In this work we analyze the structure of optimal solutions for a class of infinite-dimensional control systems. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence result that we obtain extends the result of Carlson, Haurie, and Jabrane to a situation where the trajectories are not necessarily bounded. Also, we show that an optimal trajectory defined on an interval [0, tau] is contained in a small neighborhood of the optimal steady-state in the weak topology for all t is an element of [0, tau]\E, where E subset of [0, tau] is a measurable set such that the Lebesgue measure of E does not exceed a constant which depends only on the neighborhood of the optimal steady-state and does not depend on tau.