Doctoral Thesis

Weakly Delayed Systems of Linear Discrete Equations in R^3

Final Thesis 645.22 kB Summary of Thesis 573.35 kB

Author of thesis: Mgr. et Mgr. Jan Šafařík, Ph.D.

Acad. year: 2017/2018

Supervisor: prof. RNDr. Josef Diblík, DrSc.

Reviewers: prof. Denys Khusainov, DrSc., prof. RNDr. Miroslava Růžičková, CSc.

Abstract:

The present thesis deals with the construction of a general solution of weakly delayed systems of linear discrete equations in R3 of the form
x(k+1)=Ax(k)+Bx(km)
where m>0 is a positive integer, x:\bZm\bR3, \bZm:={m,m+1,,}, k\bZ0, A=(aij) and B=(bij) are constant 3×3 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
y(k+1)=Ay(k),
where dim y=3(m+1).
Using methods of linear algebra, it is possible to find the Jordan forms of 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.

Keywords:

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

Date of defence

15.06.2018

Result of the defence

Defended (thesis was successfully defended)

znamkaPznamka

Process of defence

V práci byla vytvořena teorie pro nalezení analytického řešení systémů slabě zpožděných lineárních diskrétních rovnic v R^3. Výsledky jsou nové a zobecňují předchozí výsledky pro systémy v R^2. Hlavní výsledky už byly publikovány.

Language of thesis

English

Faculty

Department

Study programme

Electrical Engineering and Communication (EKT-PP)

Field of study

Mathematics in Electrical Engineering (PP-MVE)

Composition of Committee

prof. RNDr. Jan Chvalina, DrSc. (předseda)
prof. Ing. Pavel Jura, CSc. (člen)
prof. RNDr. Miroslava Růžičková, CSc. - oponent (člen)
prof. Denys Khusainov, DrSc. - oponent (člen)
doc. RNDr. Jaromír Baštinec, CSc. (člen)
doc. RNDr. Jaroslav Beránek, CSc. (člen)
doc. RNDr. Jiří Moučka, CSc. (člen)

Supervisor’s report
prof. RNDr. Josef Diblík, DrSc.

viz příloha PDF
File inserted by supervisor Size
Posudek vedoucího práce [.pdf] 194,75 kB

Reviewer’s report
prof. Denys Khusainov, DrSc.

viz příloha pdf
File inserted by the reviewer Size
Posudek oponenta [.pdf] 561,66 kB

viz příloha pdf
File inserted by the reviewer Size
Posudek oponenta [.pdf] 125,99 kB