A different game problem with two players (cars), in which one player (car) pursues the other, is considered. The roles of the players are fixed, and the functional to be minimized (for player I) and maximized (for player II) is the maximum value of a given scalar non-negative function (the performance index) of the phase vector along the trajectory of the dynamical system over a fairly long time interval. A zero value of the performance index corresponds to the situation in which the pursuer is behind the evader at a given distance from it, and the velocity vectors are codirectional and lie on the same straight line. A detailed investigation is presented of the special case in which the car being pursued is at rest, and the pursuer is moving in the plane at a velocity of constant magnitude subject to a certain constraint on its turning radius. The game ends when the car is moving in a circle of given radius, in which case its velocity vector must point toward the centre of the circle. The relations of the Pontryagin maximum principle characterizing optimal open-loop controls are written out and analysed. The main result of the paper is the synthesis of an optimal feedback control.