Further remarks on strict input-to-state stable Lyapunov functions for time-varying systems

We study the stability properties of a class of time-varying nonlinear systems. We assume that nonstrict input-to-state stable (ISS) Lyapunov functions for our systems are given. We posit a mild nondegeneracy condition on our given Lyapunov functions under which strict ISS Lyapunov functions for our systems are also known to exist. However, the known existence result is based on value function representations and is therefore nonconstructive. We provide simple direct constructions of explicit strict ISS Lyapunov functions for our systems by applying an integral smoothing method. We illustrate our constructions using a tracking problem for a rotating rigid body.