Optimization of a model Fokker-Planck equation

Herty, Michaël; Jörres, Christian; Sandjo, Albert N.

doi:10.3934/krm.2012.5.485

Cited by 8 publications

(7 citation statements)

References 25 publications

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“…Results in Chupin (2010) consider existence results for a special form of stationary Fokker-Planck equation in weighted Sobolev spaces. In Herty et al (2012) and Herty and Sandjo (2011) existence results are obtained in the context of dose calculation for optimal radiation treatment planning.…”

Section: Introductionmentioning

confidence: 99%

On Global Existence and Regularity of Solutions for a Transport Problem Related to Charged Particles

Tervo

2021

Journal of Computational and Theoretical Transport

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The paper considers a class of linear Boltzmann transport equations which models charged particle transport, for example in dose calculation of radiation therapy. The equation is an approximation of the exact transport equation containing hyper-singular integrals in its collision terms. The paper confines to the global case where the spatial domain G is the whole space R 3 : Existence results of solutions for the due initial value problem are formulated by applying variational methods. In addition, some regularity results of solutions are verified in scales of relevant anisotropic mixed-norm Sobolev spaces.

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Section: Introductionmentioning

confidence: 99%

On Global Existence and Regularity of Solutions for a Transport Problem Related to Charged Particles

Tervo

2021

Journal of Computational and Theoretical Transport

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“…More precisely, these equations are known to describe the time evolution of the probability density function of agents, where the motion of each of them is modelled by a stochastic differential equation. In particular, Fokker-Planck equations are present in models of mass opinion dynamics [2], tumor growth [12,27], bird flocks movement [21] and various biological events [18,36], among others. The recent survey [24] describes several applications of the Fokker-Planck equation to different socio-economic phenomena.…”

Section: Introductionmentioning

confidence: 99%

“…It is worth mentioning that some of these articles already analyze control problems associated to their models, as e.g. [2,27].…”

Section: Introductionmentioning

confidence: 99%

First and second order optimality conditions for the control of Fokker-Planck equations

Aronna

Tröltzsch

2021

ESAIM: COCV

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In this article we study an optimal control problem subject to the Fokker-Planck equation \[ \partial_t \rho - \nu \Delta \rho - {\rm div } \big(\rho B[u]\big) = 0. \] The control variable $u$ is time-dependent and possibly multidimensional, and the function $B$ depends on the space variable and the control. The cost functional is of tracking type and includes a quadratic regularization term on the control. For this problem, we prove existence of optimal controls and first order necessary conditions. Main emphasis is placed on second order necessary and sufficient conditions.

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“…In this article we prove first and second order optimality conditions for a control problem subject to the Fokker-Planck equation Fokker-Planck equations arise in many situations in which a large number of agents is involved. In particular, these equations are present in models of mass opinion dynamics [1], tumor growth [8,22], bird flocks movement [16] and various biological events [30,13], among others. The recent survey [19] describes several applications of the Fokker-Planck equation to different socio-economic phenomena.…”

Section: Introductionmentioning

confidence: 99%

“…It is worth mentioning that some of these articles already analyze control problems associated to their models, as e.g. [22,1].…”

Section: Introductionmentioning

confidence: 99%

First and Second Order Optimality Conditions for the Control of Fokker-Planck Equations

Aronna¹,

Tröltzsch²

2020

Preprint

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In this article we study an optimal control problem subject to the Fokker-Planck equationThe control variable u is time-dependent and possibly multidimensional, and the function B depends on the space variable and the control. The cost functional is of tracking type and includes a quadratic (Tikhonov regularizing) term on the control. For this problem we prove existence of optimal control, first order necessary conditions, and second order necessary and sufficient conditions.

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Optimization of a model Fokker-Planck equation

Cited by 8 publications

References 25 publications

On Global Existence and Regularity of Solutions for a Transport Problem Related to Charged Particles

On Global Existence and Regularity of Solutions for a Transport Problem Related to Charged Particles

First and second order optimality conditions for the control of Fokker-Planck equations

First and Second Order Optimality Conditions for the Control of Fokker-Planck Equations

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