On the Existence of Invariantly Differentiable Functionals for Globally Exponentially Stable Time-Delay Systems

This note argues over the existence of invariantly differentiable functionals for globally exponentially stable time-delay systems. We show that the existence of invariantly differentiable, Lipschitz on bounded sets Lyapunov-Krasovskii functional is a necessary and sufficient condition for the global exponential stability of time-invariant nonlinear systems described by retarded functional differential equations with discrete type delay. The presented converse results are in terms of the invariant derivative definition. To prove converse results, both pointwise and historywise dissipation conditions are provided. The validity of the results is illustrated by an example.