Start control problem for a class of degenerate equations with lower order fractional derivatives

Fractional order degenerate evolution equation with lower fractional derivatives is studied. The case of a relatively bounded pair of operators in the main part of the equation is considered. For linear and semilinear equations the existence of a unique strong solution of the generalized Showalter — Sidorov problem is proved.These results are used for the proof of the solvability of the start control problem in the linear and the semilinera case. The obtained results are applied to study of an optimal control problem for a fractional order in time degenerate distributed system.