Systems Satisfying the Conditions of LaSalle

Abstract

The LaSalle Invariance Principle uses non-strict Lyapunov functions to show asymptotic stability. However, even when a system is known to be asymptotically stable, it is still desirable to be able to construct a strict Lyapunov function for the system, e.g., for robustness analysis and feedback design. In this chapter, we give two more methods for constructing strict Lyapunov functions, which apply to cases where asymptotic stability is already known from the LaSalle Invariance Principle. The first imposes simple algebraic conditions on the higher order Lie derivatives of the non-strict Lyapunov functions, in the directions of the vector fields that define the systems. Our second method uses our continuous time Matrosov Theorem from Chap. 3. We illustrate our approach by constructing a strict Lyapunov function for an appropriate error dynamics involving the Lotka-Volterra Predator-Prey System.