Conditional Stability and Asymptotic Behavior of Solutions of Weakly Delayed Linear Discrete Systems in R^2

Abstract

Two-dimensional linear discrete systems $$ x(k+1)=Ax(k)+\sum\limits_{l=1}^{n}B_{l}x_{l}(k-m_{l}),\,\,\,k\ge 0 $$are analyzed, where $m_{1}, m_{2},\dots, m_{n}$ are constant integer delays, $0<m_{1}<m_{2}<\dots<m_{n}$, $A$, $B_{1},\dots, B_{n}$ are constant $2\times 2$ matrices, $A=(a_{ij})$, $B_{l}=(b^l_{ij})$, $i,j=1,2$, $l=1,2,\dots,n$ and $x: \{-m_n,-m_n+1,\dots\}\to \mathbb{R}^2$. Under the assumption that the system is weakly delayed, the asymptotic behavior of its solutions is studied and asymptotic formulas are derived.Two-dimensional linear discrete systems $$ x(k+1)=Ax(k)+\sum\limits_{l=1}^{n}B_{l}x_{l}(k-m_{l}),\,\,\,k\ge 0 $$are analyzed, where $m_{1}, m_{2},\dots, m_{n}$ are constant integer delays, $0<m_{1}<m_{2}<\dots<m_{n}$, $A$, $B_{1},\dots, B_{n}$ are constant $2\times 2$ matrices, $A=(a_{ij})$, $B_{l}=(b^l_{ij})$, $i,j=1,2$, $l=1,2,\dots,n$ and $x: \{-m_n,-m_n+1,\dots\}\to \mathbb{R}^2$. Under the assumption that the system is weakly delayed, the asymptotic behavior of its solutions is studied and asymptotic formulas are derived.