Strictification: Basic Results

Abstract

In the last three chapters, we gave general methods for constructing strict Lyapunov functions for time-invariant systems. Several of these methods have analogs for time-varying systems. In general, these involve replacing the negative semi-definite function of the state in the right side of the non-strict Lyapunov decay condition with a product of a negative semi-definite function of the state and a suitable time-varying parameter. We assume that the time-varying parameter satisfies a persistency of excitation (PE) condition. The challenge is then to transform non-strict Lyapunov functions satisfying this more complicated decay condition into explicit strict Lyapunov functions. In this chapter, we provide methods for solving this and related problems, including the construction of ISS Lyapunov functions for time-varying systems. We apply our work to stabilization problems for rotating rigid bodies and underactuated ships.