  This paper gives further insights about the Lyapunov-Krasovskii
characterization of input-tostate stability (ISS) for switching retarded
systems on the basis of the results in [I. Haidar and P. Pepe.
Lyapunov-krasovskii characterization of the input-to-state stability for
switching retarded systems. SIAM Journal on Control and Optimization,
59(4):2997-3016, 2021]. We give new characterizations of the ISS property
through the existence of a relaxed common Lyapunov-Krasovskii functional. More
precisely, we show that the existence of a continuous Lyapunov-Krasovskii
functional whose upper right-hand Dini derivative satisfies a dissipation
inequality almost everywhere is necessary and sufficient for the ISS of
switching retarded systems with measurable inputs and measurable switching
signals. Different characterization results, using different derivative
notions, are also given.
