In this thesis, a pursuit-evasion game, in which the pursuer moves with simple motion whereas the evader moves at a fixed speed but with a curvature constraint, is investigated. The game is the inverse of the usual homicidal chauffeur game. Square of the distance between the pursuer and the evader when the game is terminated is selected as the cost function. To solve such a zero-sum game, the variational approach will be employed to solve the problem. An algorithm will be proposed to determine a saddle point and the value of the game under consideration