For the problem of control of the plane motion of a wheeled robot the estimate of the attraction domain is set up, which ensures the prescribed exponential speed of asymptotic stability and the absence of overcontrol. This statement of the problem is the generalization of the problems “on the attraction domain,” “on overcontrol,” and “on monotone damping,” which were stated by A.M. Letov. To define the attractive domain estimate, periodic trajectories are considered, which exist at the boundary of the stability domain of two-dimensional linear nonstationary systems. Periodic solutions have a finite number of switchings over the period and describe the attraction domain boundary satisfying the preset geometric constraints.