Barbalat’s lemma and stability – Misuse of a correct mathematical result?

Abstract

Even though Lyapunov methodology and the use of Lyapunov functions are the basis for almost any stability analysis of nonlinear systems, the use of Lyapunov Stability theorem is limited by the fact that the derivative of the Lyapunov function is required to be negative definite, while in most cases it is hardly negative semidefinite. Because first extensions to the semidefinite case only covered autonomous or periodic systems, Barbalat’s Lemma has been adopted from the theory of functions and applied to stability analysis. Although new works of LaSalle and followers have extended the Invariance Principle to nonautonomous systems, they have remained largely unknown and so, Barbalat’s Lemma has remained the main stability analysis tool, in spite of the fact that it either requires all signals to be uniformly continuous or requires finding solutions that could prevent the assumed negative effect of discontinuities. This paper shows that the apparent need for continuity might have been the result of a pure misuse of Barbalat’s Lemma.