Existence and structure of optimal solutions of infinite-dimensional control problems

In this work we analyze the structure of optimal solutions for a class of i nfinite-dimensional control systems. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence r esult that we obtain extends the result of Carlson, Haurie, and Jabrane to a situation where the trajectories are not necessarily bounded. Also, we sh ow 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 fo r all t is an element of [0, tau]\E, where E subset of [0, tau] is a measur able set such that the Lebesgue measure of E does not exceed a constant whi ch depends only on the neighborhood of the optimal steady-state and does no t depend on tau.