On Correcting Customary Misuses of Limits and Derivatives and Eliminating Their Negative Impact on Nonlinear Systems Stability Analysis

摘要

Although Lyapunov approach has established itself as the most commonly used method for stability analysis, the realization that, in most applications, the Lyapunov derivative is at most negative semidefinite and not negative definite as desired seems to have forced stability analysis to become very complex and to require tough prior conditions. In particular, many developers work very hard to guarantee the strong continuity conditions of practically all signals involved. Even more important, no matter how hard the prior conditions are, conclusions of the analysis do not seem to guarantee more than simple stability and so, seem to be less significant than initially thought. Therefore, publications require devising a few Lyapunov functions for same system and using their combination in order to facilitate stability analysis. However, continuing along the lines of milder conditions first introduced by LaSalle, recent publications not only considerably mitigate those prior conditions and simplify stability analysis, yet also allow definitive conclusions about the ultimate behavior of the system trajectories. Nevertheless, counterexamples are continuously used to support the assumed need for continuity in the context of stability. A thorough review of those assumed counterexamples shows that they may be misusing basic limit and derivative rules and applying them where they do not belong. Eliminating those tough yet only apparently necessary prior conditions, one ends with a new Theorem of Stability, which only requires the rather trivial condition that bounded trajectories cannot pass an infinite distance in finite time and is simply formulated as a direct extension of Lyapunov Theorem for those cases when the derivative of the Lyapunov function is at most negative semidefinite.