A PDE approach to regularity of solutions to finite horizon optimal switching problems

Arnarson, Teitur; Djehiche, Boualem; Poghosyan, Michael; Shahgholian, Henrik

doi:10.1016/j.na.2009.05.063

Cited by 16 publications

(14 citation statements)

References 4 publications

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“…As in the existing models, the HJBQVIs lead to the following implications on the optimal control policy of the panel angle. ✓If ( t , X t ) ∈ D 01 and i = 0, then immediately switch the panel angle from θ 0 to θ 1 . ✓If ( t , X t ) ∈ D 10 and i = 1, then immediately switch the panel angle from θ 1 to θ 0 . ✓Otherwise, do nothing. …”

Section: Mathematical Modelmentioning

confidence: 99%

“…From a formal standpoint, this treatment along the boundaries is a sign of the resulting Fichela functions associated with the operator L along x = 0,1. 64 As in the existing models, 45,65 the HJBQVIs lead to the following implications on the optimal control policy of the panel angle.…”

Section: Hamilton-jacobi-bellman Variational Inequalitiesmentioning

confidence: 99%

“…Here, it is reasonable to consider continuous viscosity solutions by Proposition . For definitions of viscosity solutions to HJBQVIs, see the works of Arnarson et al and Pham …”

Section: Mathematical Modelmentioning

confidence: 99%

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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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Section: Mathematical Modelmentioning

confidence: 99%

Section: Hamilton-jacobi-bellman Variational Inequalitiesmentioning

confidence: 99%

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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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“…Optimal switching problems are well documented in the literature (see e.g. [6,8,3,7,14,17,12,18,20,24,27,11] etc. and the references therein), especially in connection with mathematical finance, energy market, etc.…”

Section: Introductionmentioning

confidence: 99%

Systems of integro-PDEs with interconnected obstacles and multi-modes switching problem driven by Lévy process

Hamadène¹,

Zhao

2015

Nonlinear Differ. Equ. Appl.

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In this paper we show existence and uniqueness of the solution in viscosity sense for a system of nonlinear m variational integral-partial differential equations with interconnected obstacles whose coefficients (fi)i=1,··· ,m depend on (uj)j=1,··· ,m. From the probabilistic point of view, this system is related to optimal stochastic switching problem when the noise is driven by a Lévy process. The switching costs depend on (t, x). As a by-product of the main result we obtain that the value function of the switching problem is continuous and unique solution of its associated Hamilton-Jacobi-Bellman system of equations. The main tool we used is the notion of systems of reflected BSDEs with oblique reflection driven by a Lévy process.Next let (a s ) s∈[0,T ] be the following pure jump process:where {θ j } j≥0 is an increasing sequence of stopping times with values in [0, T ] and (α j ) j≥0 are random variables with values in A := {1, . . . , m} (the set of modes to which the controller can switch) such that for any j ≥ 0, α j is F θj −measurable. The pair Υ = ((θ j ) j≥0 , (α j ) j≥0 ) is called a strategy of switching and when it satisfies P[θ n < T, ∀n ≥ 0] = 0 it is moreover said admissible. Finally we denote by A i t the set of admissible strategies such that α 0 = i and θ 0 = t. Assume next that for anyLet Υ be an admissible strategy of A i t with which one associates a payoff given by:

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“…Switching problems have recently attracted a lot of research activities, especially in connection with mathematical finance, commodities, and in particular energy markets, etc (see e.g. [3,23,4,5,11,1,9,8,10,15,16,18,20,22,24,25,26,21,30,27,28] and the references therein). Several points of view, mainly dealing with control problems, have been considered (theoritical and practical [3,23,5,9,10,15,18,26], numerics [4,15], filtering and partial information [22]).…”

Section: Introductionmentioning

confidence: 99%

On the equality of solutions of max–min and min–max systems of variational inequalities with interconnected bilateral obstacles

Djehiche

Hamadène

Morlais

et al. 2017

Journal of Mathematical Analysis and Applications

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In this paper, we deal with the solutions of systems of PDEs with bilateral inter-connected obstacles of minmax and max-min types. These systems arise naturally in stochastic switching zero-sum game problems. We show that when the switching costs of one side are regular, the solutions of the min-max and max-min systems coincide. Then, this common viscosity solution is related to a multi-dimensional doubly reflected BSDE with bilateral interconnected obstacles. Finally, its relationship with the the values of a zero-sum switching game is studied.

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A PDE approach to regularity of solutions to finite horizon optimal switching problems

Cited by 16 publications

References 4 publications

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

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

Systems of integro-PDEs with interconnected obstacles and multi-modes switching problem driven by Lévy process

On the equality of solutions of max–min and min–max systems of variational inequalities with interconnected bilateral obstacles

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