Supremum of sampling period for planar engagement in the sense of closed-loop system sampling

In nonsmooth analysis, a well known result is that every asymptotically controllable system can be stabilized by means of some (discontinuous) feedback law in the sense of closed-loop system sampling (also referred to as sample-and-hold). The key points rely on the partition of interval and the construction of the feedback law based on the partition by pointwise optimization. Regarding the latter, it is complex because it is not required to have any particular regularity properties when talking about the form of the feedback. Regarding the former, it can be calculated in a mathematical point of view. We focus on the former in this paper. In planar engagement problem, it is a time-varying system and the control law must be discontinuous and piecewise fixed constant because of the technical difficulties. Omitting the latter difficulty about construction of the feedback, i.e. without thinking about the form of the (possibly discontinuous) feedback, we can calculate the supremum of sampling period of planar engagement problem by the given initial conditions.