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.