  A fundamental task in mobile robotics is keeping an intelligent agent under
surveillance with an autonomous robot as it travels in the environment. This
work studies a version of that problem involving one of the most popular
vehicle platforms in robotics. In particular, we consider two identical Dubins
cars moving on a plane without obstacles. One of them plays as the pursuer, and
it is equipped with a limited field-of-view detection region modeled as a
semi-infinite cone with its apex at the pursuer's position. The pursuer aims to
maintain the other Dubins car, which plays as the evader, as much time as
possible inside its detection region. On the contrary, the evader wants to
escape as soon as possible. In this work, employing differential game theory,
we find the time-optimal motion strategies near the game's end. The analysis of
those trajectories reveals the existence of at least two singular surfaces: a
Transition Surface and an Evader's Universal Surface. We also found that the
barrier's standard construction produces a surface that partially lies outside
the playing space and fails to define a closed region, implying that an
additional procedure is required to determine all configurations where the
evader escapes.
