Computational methods for Lyapunov functions

Peter Giesl,Sigurdur Hafstein

doi:10.3934/dcdsb.2015.20.8i

Abstract

Lyapunov functions, introduced by Lyapunov more than 100 years ago, are to this day one of the most important tools in the stability analysis of dynamical systems. They are functions which decrease along solution trajectories of systems, and they can be used to show stability of an invariant set, such as an equilibrium, as well as to determine its basin of attraction. Lyapunov functions have been considered for a variety of dynamical systems, such as continuous-times, discrete-time, linear, non-linear, non-smooth, switched, etc. Lyapunov functions are used and studied in different communities, such as Mathematics, Informatics and Engineering, often using different notations and methods. For more information please click the “Full Text” above.

Highlights

In 1892 Lyapunov published his famous doctoral dissertation [158], where he introduced a sufficient condition for the stability of a nonlinear system, namely, the existence of a positive definite function decreasing along the solution trajectories
His work was motivated by problems in astronomy such as the stability of the motion of the planets, the concept of a Lyapunov function has turned out to be extremely fruitful in many other different contexts
We assume that x(t, t0; 0) = 0 is a solution and we study its stability by a Lyapunov function V (x, t)

Summary

Introduction

In 1892 Lyapunov published his famous doctoral dissertation [158], where he introduced a sufficient condition for the stability of a nonlinear system, namely, the existence of a positive definite function decreasing along the solution trajectories. When considering a finite time interval, stability can be studied by using Lagrangian coherent structures, see Haller 2000 [113], stable and unstable manifolds, see Berger, Doan, and Siegmund 2009 [27], or finite-time Lyapunov exponents, see Berger 2011 [26] Examples for such systems are non-autonomous differential and difference equations over infinite- and finite-time as well as random dynamical systems, see Section 2.6. As a simple example we can take the time-delayed differential equation with delay τ > 0 x = f (x(t − τ ), t), where the initial condition is given by specifying x on the interval [−τ, 0] The stability of such systems can be studied using Lyapunov-like functions, e.g. based on Razumikhin’s theorem or the Lyapunov-Krasovskii Theorem.

Ai attractor

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