An approximation for a continuous datum search game with energy constraint

This paper deals with a datum search game, where a target reveals his position (datum point) at a certain time (datum time) and a pursuer begins the search for the target by distributing his searching effort some time later. The target might move in the diffusive fashion from the datum point to evade his pursuer. His motion is restricted by its continuity in a two-dimensional space and constraints on its energy and maximum speed. The pursuer distributes searching effort to detect the evader under constraints on the amount of effort. A payoff is assumed to be the summation of searching effort weighted by the probability distribution of the target. In the previous paper, we formulated the problem as a single-stage two-person zero-sum game on continuous space and continuous time and proposed an upper bound and a lower bound for the value of the game. This paper extends the result and proposes an approximation for the value, noting that a constant-speed motion is crucial for the target.