One studies the Lagrange optimal control problem for a large class of nonlinear systems governed by Volterra integrodifferential evolution equations. The hypotheses are general enough to incorporate both parabolic and hyperbolic distributed parameter control systems. First one establishes the existence of optimal controls by considering systems in which the control appears linearly in the dynamics. Then, one considers nonlinear systems. Now in order to guarantee the existence of optimal controls, one has to pass to a larger system with measure valued controls, known as the «relaxed system». For this augmented system, one proves that optimal controls exist and in addition, under mild hypotheses the value of the relaxed problem equals that of the original one. Finally one proves a density (relaxation) result relating the trajectories of the original and relaxed systems. This result illustrates that the relaxed problem is the «closure» of the original one