Cost analysis with input extension in time optimal control

Due to the popularity of the input redundant systems, this paper focused on the problem of input extension which concerns about the effect of adding new input redundancy into a controllable system. According to the time optimal control problem, such effect is evaluated by the optimal time. Based on the assumption of the existence and uniqueness of the optimal control, this paper proves that the more input redundancy a system possesses the less time is needed to drive it from a same initial state to the origin point if at least one redundant input channel is not an idle channel. Moreover, the normality of the extended optimal control problem will guarantee that all of the input redundant channels will be used to short the optimal time. Finally, two numerical examples are deployed to demonstrate the main results in this paper.