Detail publikace

Weakly Delayed Systems of Linear Discrete Equations in $\mathbb{R}^3$

ŠAFAŘÍK, J.

Originální název

Weakly Delayed Systems of Linear Discrete Equations in $\mathbb{R}^3$

Typ

dizertace

Jazyk

angličtina

Originální abstrakt

The present thesis deals with the construction of a general solution of weakly delayed systems of linear discrete equations in ${\mathbb R}^3$ of the form \begin{equation*} x(k+1)=Ax(k)+Bx(k-m) \end{equation*} where $m>0$ is a positive integer, $x\colon \bZ_{-m}^{\infty}\to\bR^3$, $\bZ_{-m}^{\infty} := \{-m, -m+1, \dots, \infty\}$, $k\in\bZ_0^{\infty}$, $A=(a_{ij})$ and $B=(b_{ij})$ are constant $3\times 3$ matrices. The characteristic equations of weakly delayed systems are identical with those of the same systems but without delayed terms. The criteria ensuring that a system is weakly delayed are developed and then specified for every possible case of the Jordan form of matrix $A$. The system is solved by transforming it into a higher-dimensional system but without delays \begin{equation*} y(k+1)=\mathcal{A}y(k), \end{equation*} where ${\mathrm{dim}}\ y = 3(m+1)$. Using methods of linear algebra, it is possible to find the Jordan forms of $\mathcal{A}$ depending on the eigenvalues of matrices $A$ and $B$. Therefore, general the solution of the new system can be found and, consequently, the general solution of the initial system deduced.

Klíčová slova

discrete equation, linear systems of difference equations, weakly delayed system, Cayley-Hamilton theorem, Laplace theorem, Jordan form

Autoři

ŠAFAŘÍK, J.

Vydáno

1. 6. 2018

Nakladatel

VUT

Místo

Brno

Strany počet

104

URL

https://www.vutbr.cz/studenti/zav-prace?zp_id=112186

BibTex

@phdthesis{BUT168033,
  author="Jan {Šafařík}",
  title="Weakly Delayed Systems of Linear Discrete Equations in $\mathbb{R}^3$",
  publisher="VUT",
  address="Brno",
  pages="104",
  year="2018",
  url="https://www.vutbr.cz/studenti/zav-prace?zp_id=112186"
}