Modification of Barbalat's Lemma

Abstract

Stability analysis of nonlinear systems has seen many successive developments after the first Lyapunov works. While before Lyapunov, one would have practically had to solve the differential equation and try to analyze
the behavior of each one of trajectories for the various initial conditions, etc., Lyapunov's methodology required to fit to the system an appropriate function, say positive definite and to test whether its derivative “along the trajectories of the system”  is negative definite. Because, except for simple class-room examples, finding such a function with a negative definite derivative was found to look like mission-impossible, new developments tried to further develop
the Lyapunov stability analysis for those practical cases when the derivative is at most negative semidefinite. Here, some commonly accepted counterexamples seem to show that even if a function ends with a constant limit,
its derivative may keep moving up-and-down forever, without reaching any limit. Then, a mathematical result known as Barbalat Lemma was very important at the time, as it allowed for stability analysis to end with useful
conclusions, yet it required tough continuity conditions from the system under analysis. Because those condition may be hard to satisfy in real world systems, we decided to review the existent concepts and the counterexamples,
and this review seems to show that some errors were made while taking functions to limit as time tends to infinity. These results allowed simplifying stability analysis and here we show that the Barbalat Lemma itself can be simplified
and that the tough continuity conditions can be eliminated.