  We revisit the two-player planar target-defense game posed in [1], a special
class of pursuit-evasion games in which the pursuer (or defender) strives to
defend a stationary target area from the evader (or intruder) who desires to
reach it, if possible, or approach it as close as possible. In this paper, the
target area is assumed to be a compact and convex set. Unlike classical
two-player pursuit-evasion games, this game involves two subgames: a capture
game and an escape game. In the capture game, where capture is assured, the
evader attempts to minimize the distance between her final position and the
target area whereas the pursuer tries to maximize the same distance. In the
escape game, where capture is not guaranteed, the evader attempts to maximize
the distance between herself and the pursuer at the moment that she reaches the
target for the first time. Our solution approach is based on Isaacs classical
method in differential games. We first identify the barrier surface that
demarcates the state space of the game into two subspaces, each of which
corresponds to the two aforementioned subgames, by means of geometric
arguments. Thereafter, we derive the optimal strategies for the players in each
subspace. We show that, as long as the target area is compact and convex, the
value of the game in each subspace is always continuously differentiable, and
the proposed optimal strategies correspond to the unique saddle-point
state-feedback strategies for the players. We illustrate our proposed solutions
by means of numerical simulations.
