Distributed control for semilinear equations with Gerasimov–Caputo derivatives

We consider the optimal control problem for semilinear evolution equations with lower fractional derivatives, resolved with respect to the higher fractional derivative, as well as having a degenerate linear operator at it. The nonlinear operator depends on the Gerasimov–Caputo fractional derivatives of lower orders. For the degenerate equation, a nonlinear operator is considered in two cases: if its image lies in the subspace without degeneration and if this operator depends only on the elements of the subspace without degeneration. It is shown that in the case when the solvability of the initial problem, for at least one admissible control, is obvious or can be shown directly, it is possible to prove the existence of an optimal control under a weaker condition of uniform in time local Lipschitz continuity with respect to the phase variables of the nonlinear operator, instead of the condition of its Lipschitz continuity. The theoretical results are applied to an optimal control problem for a system of partial differential equations with fractional time derivatives.