LaSalle Invariance Principle for Discrete-time Dynamical Systems: A Concise and Self-contained Tutorial

As a method to establish the Lyapunov stability of differential/difference equations systems, the LaSalle invariance principle was originally proposed in the 1950’s and has become a fundamental mathematical tool in the area of dynamical systems and control. In both theoretical research and engineering practice, discrete-time dynamical systems have been at least as extensively studied as continuous-time systems. For example, model predictive control is typically studied in discrete-time via Lyapunov methods. However, there is a peculiar absence in the standard literature of standard treatments of Lyapunov functions and LaSalle invariance principle for discrete-time nonlinear systems. Most of the textbooks on nonlinear dynamical systems focus only on continuous-time systems. For example, the classic textbook by Khalil on nonlinear systems [10] relegates discrete-time systems to a few exercises at the end of Chapter 4. The textbook by Vidyasagar [14] does not present the discrete-time LaSalle invariance principle. In Chapter 1 of the book by LaSalle [11], the author establishes the LaSalle invariance principle for difference equation systems. However, all the useful lemmas in [11] are given in the form of exercises with no proof provided. In this document, we provide the proofs of all the lemmas proposed in [11] that are needed to derive the main theorem on the LaSalle invariance principle for discrete-time dynamical systems. We organize all the materials in a self-contained manner. We first introduce some basic concepts and definitions in Section 1, such as dynamical systems, invariant sets, and limit sets. In Section 2 we present and prove some useful lemmas on the properties of invariant sets and limit sets. Finally, we establish the original LaSalle invariance principle for discrete-time dynamical systems and a simple extension in Section 3. In Section 4, we provide some references on extensions of LaSalle invariance principles for further reading. This document is intended for educational and tutorial purposes and contains lemmas that might be useful as a reference for researchers.