Optimal control of mixed local-nonlocal parabolic PDE with singular boundary-exterior data

<p style='text-indent:20px;'>We consider parabolic equations on bounded smooth open sets <inline-formula><tex-math id="M1">\begin{document}$ {\Omega}\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ N\ge 1 $\end{document}</tex-math></inline-formula>) with mixed Dirichlet type boundary-exterior conditions associated the elliptic operator id="M3">\begin{document}$ \mathscr{L} : = - \Delta + (-\Delta)^{s} id="M4">\begin{document}$ 0<s<1 $\end{document}</tex-math></inline-formula>). Firstly, we prove several well-posedness and regularity results of problems smooth, then singular data. Secondly, show existence optimal solutions control problems, characterize optimality conditions. This is first time that such topics have been presented studied in a unified fashion for local-nonlocal PDEs data.</p>