A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs

Abstract

<p style='text-indent:20px;'>We consider optimal control of fractional in time (subdiffusive, i.e., for <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;\gamma &lt;1 $\end{document}</tex-math></inline-formula>) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we <inline-formula><tex-math id="M2">\begin{document}$\mathsf{first\;show}$\end{document}</tex-math></inline-formula> the existence and regularity of solutions to the forward and the associated <inline-formula><tex-math id="M3">\begin{document}$\mathsf{backward\;(adjoint)}$\end{document}</tex-math></inline-formula> problems. In the second part, we prove existence of optimal <inline-formula><tex-math id="M4">\begin{document}$\mathsf{controls }$\end{document}</tex-math></inline-formula> and characterize the associated <inline-formula><tex-math id="M5">\begin{document}$\mathsf{first\;order}$\end{document}</tex-math></inline-formula> optimality conditions. Several examples involving fractional in time (and some fractional in space diffusion) equations are described in detail. The most challenging obstacle we overcome is the failure of the semigroup property for the semilinear problem in any scaling of (frequency-domain) Hilbert spaces.</p>