This contribution constitute an theoretic work devoted to the class of optimal control problems (OCPs) involving a specific dynamics described by Volterra integro-differential equations. We study OCPs associated with the Volterra integro-differential systems and establish the solvability property of this class of problems. A special structure of the abstract dynamic optimization problem under consideration makes it possible to interpret the initially given sophisticated OCP as a separate convex optimization problem in a suitable Hilbert space. This fact implies effective splitting type solution schemes in combination with the first-order computational methods for the numerical treatment of the initially given OCPs. We concretely consider the celebrated Armijo-type gradient method for this purpose. Finally, we give a mathematically rigorous proof of the numerical consistence of splitting algorithm applied to the main OCP.