We consider the simultaneous slewing and vibration suppression control problem of an idealized structural model which has a rigid hub with two cantilevered flexible appendages and finite tip masses. The finite clement method(FEM) is used to obtain linear finite dimensional equations of motion for the model. In the linear quadratic regulator(LQR) problem, a simple method is introduced to provide a physically meaningful performance index for space structure models. This method gives us a mathematically minor but physically important modification of the usual energy type performance index. A numerical procedure to solve a time-variant LQR problem with inequality control constraints is presented using the method of particular solutions.