This paper presents an iterative method for determining control trajectories for systems subject to state and control constraints. By maximizing in the Lagrange multiplier, or dual variable, as opposed to minimizing in the control, or primal variable, consistent updating sets are determined to converge upon the optimal solution. The method is applied to the control of damping forces in an equipment isolation system subjected to constraints imposed by the physical implementation of a particular controllable damper. Control trajectories are designed to minimize peak total response accelerations when the system is excited by a pulse-like acceleration at the base. Potential parameterized feedback control laws are deduced from optimal control trajectories for different excitation frequencies. Performance metrics are compared between the proposed method and a passively damped system for a single-degree-of-freedom isolation system. © 2012 AACC American Automatic Control Council).