Degenerate distributed control systems with fractional time derivative

The existence of a unique strong solution for the Cauchy problem to semilinear nondegenerate fractional dierential equation and for the generalized Showalter - Sidorov problem to semilinear fractional dierential equation with degenerate operator at the Caputo derivative in Banach spaces is proved. These results are used for search of solution existence conditions for a class of optimal control problems to a system described by the degenerate semilinear fractional evolution equation. Abstract results are applied to the research of an optimal control problem solvability for the equations system of Kelvin-Voigt fractional viscoelastic fluids.