Optimal control of fractional semilinear PDEs
In this paper, we consider the optimal control of semilinear fractional PDEs with both spectral and integral fractional diffusion operators of order 2 s with s ∈ (0, 1). We first prove the boundedness of solutions to both semilinear fractional PDEs under minimal regularity assumptions on domain and data. We next introduce an optimal growth condition on the nonlinearity to show the Lipschitz continuity of the solution map for the semilinear elliptic equations with respect to the data. We further apply our ideas to show existence of solutions to optimal control problems with semilinear fractional equations as constraints. Under the standard assumptions on the nonlinearity (twice continuously differentiable) we derive the first and second order optimality conditions.
Authors:
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Award ID(s):
Publication Date:
NSF-PAR ID:
10175696
Journal Name:
ESAIM: Control, Optimisation and Calculus of Variations
Volume:
26
Page Range or eLocation-ID:
5
ISSN:
1292-8119
1. 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.