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

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). 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.