On strict Lyapunov functions for rapidly time-varying nonlinear systems

We explicitly construct Lyapunov functions for rapidly time-varying nonlinear 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 globally asymptotically stable. This leads to new sufficient conditions for global exponential, global asymptotic, and input-to-state stability of fast time-varying dynamics. We apply our results to two examples