Optimal control problems of hereditary systems have been given a great deal of attention in recent years. In this paper a class of non-linear systems with finite heredity in both the state and the control variables is considered. Using Pontryagin's geometric approach necessary conditions not only for the optimal control but also for the optimal initial control are derived. In the linear time-optimal case theorems for the form and uniqueness of the optimal control and optimal initial control are given. An example of a Fixed-time energy-optimal problem demonstrates the effect of the optimization of the initial control.