Abstract Optimal control problems solvability are researched for infinite dimensional control systems, described by semilinear evolution equations in Banach spaces with degenerate linear operator at the Caputo fractional derivative. The pair of linear operators in the equation is relatively bounded and the nonlinear operator satisfies some smoothness conditions, in particular the condition of uniform Lipschitz continuity, and one of two types additional conditions: independence of degeneracy subspace elements or non-belonging of the operator image to the degeneracy subspace. The control system is endowed by the generalized Showalter — Sidorov initial conditions, which are natural for degenerate evolution equations. Optimal control have to belong to a convex closed set of admissible controls and to minimize a convex, bounded from below, lower semicontinuous and coercive cost functional. Solvability conditions are found for the optimal control problem of this class. If the existence of the initial problem solution with an admissible control is obvious, it is shown that the local Lipschitz continuity in phase variables that uniform with respect to time is sufficient for the optimal control existence. Abstract results are illustrated by optimal control problem for the equations system of the fractional viscoelastic Kelvin — Voigt fluid dynamics.