On asymptotic stability properties of direct control systems with a feedback characteristic lying partly outside the Hurwitz sector

In his original work,Popov developped a simple frequency domain criterion by which global asymptotic stability of a nonlinear control system is guaranteed for all nonlinearities lying inside a certain sector called the Popov sector. Following this result, particular attention has been devoted in the scientific literature to the cases in which the Popov sector differs from the Hurwitz sector, this with a view to establishing additional conditions on the nonlinearity that may permit the so-called Aizerman conjecture to be verified. In the present work, second order nonlinear control problems, with finite Popov sector, are investigated with respect to asymptotic stability of the origin for nonlinear feedback characteristics lying partly outside the Popov-Hurwitz sector, which seems to be a new viewpoint. A new kind of stability ,near asymptotic stability is introduced and studied for second order control systems with a piecewise linear feedback characteristic and is extended to systems with a general feedback characteristic of the type mentioned above. A theorem is given which permits to predict near asymptotic stability for such second order systems. All the studies carried out in this work are obtained through the use of a state space approach and their validity is illustrated both by analog and digital computer solutions