Converse Lyapunov–Krasovskii theorems for systems described by neutral functional differential equations in Hale's form

Abstract

In this article, we show that the existence of a Lyapunov–Krasovskii functional is necessary and sufficient condition for the uniform global asymptotic stability and the global exponential stability (GES) of time-invariant systems described by neutral functional differential equations in Hale's form. It is assumed that the difference operator is linear and strongly stable, and that the map on the right-hand side of the equation is Lipschitz on bounded sets. A link between GES and input-to-state stability is also provided.

Keywords:

• converse Lyapunov theorem
• Lyapunov–Krasovskii functionals
• retarded functional differential equation
• neutral functional differential equation

Acknowledgement

The work of P. Pepe has been supported in part by the MIUR-PRIN 2009 grant N. 2009J7FWLX002 and by the Center of Excellence for Research DEWS, Italy.