| Issue |
ESAIM: COCV
Volume 28, 2022
|
|
|---|---|---|
| Article Number | 77 | |
| Number of page(s) | 30 | |
| DOI | https://doi.org/10.1051/cocv/2022066 | |
| Published online | 22 December 2022 | |
Second-order analysis of Fokker–Planck ensemble optimal control problems*
Institut für Mathematik, Universität Würzburg,
Emil-Fischer-Strasse 30,
Würzburg
97074,
Germany
** Corresponding author: jacob.koerner@mathematik.uni-wuerzburg.de
Received:
22
October
2021
Accepted:
8
October
2022
Ensemble optimal control problems governed by a Fokker-Planck equation with space-time dependent controls are investigated. These problems require the minimisation of objective functionals of probability type and aim at determining robust control mechanisms for the ensemble of trajectories of the stochastic system defining the Fokker-Planck model. In this work, existence of optimal controls is proved and a detailed analysis of their characterization by first- and second-order optimality conditions is presented. For this purpose, the well-posedness of the Fokker-Planck equation, and new estimates concerning an inhomogeneous Fokker-Planck model are discussed, which are essential to prove the necessary regularity and compactness of the control-to-state ma p appearing in the first-and second-order analysis.
Mathematics Subject Classification: 35Q84 / 35Q93 / 49J20 / 49K20
Key words: Optimal control theory / Fokker-Planck equation / ensemble optimal control problems / second-order analysis / optimality conditions / stochastic drift-diffusion process
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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