Further Results on Strict Lyapunov Functions for Rapidly Time-Varying   Nonlinear Systems
  We explicitly construct global strict Lyapunov functions for rapidly time-varying nonlinear control systems. The Lyapunov functions we construct are expressed in terms of oftentimes more readily available Lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable. This leads to new sufficient conditions for uniform global exponential, uniform global asymptotic, and input-to-state stability of fast time-varying dynamics. We also construct strict Lyapunov functions for our systems using a strictification approach. We illustrate our results using a friction control example. 
Frederic Mazenc, Michael Malisoff, and Marcio S. de Queiroz