The synthesis control problem for the plane motion of a wheeled robot with constrained control resource is studied. The goal of the control is to bring the robot to an assigned curvilinear trajectory and to stabilize its motion along it. For a synthesized control law, the problem of finding the best in the sense of volume ellipsoidal approximation of the attraction domain of the target path is posed. To take into account constraints on the control, an approach based on methods of absolute stability theory is used, in the framework of which construction of an approximating ellipsoid reduces to solving a system of linear matrix inequalities. It is shown that the desired maximum-volume approximating ellipsoid can be found by solving a standard constrained optimization problem for a function of two variables.