  This paper concerns some time optimal control problems of three different
ordinary differential equations in $\mathbb{R}^2$. Corresponding to certain
initial data and controls, the solutions of the systems quench at finite time.
The goal to control the systems is to minimize the quenching time. To our best
knowledge, the study on quenching time optimal control problems has not been
touched upon. The purpose of this study is to obtain the existence and the
Pontryagin maximum principles of optimal controls. We hope that our methods
could hint people to study the same problems with more general vector fields in
$\mathbb{R}^d$ with $d\in \mathbb{N}$. We also wish that our results could be
extended to the same issue for parabolic equations.
