In this article, we address the question of relating the stability properties of an operator with the stability properties of its associate symmetric operator. The linear-algebra results of Bendixson and Hirsch indicate that the symmetric part of a matrix is always less stable than the matrix itself. We show that in a variety of cases, including infinite dimensional cases associated to systems of PDEs, the same result is valid. We also discuss the applicability to non-autonomous systems, and we show that, in general, this result is not valid. We also review some of the literature that in these years has appeared on the subject.

http://paduaresearch.cab.unipd.it/5064/1/2012.AAM.pdf

Does Symmetry of the Operator of a Dynamical System Help Stability

In this paper a new general method is developed by means of which one can ascertain whether a nonlinear autonomous system is asymptotically stable. The method is essentially an extension to nonlinear systems of a theorem developed earlier by the first author for linear autonomous systems. Necessary and sufficient conditions are specified, the satisfaction of which guarantees that the system being studied is asymptotically stable. The new method, by design, always uses a positive-definite function which satisfies Lyapunov?s stability theorem. However, the new method uses only one positive-definite function, in contrast to Lyapunov?s stability theorem, which requires two functions to be definite at the same time. In addition, the new method specifies the stability function at the outset once a mathematical system model has been obtained.

The asymptotic stability of nonlinear autonomous systems

In this paper a new method is developed for obtaining necessary and sufficient conditions which, when satisfied, guarantee that a linear non-autonomous system will be uniformly asymptotically stable. The method is essentially an extension of two earlier theorems on the asymptotic stability of linear autonomous systems; these theorems, developed by the same author, are here extended to non-autonomous systems. Basically, the method consists of ascertaining whether a certain matrix is positive definite. In general, it is shown that this knowledge is necessary and sufficient to determine whether a linear non-autonomous system is exponentially stable and hence uniformly asymptotically stable.

Uniform asymptotic stability of linear non-autonomous systems

This thesis introduces the notion of using stability analysis in terms of two measures for impulsive switched systems. Impulsive switched systems are defined in the context of hybrid system theory and the motivation for the study of these systems is presented. The motivation for studying stability in two measures is also given, along with the definitions of stability, uniform stability, and uniform asymptotic stability in one and two measures. The results presented are a sets of sufficient stability criteria for linear and nonlinear systems. For autonomous linear systems, there are criteria for stability and asymptotic stability using a particular family of choices for the two measures. There is an additional stronger set of criteria for asymptotic stability using one measure, for comparison. There is also a proposed method for finding the asymptotic stability of a non-autonomous system in one measure. The method for extending these criteria to linearized systems is also presented, along with stability criteria for such systems. The criteria for nonlinear systems cover stability, uniform stability, and uniform asymptotic stability, considering state-based and time-based switching rules in different ways. The sufficient stability criteria that were found were used to solve four instructive examples. These examples show how the criteria are applied, how they compare, and what the shortcomings are in certain situations. It was found that the method of using two measures produced stricter stability requirements than a similar method for one measure. It was still found to be a useful result that could be applied to the stability analysis of an actual impulsive switched system.

Stability of Impulsive Switched Systems in Two Measures

Uniform asymptotic stability of linear non-autonomous systems Stabilit´ e asymptotique uniforme des syst` emes lin´ eaires non autonomes

The method introduced here determines explicitly whether a given system has only eigenvalues with negative real parts, without determining these eigenvalues, but rather by determining the eigenvalues of a related symmetric matrix which are always real and hence easy to calculate. The new method is similarly applied to linear autonomous discrete systems where all the eigenvalues of the system matrix are required to be positioned within the unit circle in the z-plane.

G.S. Christensen

Papers

A simple and fast algorithm for determining asymptotic stability of linear autonomous systems