Linear systems with quadratic terminal cost function and inputs with bounded amplitudes are considered. The optimal control and optimal terminal state are associated with the minimum value of the cost function. An iteration procedure is developed whereby a sequence of subsets of the reachable set are generated such that the sequence converges to a subset containing the optimal terminal state. The control yielding a minimum value for the cost function is obtained along with the terminal subset. The optimization processes on the various subsets are far simpler than the optimization of the original continuous system. An example of a third-order system is included for illustration.