Optimal control problems for a class of degenerate evolution equations with delay

The existence of a solution to an optimal control problem for a wide class of equations with delay not solvable with respect to the time derivative is investigated. The set of admissible controls is assumed to be convex and closed in the space of control functions, the minimized functional is quadratic. Under the condition of strong relative $p$-radiality of the pair of operators in the equation, the theorems on the unique solvability of the optimal control problem for the case of an abstract delay operator and for the case when this operator has an integral form are proved. The obtained general results are used in the study of the optimal control problem for the system of gravitational-gyroscopic waves perturbed by an integral delay operator.