The purpose of the present paper is to study the robustness of the optimality of an optimal control for a linear single-input system with respect to a quadratic cost. The robust optimality of an optimal control is defined as its invariant optimality in the usual sense of minimizing some quadratic cost under the presence of small plant parameter variations. A necessary and sufficient condition for robust optimality of a given optimal control is derived first. Based on this result, secondly, a choice of the weighting matrix is proposed such that the resulting optimal control is robustly optimal. The former condition expressed in terms of a given optimal control law can also be expressed as some constraints on the Nyquist plot of the loop transfer function for the optimal regulator. Similarly, the latter condition on the choice of weighting matrices that yield robustly optimal controls is also expressed as some constraints on the transfer function from the input to the controlled variables associated with the weighting matrix to be chosen.