Families of Liapunov-Krasovskiį functionals and stability for functional differential equations

Abstract

Among the methods for studying stability and related concepts for ordinary differential equations, Liapunov's second (or direct) method has become the most widely used. It is characterized by the use of certain (originally real-valued) auxiliary functions, now usually called Liapunov functions. Because of the difficulties in constructing suitable Liapunov functions, many authors have investigated various kinds of variations of Liapunov's second method, and new approaches to stability problems have emerged. One of the important approaches is that of considering finite or infinite families of continuous or discontinuous Liapunov functions. Such an approach leads to the generalization and improvement of existing results and proof, as well as finding new extensions and opening up of new fields (cf. [1,2], and the literature there).(*) Entrata in Redazione il 25 settembre 1997.

On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals

“…). These properties were used by other authors [1,2,8,14,15,16,18,19,23,24,20,25,26]. Parts B in Theorems 2.2 and 2.3 show that these conditions can be essentially weakened if the Lyapunov functional is decrescent in Krasovskiȋ's sense.…”