On the spectral set of a~linear discrete system with stable Lyapunov exponents

I. N. Banshchikova ; S. N. Popova

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 26 (2016) no. 1, pp. 15-26

Résumé

Let us fix a certain class of perturbations of the coefficient matrix $A(\cdot)$ for a discrete time-varying linear system $$ x(m+1)=A(m)x(m),\quad m\in\mathbb Z,\quad x\in\mathbb R^n, $$ where $A(\cdot)$ is completely bounded on $\mathbb Z$, i.e., $\sup_{m\in\mathbb Z}(\|A(m)\|+\|A^{-1}(m)\|)\infty$. The spectral set of this system, corresponding to a given class of perturbations, is a collection of all Lyapunov spectra (with multiplicities) for perturbed systems, when the perturbations range over this class all. The main attention is paid to the class $\mathcal R$ of perturbed systems $$ y(m+1)=A(m)R(m)y(m),\quad m\in\mathbb Z,\quad y\in\mathbb R^n, $$ where $R(\cdot)$ is completely bounded on $\mathbb Z$, as well as its subclasses $\mathcal R_\delta$, where $\sup_{m\in\mathbb Z}\|R(m)-E\|\delta$, $\delta>0$. For an original system with stable Lyapunov exponents, we prove that the spectral set $\lambda(\mathcal R)$ of class $\mathcal R$ coincides with the set of all ordered ascending sets of $n$ numbers. Moreover, for any $\Delta> 0$ there exists an $\ell=\ell(\Delta)>0$ such that for any $\delta\Delta$ the spectral set $\lambda(\mathcal R_{\ell\delta})$ contains the $\delta$-neighborhood of the Lyapunov spectrum of the unperturbed system.

Détail

Keywords: discrete time-varying linear system, lyapunov exponents
Mots-clés : perturbations of coefficients.