On the problem of Aizerman

The well known problem stated by M. A. Aizerman several decades ago consists of the comparison of the linear stability (Hurwitz) and absolute stability sectors in systems with sector restricted nonlinearities. This problem remains a test of the sharpness (“degree of conservatism”) of various sufficient absolute stability criteria including frequency domain inequalities and Liapunov function(al)s. The present paper contains some analysis of various case of Aizerman problems in systems with lumped parameters and with time delays. In particular the Aizerman problem gets a positive answer in the application of the gyroscopic pendulum, where two nonlinearities are present.