Lyapunov-like functions for almost global convergence in discrete-time systems

This paper presents new results on discrete-time almost global convergence to an invariant set. First, we present new sufficient conditions for almost global convergence using density functions. In contrast to existing density-function results, the new results in this paper do not rely on a local convergence assumption, and they do not require knowledge of the inverse map of the difference equation. Next, we present new sufficient conditions for almost global convergence using Lyapunov-like functions rather than density functions. These Lyapunov-like results can be useful because constructing and analyzing density functions is often difficult in comparison to Lyapunov-like analysis. This paper also presents a variety of simple examples to illustrate these new methods for almost global convergence analysis.