Krasovskii functional, Razumikhin function and backstepping

Recently, a renewal interesting is focused on time-delay systems, particularly nonlinear time-delay systems. Krasovskii functional or Razumikhin function based methods have been investigated widely, since the time-delay system is described by the functional differential equation, and the stability analyzing tools are provided early with the functional or function as a parallel way with the Lyapunov stability theory. In the spirit of Lyapunov function based design method, some interesting results on backstepping design for the time-delay systems have been given, however, this is an issue that easily leads a misunderstanding. This paper would show that the problem to construct a Krasovskii functional or Razumikhin function is not a trivial problem.