Can stability analysis be really simplified? (From Lyapunov to the new theorem of stability - Revisiting Lyapunov, Barbalat, LaSalle and all that)

Abstract

A review of most relevant publications seems to show that the nonlinear systems stability analysis is very complex and with less then satisfactory conclusions. 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 not negative definite as desired. Many different approaches have been used in an attempt to overcome these difficulties. One of them is the vectorial Lyapunov function, where the use of a few Lyapunov functions and of their derivative and their combined conclusions might end in more conclusive results.  Still, other approaches have been developed attempting to get more than mere stability for the case when the single Lyapunov derivative is negative semidefinite. Many apparent limitations and the apparent need for tough prior conditions make the stability analysis in this case both complex and difficult to satisfy. While some methodologies seem to be limited to autonomous systems only, the most widely accepted stability analysis for non-autonomous systems has been based on Barbalat's Lemma which seems to require uniform continuity of practically all signals involved. Recently, stability analysis methods for nonautonomous nonlinear systems have been revisited. Nevertheless, although recent developments based on unknown works of LaSalle attempted to mitigate these continuity conditions, counterexamples are continuously suggested to contradict these results. New analysis shows that these counterexamples, which are making use of well-known mathematical expressions, are actually using them beyond their  domain of validity. Therefore, the restrictive condition of uniform continuity required by Barbalat's Lemma and even the milder conditions required by LaSalle's extension of the Invariance Principle to nonautonomous systems can be further mitigated. Moreover, no matter what the prior conditions are, either hard or mild, and in spite of the great efforts invested towards more  satisfactory conclusions, a deeper analysis seems to show that the final conclusion does not guarantee anything more than mere stability, which was already there in Lyapunov's original Theorem for the semidefinite derivative case. Therefore, even when the Lyapunov derivative is negative semidefinite, the Matrosov school of stability approach requires using a few Lyapunov functions. However, a new Invariance Principle, which only required that bounded trajectories cannot pass an infinite distance in finite time first showed how one can obtain more conclusive results. Finally, a new Theorem of Stability, which looks like a direct extension and a generalization of Lyapunov's Theorem, not only simplifies the stability analysis of nonlinear systems, but also leads to really conclusive results about the system under analysis.