On the effects of redundant control inputs

For a stabilizable system, the extension of the control inputs has no use for stabilizability, but it is important for optimal control. In this paper, the necessary and sufficient condition is discussed for the strict decrease of the quadratic optimal performance index with control input extensions. The initial state set is determined for the case of non-strict decrease. A similar result is also provided for discrete-time systems. These results show an essential difference between single-input and multi-input control systems. A satellite launch vehicle system is finally discussed for application.