This paper studies a time optimal control problem governed by a classical controlled system of nonlinear ordinary differential equations. Corresponding to certain controls, the solutions of the system have the behavior of blowup; namely, they blow up at a finite time. The goal for controlling the system is to minimize the blowup time. Therefore, the time optimal control problem studied in this paper holds its target set outside of its state space. From this perspective, it essentially differentiates from the most time optimal control problems in studies from past publications. The purpose of this study is to develop a new method for proving the existence of time optimal controls for such kinds of time optimal control problems through analyzing a typical problem. We also aim to create a new method for establishing the Pontryagin maximum principle of optimal controls. The main results derived from this paper can be extended to controlled systems, where nonlinear terms have a more general form, by the two methods mentioned above.