Results on Lyapunov-Like Functions for Almost Global Convergence in Discrete-Time Systems

This paper presents new results on almost global convergence to an invariant set for discrete-time dynamic systems. The evolution of a discrete-time dynamic system is governed by the function on the right-hand side of the state-variable difference equation. Existing results on almost global convergence in discrete time require that this function is nonsingular with respect to a measure µ. However, for many dynamic systems, this function is either not nonsingular with respect to µ, or its nonsingularity is difficult to examine analytically. This paper presents new results (specifically, sufficient conditions) for demonstrating almost global convergence in discrete-time systems. Notably, these results do not require nonsingularity of the right-hand side of the state-variable difference equation with respect to µ. The new results are developed using density functions, which can be viewed as a dual to Lyapunov functions. However, constructing density functions can be difficult— particularly in the case where the right-hand-side function is not nonsingular with respect to µ. To overcome this difficulty, this paper also presents sufficient conditions for almost global convergence using Lyapunov-like functions rather than density functions. Finally, we demonstrate these new results using several simple examples.