Asymptotic Stability of Non-Autonomous Systems and a Generalization of Levinson’s Theorem

Abstract

We study the asymptotic stability of non-autonomous linear systems with time dependent coefficient matrices { A ( t ) } t ∈ R . The classical theorem of Levinson has been an indispensable tool for the study of the asymptotic stability of non-autonomous linear systems. Contrary to constant coefficient system, having all eigenvalues in the left half complex plane does not imply asymptotic stability of the zero solution. Levinson’s theorem assumes that the coefficient matrix is a suitable perturbation of the diagonal matrix. Our objective is to prove a theorem similar to Levinson’s Theorem when the family of matrices merely admits an upper triangular factorization. In fact, in the presence of defective eigenvalues, Levinson’s Theorem does not apply. In our paper, we first investigate the asymptotic behavior of upper triangular systems and use the fixed point theory to draw a few conclusions. Unless stated otherwise, we aim to understand asymptotic behavior dimension by dimension, working with upper triangular with internal blocks adds flexibility to the analysis.

New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

“…The stability of nonautonomous perturbed dynamical systems has attracted the attention of many researchers and has produced several important results. [1][2][3][4][5] It turns out that time-varying differential equations appear as a natural description of observed evolution phenomena of various real-world problems where the study of asymptotic stability is more interesting than stability. Indeed, the study of asymptotic stability of dynamical systems is one of the most important research area in system design.…”