Lyapunov-like Vector Functions Using Pointwise Degenerate Systems as Comparison Functions

Abstract

**Please note that the full text is embargoed** ABSTRACT: The use of Lyapunov-like vector functions is recognized as an important tool for estimating the behavior of a dynamical system. In applications, one needs to determine a suitable comparison function which contains information or properties that can be used to obtain some information about the behavior of the dynamical system. In recent years, the remarkable property of pointwise degeneracy has been discovered for linear, autonomous, delay-differential equations. For this class of equations one knows that all solutions will reach a subspace in finite time and remain on the subspace thereafter. As this class of equations is particularly simple, it becomes an obvious candidate for use as a comparison function. In this paper (Section 3), we show how one may effectively use pointwise degenerate systems as comparison functions. Section 4 treats functional differential equations which possess certain monotonic features. These monotonic properties do not appear to allow us to select pointwise degenerate systems as natural comparison functions.