  In this paper, we study 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. But one observation
of another player's state comes with two costs: the direct cost of observing
and the implicit cost of exposing his/her 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. We conduct 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/her stealthiness.
