Constructions of Strict Lyapunov Functions for Discrete Time and Hybrid   Time-Varying Systems
  We provide explicit closed form expressions for strict Lyapunov functions for time-varying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete time analog of our previous continuous time Lyapunov function constructions. We also construct explicit strict Lyapunov functions for systems satisfying nonstrict discrete time analogs of the conditions from Matrosov's Theorem. We use our methods to build strict Lyapunov functions for time-varying hybrid systems that contain mixtures of continuous and discrete time evolutions. 
Michael Malisoff and Frederic Mazenc