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

The first author is partially supported by NSF grants DMS-2110263, DMS-1913004 and the Air Force Office of Scientific Research under Award NO: FA9550-19-1-0036. The last author is partially supported by the Air Force Office of Scientific Research under Award NO: FA9550-18-1-0242 and the Army Research Office (ARO) under Award NO: W911NF-20-1-0115

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  • We consider optimal control of fractional in time (subdiffusive, i.e., for $ 0<\gamma <1 $) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we $\mathsf{first\;show}$ the existence and regularity of solutions to the forward and the associated $\mathsf{backward\;(adjoint)}$ problems. In the second part, we prove existence of optimal $\mathsf{controls }$ and characterize the associated $\mathsf{first\;order}$ 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.

    Mathematics Subject Classification: Primary: 49J20, 49K20; Secondary: 35S15, 65R20.


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