MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
——————–o0o———————
LE HUY VU
STABILITY OF DISCRETE-TIME
2-D SINGULAR SYSTEMS WITH DELAYS
DISSERTATION OF
DOCTOR OF PHILOSOPHY IN MATHEMATICS
HA NOI-2024
MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
——————–o0o———————
LE HUY VU
STABILITY OF DISCRETE-TIME
2-D SINGULAR SYSTEMS WITH DELAYS
Speciality: Differential and Integral Equations
Code: 9.46.01.03
A dissertation submitted to
Hanoi National University of Education
for fulfilled requirements of the degree
of
Doctor of Philosophy in Mathematics
Under the guidance of
Associate Professor Le Van Hien
HA NOI-2024
DECLARATION
The author declares that this dissertation has been conducted by the PhD can-
didate whose name displayed below at the Faculty of Mathematics and Informatics,
Hanoi National University of Education, under the supervision of Associate Professor
Le Van Hien.
I hereby affirm that the results presented in this dissertation are correct and
have not been included in any other dissertations or theses submitted to any other
universities or institutions for a degree or diploma.
“I certify that I am the PhD student named below and that the information provided
is correct”
Full name: Le Huy Vu
Signed:
Date:
1
ACKNOWLEDGMENT
This dissertation has been completed at the Faculty of Mathematics and In-
formatics, Hanoi National University of Education, under the guidance of Associate
Professor Le Van Hien. Let me take this opportunity to express my sincere thanks
to my supervisor, Associate Professor Le Van Hien, for his enlightening guidance, in-
sightful ideas, and endless support during my candidacy at Hanoi National University
of Education. His rigorous research ethics, diligent work attitude, and wholehearted
dedication to his students have been an inspiration to me and will influence me forever.
The author is grateful to Professor Cung The Anh, Associcate Professor Tran
Dinh Ke and other members of the weekly seminar at the Division of Mathemati-
cal Analysis, Faculty of Mathematics and Informatics, Hanoi National University of
Education, for their encouragement, valuable discussions and comments. I am also
grateful to Hong Duc University and the Division of Mathematics and Teaching meth-
ods, Faculty of Natural Science, for unlimited help and support during the time of my
postgraduate study.
I would like to express my gratitude to my parents for the encouragement, endless
love, and unconditional support they have been giving me. Last but not least, I am
indebted to my beloved wife, Mrs Le Thi Dao, our lovely son, Le Huy Nghia Linh, and
beautiful daughters, Le Thi Thu Hang, Le Thu Khanh Ngoc, who always trust and
stay beside me. None of this would have been possible without their continuous and
unconditional love, kindness and comfort through my journey.
The author
2
TABLE OF CONTENTS
Page
Declaration ..................................... 1
Acknowledgment .................................. 2
List of notations and acronyms ......................... 5
INTRODUCTION ................................. 7
1.AUXILIARYRESULTS ............................. 19
1.1. Stability of discrete-time 1-D singular systems . . . . . . . . . . . . . . . 19
1.2. Finite-time stability of discrete-time 1-D systems . . . . . . . . . . . . . 21
1.3. Reachable set bounding for discrete-time 1-D systems . . . . . . . . . . 23
1.3.1. Reachable set estimation . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.2. Reachable set estimation for discrete-time singular systems . . . 24
1.4.Auxiliarylemmas............................... 25
2. STABILITY OF 2-D SINGULAR SYSTEMS WITH
GENERALIZED DIRECTIONAL DELAYS . . . . . . . . . . . . . . . . . . 27
2.1. Model description and preliminaries . . . . . . . . . . . . . . . . . . . . 27
2.2. Stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4. Conclusions of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3. FINITE-REGION STABILITY AND DISSIPATIVITITY OF 2-D SINGU-
LAR ROESSER SYSTEMS WITH DELAYS . . . . . . . . . . . . . . . . . . 45
3.1. Finite-region stability of 2-D singular systems with delays . . . . . . . . 45
3.1.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.2. Regularity, causality and finite-region stability . . . . . . . . . . . 46
3.1.3. An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . 55
3