primitive model for stochastic regular‐impulse population control and its application to ecological problems

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