  This paper studies a two-person linear-quadratic-Gaussian pursuit-evasion
differential game with costly but controlled information. One player can decide
when to observe the other player's state. However, one observation of another
player's state comes with two costs: the direct cost of observing and the
implicit cost of exposing his state. We call games of this type a
Pursuit-Evasion-Exposure-Concealment (PEEC) game. The PEEC game constitutes two
types of strategies: The control strategies and the observation strategies. We
fully characterize the Nash control strategies of the PEEC game using
techniques such as completing squares and the calculus of variations. We show
that the derivation of the Nash observation strategies and the Nash control
strategies can be decoupled. We develop a set of necessary conditions that
facilitate the numerical computation of the Nash observation strategies. We
show, in theory, that players with less maneuverability prefer concealment to
exposure. We also show that when the game's horizon goes to infinity, the Nash
observation strategy is to observe periodically, and the expected distance
between the pursuer and the evader goes to zero with a bounded second moment.
We conducted a series of numerical experiments to study the proposed PEEC game.
We illustrate the numerical results using both figures and animation. Numerical
results show that the pursuer can maintain high-grade performance even when the
number of observations is limited. We also show that an evader with low
maneuverability can still escape if the evader increases his stealthiness.
