MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
NGUYEN THI LAN HUONG
STABILITY AND STABILIZATION OF DISCRETE-TIME
2-D SYSTEMS WITH STOCHASTIC PARAMETERS
DISSERTATION OF
DOCTOR OF PHILOSOPHY IN MATHEMATICS
HA NOI-2020
MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
NGUYEN THI LAN HUONG
STABILITY AND STABILIZATION OF DISCRETE-TIME
2-D SYSTEMS WITH STOCHASTIC PARAMETERS
Speciality: Differential and Integral Equations
Code: 9 46 01 03
DISSERTATION OF
DOCTOR OF PHILOSOPHY IN MATHEMATICS
Supervisors:
1. Assoc.Prof. LE VAN HIEN
2. Assoc.Prof. NGO HOANG LONG
HA NOI-2020
DECLARATION
I am the creator of this dissertation, which has been conducted at the Faculty
of Mathematics and Informatics, Hanoi National University of Education, under the
guidance and direction of Associate Professor Le Van Hien and Associate Professor
Ngo Hoang Long.
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: Nguyen Thi Lan Huong
Signed:
Date:
1
ACKNOWLEDGMENT
First and foremost, I would like to express my deep gratitude and great appre-
ciation to my supervisors, Associate Professor Le Van Hien and Associate Professor
Ngo Hoang Long, for their valuable support, enthusiastic encouragement and useful
critiques for this research work. It is my great pleasure having a chance to work with
them who are amazing researchers. Especially, I would like to express sincere thanks
to Associate Professor Le Van Hien for his professional guidances and valuable sugges-
tions.
The wonderful working environment of Hanoi National University of Education
and its excellence staff have assisted me throughout my PhD candidature. In partic-
ular, I am grateful to Associcate Professor Tran Dinh Ke and other members of the
weekly seminar at the Division of Mathematical Analysis, Faculty of Mathematics and
Informatics, as well as members of the research group of Professor Vu Ngoc Phat at
the Institute of Mathematics, Vietnam Academy of Science and Technology, for their
valuable comments and fruitful discussions on my research results.
I am also grateful to my colleagues at Faculty of Mathematics and Informatics,
Hanoi National University of Education, for their help and support during the time of
my postgraduate study.
Lastly, I would like to thank all members in my big family, especially my wonderful
parents, for the encouragement, endless love and unconditional support they have been
giving me throughout my entire life. Special thanks to my beloved husband, Mr Tran
Minh Duc, and my daughter, Miss Tran Hong Anh, who always trust and stay beside
me.
The author
2
TABLE OF CONTENTS
Page
Declaration ..................................... 1
Acknowledgment .................................. 2
List of Notations and Abbreviations . . . . . . . . . . . . . . . . . . . . . 5
INTRODUCTION ................................. 7
1. AUXILIARY RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.1. Random variables and random vectors . . . . . . . . . . . . . . . . . . . 22
1.2. Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3. Conditional expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4. Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5. Stability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.1. Stability concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.2. Stability of linear systems . . . . . . . . . . . . . . . . . . . . . . 30
1.6. Lyapunov’s direct method . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.7. Lyapunov theory for stochastic discrete-time 1-D systems . . . . . . . . 36
1.8. Auxiliary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2. OBSERVER-BASED ℓ2-ℓ∞CONTROL OF 2-D LINEAR
ROESSER SYSTEMS WITH RANDOM PACKET DROPOUT . . . . . . . 39
2.1. Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2. Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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