Semi-active damping control represents a class of parametric control problems in which control decisions affect the value of positively-constrained damping parameters. Transfer functions from control to performance can not be written in closed form and performance is typically assessed in terms of the peak values of transient responses. In this study optimal control trajectories are designed to satisfy the Hamilton-Jacobi-Bellman conditions. The performance associated with such control trajectories may be viewed as the performance limit of a vibrating system controlled by adjusting one or more damping parameters. © 2010 AACC.