Open AccessOptimal control of mixed local-nonlocal parabolic PDE with singular boundary-exterior data
Author(s) -
Jean-Daniel Djida,
Gisèle Mophou,
Mahamadi Warma
Publication year - 2022
Publication title -
evolution equations and control theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.665
H-Index - 19
eISSN - 2163-2480
pISSN - 2163-2472
DOI - 10.3934/eect.2022015
Subject(s) - mathematics , bounded function , dirichlet boundary condition , boundary (topology) , elliptic operator , operator (biology) , omega , dirichlet distribution , combinatorics , mathematical analysis , boundary value problem , physics , biochemistry , chemistry , repressor , quantum mechanics , transcription factor , gene
We consider parabolic equations on bounded smooth open sets \begin{document}$ {\Omega}\subset \mathbb{R}^N $\end{document} ( \begin{document}$ N\ge 1 $\end{document} ) with mixed Dirichlet type boundary-exterior conditions associated with the elliptic operator \begin{document}$ \mathscr{L} : = - \Delta + (-\Delta)^{s} $\end{document} ( \begin{document}$ 0<s<1 $\end{document} ). Firstly, we prove several well-posedness and regularity results of the associated elliptic and parabolic problems with smooth, and then with singular boundary-exterior data. Secondly, we show the existence of optimal solutions of associated optimal control problems, and we characterize the optimality conditions. This is the first time that such topics have been presented and studied in a unified fashion for mixed local-nonlocal PDEs with singular data.