In this paper, the strong stabilization of the Acrobot at the Down-Up equilibrium point is addressed, with the first link being downward and the second link being upright. By converting the strong stabilization of the Acrobot at the Down–Up equilibrium point equivalently to the existence and design of a stable stabilizing controller for a fourth-order single-input single-output linear plant with adjustable zeros, a pair of poles on the imaginary axis, and a pair of real poles located symmetrically with respect to the origin, this paper has three main contributions. Firstly, the existence of a stable stabilizing controller for any Acrobot which is linearly controllable at the Down–Up equilibrium point is proved by showing the range of the adjustable zero via adjusting a parameter of the output signal. Secondly, a necessary and sufficient condition on the mechanical parameters of the Acrobot is provided to guarantee the existence of a second-order stable stabilizing controller for the Acrobot around its Down– Up equilibrium point. Thirdly, a direct method is presented to design a second-order stable controller, whose transfer function is preset with three parameters. By utilizing the Liénard– Chipart criterion for a fourth-order polynomial, the necessary and sufficient conditions on these parameters for achieving the strong stabilization are obtained, which are expressed in a cascade form for obtaining these parameters conveniently. A numerical example is presented to validate the effectiveness of the proposed method.