General solution to a weakly delayed planar linear discrete system, the case of real different eigenvalues of the matrix of nondelayed terms

General solution to a weakly delayed planar linear discrete system, the case of real different eigenvalues of the matrix of nondelayed terms

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The paper considers a linear discrete system with a single delay $$ x(k+1)=Ax(k)+B(k)x(k-m) $$ where $k\in\mathbb{Z}_0^{\infty}:=\{0,1,\dots,\infty\}$, $x\colon {\mathbb{Z}}_0^{\infty}\to\mathbb{R}^2$, $m$ is a positive fixed integer, $A=\{a_{ij}\}_{i,j=1}^2$ and the entries of matrix $B=\{b_{ij}(k)\}_{i,j=1}^2$ are defined for every $k\in\mathbb{Z}_0^{\infty}$. It is assumed that the system is weakly delayed and the eigenvalues of the matrix $A$ are real and different. Analyzing and simplifying a formula for the general solution derived recently, it is shown that, for $k\ge m$, the number of arbitrary constants in this solution can be reduced to two. %rather than to $2(m + 1)$. Conditional stability of a given system is considered. In addition, a~non-delayed planar linear discrete system is constructed such that, for $k\ge m$ and after a transformation, we get the same solutions as those of the delayed system.

The paper considers a linear discrete system with a single delay $$ x(k+1)=Ax(k)+B(k)x(k-m) $$ where $k\in\mathbb{Z}_0^{\infty}:=\{0,1,\dots,\infty\}$, $x\colon {\mathbb{Z}}_0^{\infty}\to\mathbb{R}^2$, $m$ is a positive fixed integer, $A=\{a_{ij}\}_{i,j=1}^2$ and the entries of matrix $B=\{b_{ij}(k)\}_{i,j=1}^2$ are defined for every $k\in\mathbb{Z}_0^{\infty}$. It is assumed that the system is weakly delayed and the eigenvalues of the matrix $A$ are real and different. Analyzing and simplifying a formula for the general solution derived recently, it is shown that, for $k\ge m$, the number of arbitrary constants in this solution can be reduced to two. %rather than to $2(m + 1)$. Conditional stability of a given system is considered. In addition, a~non-delayed planar linear discrete system is constructed such that, for $k\ge m$ and after a transformation, we get the same solutions as those of the delayed system.

planar linear discrete system; constant coefficients; weakly delayed system

planar linear discrete system; constant coefficients; weakly delayed system

HALFAROVÁ, H.; DIBLÍK, J.; ŠAFAŘÍK, J.

2022

06.04.2022

American Institute of Physics

Melville (USA)

978-0-7354-4182-8

INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2020

0094-243X

AIP conference proceedings

2245

1

Spojené státy americké

270009-1

270009-4

4

https://doi.org/10.1063/5.0081842