Abstract: The problems of stability in the whole, stability in the large with its attracting region, and the quality problem of nonlinear control systems are studied by the Lyapunov’s method. The functional relation of the nonlinear element is assumed to be piece-wise linear, continuous or not, and may possess two-valued intervals, but is linear at the origin. Thus the whole phase space can be divided into three simple regions in each of which the differential equations of the system are essentially linear. As a result the stability conditions contain also parameters of the nonlinearities.