Representation of the Solutions of Linear Discrete Systems with Constant Coefficients and Two Delays

Abstract

The purpose of this paper is to develop a method for the construction of solutions to initial problems of linear discrete systems with constant coefficients and with two delays $\mathrm{\Delta }x(k)=Bx(k-m)+Cx(k-n)+f(k),$ where $m,n\in \mathbb{N}$, $m\ne \mathrm{n,}$ are fixed, $k=0,\dots ,\mathrm{\infty }$, $B=(b_{ij})$, $C=(c_{ij})$ are constant $r\times r$ matrices, $f$ is a given $r\times 1$ vector, and $x$ is an $r\times 1$ unknown vector. Solutions are expressed with the aid of a special function called the discrete matrix delayed exponential for two delays. Such approach results in a possibility to express an initial Cauchy problem in a closed form. Examples are shown illustrating the results obtained.