In optimal control design, two of the most popular strategies used are linear quadratic regulator (LQR) control and the H2 norm optimization. This paper obtains a mathematical equivalence between the performance index of an LQR control algorithm and the H2 norm of a linear system. For a single-input excitation such as an earthquake, the optimal gain for an LQR control algorithm is the same as that of an H2 norm minimization of the system. This observation leads to the inference that for a linear system, quadratic optimal gain for a free vibration of the system can also be used as an optimal gain for the forced vibration of the system. However, the same results are not true for multi-input excitations, such as wind or blast loads. In these cases, optimal gains must be obtained separately for the two algorithms. © 2016 American Society of Civil Engineers.