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.
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Regularity of solutions to space–time fractional wave equations: A PDE approach
Abstract We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic.
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- Award ID(s):
- 1720213
- NSF-PAR ID:
- 10093769
- Date Published:
- Journal Name:
- Fractional Calculus and Applied Analysis
- Volume:
- 21
- Issue:
- 5
- ISSN:
- 1311-0454
- Page Range / eLocation ID:
- 1262 to 1293
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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\begin{document}$ 0<\gamma <1 $\end{document} \begin{document}$\mathsf{first\;show}$\end{document} \begin{document}$\mathsf{backward\;(adjoint)}$\end{document} \begin{document}$\mathsf{controls }$\end{document} \begin{document}$\mathsf{first\;order}$\end{document}
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/fca-2018-0067
