Classical converse theorems in Lyapunov's second method

Abstract

Lyapunov's second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems. This technique makes use of an auxiliary function, called a Lyapunov function, to ascertain stability properties for a specific system without the need to generate system solutions. An important question is the converse or reversability of Lyapunov's second method; i.e., given a specific stability property does there exist an appropriate Lyapunov function? We survey some of the available answers to this question.2010 Mathematics Subject Classification. Primary: 93D05, 93D30, 93D20; Secondary: 93D10.

Existence of non-coercive Lyapunov functions is equivalent to integral uniform global asymptotic stability

“…Furthermore, we give a characterization of uniform global asymptotic stability in terms of the integral stability properties and analyze which stability properties can be ensured by the existence of a non-coercive Lyapunov function, provided either the flow has a kind of uniform continuity near the equilibrium or the system is robustly forward complete.Keywords nonlinear control systems · infinite-dimensional systems · Lyapunov methods · global asymptotic stability 1 IntroductionThe theory of Lyapunov functions is one of the cornerstones in the dynamical and control systems theory. Numerous applications of Lyapunov theory include characterization of stability properties of fixed points and more complex attractors [28,5,14,11], conditions for forward completeness of trajectories [1], criteria for the existence Andrii Mironchenko…”

“…The theory of Lyapunov functions is one of the cornerstones in the dynamical and control systems theory. Numerous applications of Lyapunov theory include characterization of stability properties of fixed points and more complex attractors [28,5,14,11], conditions for forward completeness of trajectories [1], criteria for the existence Andrii Mironchenko…”

Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach

“…In other works, 32,[34][35][36] some GFs have been provided as LFs for the differential inclusion associated to (6), for example, the GF V ∶ ℝ 3 → ℝ given by V(e) = 1 |e 1 | 5 3 − 12 e 1 e 2 + 2 |e 2 | 5 2 − 23 e 2 e 3 3 + 3 |e 3 | 5 .…”

A Lyapunov Function Construction for a Non-convex Douglas–Rachford Iteration

“…The other reason is the added smoothness (compared to simply using the distance) at the reference point, which is beneficial in robustness analysis (especially in continuoustime systems, cf. [22]) and control design (see, e.g., [28]).…”

“…provided a sufficient condition on the angles θ 1 and θ 2 is met. It is common in Lyapunov stability analysis that conditions are only sufficient and not necessary (see [22] on the concept of converse Lyapunov functions; their existence proofs are usually non-constructive). This global Lyapunov function in turn is a certificate for the global asymptotic stability of the set {p 1 , p 2 } of fixed points for the iterative Figure 3.…”