Optimal control theory is employed to analyze the transient response of discrete linear systems to forcing functions whose dependence on time is not known, but which lie within specified bounds. Attention is focused on the complete determination of the forcing function that will induce (a) maximum displacement of any specified mass element; (b) maximum relative displacement of any two adjacent masses; and (c) maximum acceleration of a given mass. Undamped mechanical systems with an arbitrary number of degrees of freedom are considered wherein only a single forcing function acts. In general, the desired forcing function is found to be a bang‐bang type function, that is, a function which switches from the upper to the lower bound and vice versa at certain instants of time. The general procedure for finding these switching times is presented.