2014 General Explicit Solution of Planar Weakly Delayed Linear Discrete Systems and Pasting Its Solutions
Josef Diblík, Hana Halfarová
Abstr. Appl. Anal. 2014(SI27): 1-37 (2014). DOI: 10.1155/2014/627295

## Abstract

Planar linear discrete systems with constant coefficients and delays $x(k+\mathrm{1})=Ax(k)+{\sum }_{l=\mathrm{1}}^{n}\mathrm{‍}{B}^{l}{x}_{l}(k-{m}_{l})$ are considered where $k{\in \Bbb Z}_{\mathrm{0}}^{\mathrm{\infty }}:=\{\mathrm{0,1},\dots ,\mathrm{\infty }\}$, ${m}_{\mathrm{1}},{m}_{\mathrm{2}},\dots ,{m}_{n}$ are constant integer delays, $\mathrm{0}<{m}_{\mathrm{1}}<{m}_{\mathrm{2}}<\cdots <{m}_{n}$, $A,{B}^{\mathrm{1}},\dots ,{B}^{n}$ are constant $\mathrm{2}{\times}\mathrm{2}$ matrices, and $x:{\Bbb Z}_{{-m}_{n}}^{\mathrm{\infty }}{\to \Bbb R}^{\mathrm{2}}$. It is assumed that the considered system is weakly delayed. 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 $\mathrm{2}({m}_{n}+\mathrm{1})$ 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 special 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.

## Citation

Josef Diblík. Hana Halfarová. "General Explicit Solution of Planar Weakly Delayed Linear Discrete Systems and Pasting Its Solutions." Abstr. Appl. Anal. 2014 (SI27) 1 - 37, 2014. https://doi.org/10.1155/2014/627295

## Information

Published: 2014
First available in Project Euclid: 2 October 2014

zbMATH: 07022764
MathSciNet: MR3206806
Digital Object Identifier: 10.1155/2014/627295