Uncertainty in dynamical systems may arise due to inaccuracies in modelling, parameter variations and external disturbances. As a result of this uncertainty, the performance index of an optimal control system deviates from its optimum value, which is referred to as the deterioration. A technique is presented to find a bound on the deteriorated performance index of optimum linear systems subject to bounded uncertainty. Uncertainty is incorporated as a forcing term in the system equations. To find the deteriorated performance bound, the performance index subject to the uncertain system is to be maximized within a specified time interval. The interchange theorem is used to interchange the maximizing and integral operations in the performance index functional to obtain a pointwise problem. Then, a Lyapunov technique, used to find reachable sets of uncertain systems, is applied to find the pointwise maximum values. The method serves as a measure of performance robustness or a measure of sensitivity to uncertainties. The problem is analytically solved for first-order systems. Finally, examples concerning first- and second-order systems are given as applications of the technique.