Slowly Time-Varying Systems

Abstract

In Chap. 10, we discussed methods for building explicit global strict Lyapunov functions for rapidly time-varying systems. We turn next to the complementary problem of explicitly constructing strict Lyapunov functions for slowly time-varying continuous time systems. As in the case of rapidly time-varying systems, slowly time-varying systems involve two continuous time scales, one faster than the other. However, the methods for constructing strict Lyapunov functions for rapidly time-varying systems do not lend themselves to slowly time-varying systems, so our techniques in this chapter are completely different from the ones in Chap. 10. Instead of using limiting dynamics or averaging, we use a “frozen dynamics” approach, whereby Lyapunov functions for the corresponding frozen dynamics are used to build strict Lyapunov functions for the original slowly time-varying dynamics. We illustrate our results using friction and pendulum models.