A necessary and sufficient stability theorem and its applications

A new-type necessary and sufficient stability theorem for general retarded dynamical systems together with new stability analysis techniques are presented based on the author´s previous works in the literature. Linear systems with multiple time-varying delays and linear uncertain systems with one time-varying bounded parameter are considered to show the applications of the new stability theorem and the new stability analysis techniques. The delay-independent and delay-dependent stability criteria are established. Unlike some results in the literature, all of the established results do not depend on the derivatives of the time-varying delays and the time-varying parameter. Therefore, the proposed methods are suitable for the cases with very fast time-varying delays and parameter.