Mixed control for semilinear fractional equations

In this work, we consider problems in which two types of controls (distributed and starting control functions) are used simultaneously. The main results concern the solvability of a class of optimal control problems for systems whose states are described by equations in Banach spaces that are resolved with respect to the Gerasimov–Caputo fractional derivative and nonlinear in the lowest fractional derivatives. We consider convex lower semicontinuous, coercive functionals, which are compromise or control-independent. Abstract results are demonstrated by an example of a control problem for a fractional model of metastable states in semiconductors.