The new theorem of stability - Direct extension of Lyapunov theorem

Abstract

Even though Lyapunov approach is the most commonly used method for stability analysis, its use has been hindered by the realization that in most applications the so-called Lyapunov derivative is at most negative semidefinite and not negative definite as desired. Many different approaches have been used in an attempt to overcome these difficulties. Recently, stability analysis methods for
nonautonomous nonlinear systems have been revisited. New analysis showed that the restrictive condition of uniform continuity required by Barbalat Lemma and even the milder conditions required by
LaSalle’s extension of the Invariance Principle to nonautonomous systems can be further mitigated. A new Invariance Principle only required that bounded trajectories cannot pass an infinite distance
in finite time. New analysis shows that even this relaxed condition is not required and that a new Theorem of Stability can be formulated as a direct extension of Lyapunov Theorem for those cases
when the derivative of the Lyapunov function is negative semidefinite.