Open AccessA unified framework for optimal control of fractional in time subdiffusive semilinear PDEs
Author(s) -
Harbir Antil,
Ciprian G. Gal,
Mahamadi Warma
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series s
Language(s) - English
Resource type - Journals
eISSN - 1937-1632
pISSN - 1937-1179
DOI - 10.3934/dcdss.2022012
Subject(s) - mathematics , semigroup , order (exchange) , space (punctuation) , hilbert space , combinatorics , pure mathematics , computer science , operating system , finance , economics
We consider optimal control of fractional in time (subdiffusive, i.e., for \begin{document}$ 0<\gamma <1 $\end{document} ) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we \begin{document}$\mathsf{first\;show}$\end{document} the existence and regularity of solutions to the forward and the associated \begin{document}$\mathsf{backward\;(adjoint)}$\end{document} problems. In the second part, we prove existence of optimal \begin{document}$\mathsf{controls }$\end{document} and characterize the associated \begin{document}$\mathsf{first\;order}$\end{document} 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.