The paper considers a new stabilization problem for the Pendubot, how to construct and stabilize via feedback the following trajectory: the first (controlled) link of the Pendubot remains at rest having a given angle with horizontal, while the second (freely moving) link duplicates a given motion of the 1-d.o.f. plane pendulum. Such a stabilization could be also seen as re-distributing the energy of the Pendubot between actuated and non-actuated parts, or saving the energy in the motion of the non-actuated part. The main result of the paper suggests a wide family of the state feedback controllers, which solve the problem. In addition, the delicate issue of the convergence’s rate of the closed loop system solutions to the desired trajectory is discussed in details.