In this paper, we address the design of a minimum variance controller (MVC) for the mitigation of vibrations in modern telescope adaptive optics (AO) systems. It is widely accepted that a main source of non-turbulent perturbations is the mechanical resonance induced by the wind or the instrumentation systems, such as fans and cooling pumps. To adequately mitigate vibrations, the application of frequency-based controllers has been considered in the past decade. In this work, we express the system model in terms of the tracking of a zero-input signal via the MVC. We show that the MVC is an equivalent representation of the linear quadratic Gaussian (LQG) controller for the AO system. We also show that by developing the MVC, we can obtain different expressions, in terms of transfer functions, that offer insights into the behavior and expected performance of the controller in the frequency domain. In addition, we analyze the impact of the accuracy of the system and perturbations model on the mitigation of vibrations.