Publication detail

# Explicit general solution of planar linear discrete systems with constant coefficients and weak delays

DIBLÍK, J. HALFAROVÁ, H.

Original Title

Explicit general solution of planar linear discrete systems with constant coefficients and weak delays

English Title

Explicit general solution of planar linear discrete systems with constant coefficients and weak delays

Type

journal article - other

Language

en

Original Abstract

In this paper, planar linear discrete systems with constant coefficients and two delays $$x(k+1)=Ax(k)+Bx(k-m)+Cx(k-n)$$ are considered where $k\in\bZ_0^{\infty}:=\{0,1,\dots,\infty\}$, $x\colon \bZ_0^{\infty}\to\mathbb{R}^2$, $m>n>0$ are fixed integers and $A=(a_{ij})$, $B=(b_{ij})$ and $C=(c_{ij})$ are constant $2\times 2$ matrices. It is assumed that the system considered system is one with weak delays. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension $2(m+1)$ is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and weak delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.

English abstract

In this paper, planar linear discrete systems with constant coefficients and two delays $$x(k+1)=Ax(k)+Bx(k-m)+Cx(k-n)$$ are considered where $k\in\bZ_0^{\infty}:=\{0,1,\dots,\infty\}$, $x\colon \bZ_0^{\infty}\to\mathbb{R}^2$, $m>n>0$ are fixed integers and $A=(a_{ij})$, $B=(b_{ij})$ and $C=(c_{ij})$ are constant $2\times 2$ matrices. It is assumed that the system considered system is one with weak delays. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension $2(m+1)$ is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and weak delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.

Keywords

Discrete equation, weak delays, explicit solution, dimension of the solutions space.

RIV year

2013

Released

06.03.2013

Publisher

Springer Nature

ISBN

1687-1847

Periodical

Year of study

2013

Number

1

State

US

Pages from

1

Pages to

29

Pages count

37

URL

Full text in the Digital Library

Documents

BibTex


@article{BUT98366,
author="Josef {Diblík} and Hana {Halfarová}",
title="Explicit general solution of planar linear discrete systems with constant coefficients and weak delays",
annote="In this paper, planar linear discrete systems with constant coefficients and two delays $$x(k+1)=Ax(k)+Bx(k-m)+Cx(k-n)$$ are considered where $k\in\bZ_0^{\infty}:=\{0,1,\dots,\infty\}$, $x\colon \bZ_0^{\infty}\to\mathbb{R}^2$, $m>n>0$ are fixed integers and $A=(a_{ij})$, $B=(b_{ij})$ and $C=(c_{ij})$ are constant $2\times 2$ matrices. It is assumed that the system considered system is one with weak delays. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension $2(m+1)$ is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and weak delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.",
}