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

We consider optimal control of fractional in time (subdiffusive, i.e., for % 0<γ <1) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we~textsffirst show the existence and regularity of solutions to the forward and the associated textsfbackward (adjoint) problems. In the second part, we prove existence of optimal textsfcontrols% and characterize the associated textsffirst 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.