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Optimal Control of the Fokker-Planck Equation with State-Dependent Controls

URN to cite this document: urn:nbn:de:bvb:703-epub-2794-5

Title data

Fleig, Arthur ; Guglielmi, Roberto:
Optimal Control of the Fokker-Planck Equation with State-Dependent Controls.
Bayreuth ; Linz , 2016 . - 21 S.

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Project information

Project title:
Project's official titleProject's id
Marie Curie Initial Training Network FP7-PEOPLE-2010-ITN SADCOGA 264735-SADCO
Model Predictive Control for the Fokker-Planck EquationGR 1569/15-1

Project financing: 7. Forschungsrahmenprogramm für Forschung, technologische Entwicklung und Demonstration der Europäischen Union
Deutsche Forschungsgemeinschaft
Istituto Nazionale di Alta Matematica (INdAM)

Abstract

For a large class of stochastic processes, the evolution of the underlying probability density function is prescribed by a forward Kolmogorov equation, also called Fokker-Planck equation, a second-order parabolic partial differential equation. In this manner, an optimal control problem subject to an Itô stochastic differential equation can be rendered deterministic by recasting it as an optimal control problem for the Fokker-Planck equation, which we study in this work. In this setting, the control acts as a coefficient of the state variable in the advection term, i.e., it is of bilinear type. This optimal control problem has been firstly studied by Annunziato and Borzì (2010, 2013) for constant or time dependent controls. We extend the analysis to the case of time and space dependent controls, which allows to consider a wider variety of possible objectives. In order to deduce existence of nonnegative solutions for the state equation we require suitable integrability assumptions on the coefficients of the Fokker-Planck equation and thus on the control function. Therefore, the optimization takes place in a Banach space. We develop a systematic analysis of the existence of optimal controls and derive the system of first order necessary optimality conditions.

Further data

Item Type: Preprint, postprint
Keywords: bilinear control; Fokker-Planck equation; optimal control
theory; optimization in Banach spaces; probability density function; stochastic
process
Subject classification: Mathematics Subject Classification (2010): 35Q84 35Q93 49J20 49K20
DDC Subjects: 500 Science > 510 Mathematics
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Profile Fields > Advanced Fields > Nonlinear Dynamics
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Profile Fields
Profile Fields > Advanced Fields
Language: English
Originates at UBT: Yes
URN: urn:nbn:de:bvb:703-epub-2794-5
Date Deposited: 12 Apr 2016 07:46
Last Modified: 12 Apr 2016 07:46
URI: https://epub.uni-bayreuth.de/id/eprint/2794

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