  Stability of retarded differential equations is closely related to the
existence of Lyapunov-Krasovskii functionals. Even if a number of converse
results have been reported regarding the existence of such functionals, there
is a lack of constructive methods for their selection. For certain classes of
time-delay systems for which such constructive methods are lacking, it was
shown that Lyapunov-Krasovskii functionals that are also allowed to depend on
the time-derivative of the state-trajectory are efficient tools for the study
of the stability properties. However, in such an approach the initial condition
needs to be assumed absolutely continuous with a square integrable weak
derivative. In addition, the stability results hold for initial conditions that
are evaluated based on the magnitude of both the initial condition and its
time-derivative. The main objective of this paper is to show that, for certain
classes of time-delay systems, the aforementioned stability results can
actually be extended to initial conditions that are only assumed continuous and
that are evaluated in uniform norm.
