Second-order analysis of Fokker–Planck ensemble optimal control problems

Körner, Jacob; Borzì, Alfio

doi:10.1051/cocv/2022066

Körner, Jacob ; Borzì, Alfio

ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 77

Résumé

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.

MR Zbl

DOI : 10.1051/cocv/2022066

Classification : 35Q84, 35Q93, 49J20, 49K20
Keywords: Optimal control theory, Fokker-Planck equation, ensemble optimal control problems, second-order analysis, optimality conditions, stochastic drift-diffusion process

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     author = {K\"orner, Jacob and Borz{\`\i}, Alfio},
     title = {Second-order analysis of {Fokker{\textendash}Planck} ensemble optimal control problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     year = {2022},
     publisher = {EDP-Sciences},
     volume = {28},
     doi = {10.1051/cocv/2022066},
     mrnumber = {4524413},
     zbl = {1505.35336},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv/2022066/}
}

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Körner, Jacob; Borzì, Alfio. Second-order analysis of Fokker–Planck ensemble optimal control problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 77. doi: 10.1051/cocv/2022066

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Partially supported by the BMBF-Project iDeLIVER: Intelligent MR Diagnosis of the Liver by Linking Model and Data-driven Processes.