Some equivalences of optimal time control problems dominated by impulsive ordinary differential equation

This paper mainly talks about some equivalences of the optimal time control problems dominated by the impulsive ordinary differential equation. As the pulse has equipped with the quality of instantaneous effect, so, there is no fundamental difference between integral optimal control problems dominated by impulsive differential system and corresponding differential equation, which is also dominated by integral optimal control problems. And this is well proved by current studies, thus the instantaneous effect has been averaged. However, in terms of the optimal time control problems, the pulse has definite impacts on the optimal time because the optimal time is instantaneous value. Thus, one of the purposes, to study optimal time control problems dominated by the impulsive differential system, is to discover the influences of the pulse puts on the default properties of differential system from the aspect of optimal time control. In this paper, we offered a method to define optimal time and made a further study on the equivalence relation among optimal time control problems, optimal norm control problems and optimal control problems. During this process, we had finished some relevant targets which mentioned in this paper.