To the question about asymptotic stability of linear non-stationary systems

A. P. Zhabko ; S. K. Myshkov

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 2 (2015), pp. 166-175

Résumé

Some topics of asymptotic stability for the transposed system which differs from the initial linear non-stationary system because of transpose operation are studied. In the stationary case the operation of transposition leaves unchanged not only the eigenvalues, but also the elementary divisors. That remains for transposed systems the type of stability and even its basic characteristics. Unlike the stationary case, for time-varying linear systems the comparison dynamics of the original and transposed systems is more difficult and the same or different types of stability are possible. In the general case, the eigenvalues of time-varying matrix do not define the type of stability. Properties of linear systems are completely characterized by the fundamental matrix. For autonomous systems, this matrix is defined in analytical form. In the non-stationary case, it can be found numerically on a finite time interval with suitable accuracy, but it does not guarantee against erroneous conclusions about stability of the system. Therefore, obtaining information on the stability of the transposed system when the initial system dynamics is known, is very important. A brief overview of some theoretical results on and criteria for the asymptotic stability which is presented in the paper is used to study the relationship of the initial system and it's transposed system. A few examples illustrate the possible variants of the problem. Bibliogr. 9.

Détail

Keywords: linear non-stationary system, asymptotic stability, exponential stability, exponential instability, transposed system.