An Invariance Principle in the Theory of Stability

Abstract¶

The purpose of this paper is to give a unified presentation of Liapunov's theory of stability that includes the classical Liapunov theorems on stability and instability as well as their more recent extensions. The idea being exploited here had its beginnings some time ago. It was, however, the use made of this idea by Yoshizawa in [1] in his study of nonautonomous differential equations and by Hale in [2] in his study of autonomous functional differential equations that caused the author to return to this subject and to adopt the general approach and point of view of this paper. This produces some new results for dynamical systems defined by ordinary differential equations which demonstrate the essential nature of a Liapunov function and which may be useful in applications. Of greater importance, however, is the possibility, as already indicated by Hale's results for functional differential equations, that these ideas can be extended to more general classes of dynamical systems. It is hoped, for instance, that it may be possible to do this for some special types of dynamical systems defined by partial differential equations. In section 2 we present some basic results for ordinary differential equations. Theorem 1 is a fundamental stability theorem for nonautonomous systems and is a modified version of Yoshizawa's Theorem 6 in [1]. A simple example shows that the conclusion of this theorem is the best possible. However, whenever the limit sets of solutions are known to have an invariance property then sharper results can be obtained. This "invariance principle" explains the title of this paper. It had its origin for autonomous and periodic systems in [3] - [5], although we present here improved versions of those results. Miller in [6] has established an invariance property for almost periodic systems and obtains thereby a similar stability theorem for almost periodic systems. Since little attention has been paid to theorems which make possible estimates of regions of attraction (regions of asymptotic stability) for nonautonomous systems results of this type are included. Section 3 is devoted to a brief discussion of some of Hale's recent results [2] for autonomous functional differential equations.

Summary¶

La Salle gives a unified Lyapunov-stability treatment centered on invariance properties of limit sets. The report is a key source for the LaSalle invariance principle, which strengthens Lyapunov methods by showing convergence to the largest invariant set where the Lyapunov derivative vanishes rather than merely proving boundedness or nonincrease.

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