ISS characterization of retarded switching systems with relaxed Lyapunov–Krasovskii functionals

This paper gives further insights about the Lyapunov–Krasovskii characterization of input-to-state 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.