Tính ổn định của một số lớp hệ sai phân hai pha suy biến có trễ

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. AUXILIARY RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. Auxiliary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

3.2. Finite-region boundedness and dissipativity of 2-D singular Roesser sys-
tems with mixed time-varying delays . . . . . . . . . . . . . . . . . . . . 56
3.2.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.2. Finite-region boundedness . . . . . . . . . . . . . . . . . . . . . . 61
3.2.3. Finite-region dissipativity . . . . . . . . . . . . . . . . . . . . . . 75
3.2.4. Ilustrative examples . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3. Conclusions of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4. REACHABLE SET ESTIMATION AND CONTROLLER

DESIGN FOR 2-D SINGULAR ROESSER WITH
TIME-VARYING DELAYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.1. Reachable set estimation of 2-D singular systems . . . . . . . . . . . . . 84

4.1.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.1.2. Technical lemmas for 2-DSSs reachable set estimation . . . . . . . 86
4.1.3. Reachable set analysis . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2. Ellipsoidal reachable set bounding via controller design . . . . . . . . . . 100
4.3. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.4. Conclusions of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4

NOTATIONS AND ACRONYMS

Z, N the set of integers and positive integers, respectively

N0 N ∪ {0}
Z[a, b] = [a, b] ∩ Z the set of integers between a and b
Z ([a, b] × [c, d]) Z[a, b] × Z[c, d]
Rn the n-dimensional Euclidean space
Rn×m the set of n × m real matrices
Identity matrix of n dimension
In the transpose of a matrix A
A⊤ the null-space of matrix A
A⊥ the inverse of matrix A
A−1 the symmetry matrix operator
Sym{.} the block diagonal matrix
diag(· · · ) the column matrix
col{· · · } the maximal real part of eigenvalues of A ∈ Rn×n
λmax(A) the minimal real part of eigenvalues of A ∈ Rn×n
λmin(A) the generalized spectral radius of pair (E, A) defined as
ρ(E, A) maxλ∈{z| det(zE−A)=0} |λ|
the norrm of matrix A ∈ Rn×m defined by A =
A
λmax(A⊤A)
M >0 M is a symmetric positive definite matrix
Q is symmetric and semi-positive definite
Q≥0 the set of symmetric positive definite matrices in Rn×n
the set {diag(Mp, Mq) : Mp ∈ S+p , Mq ∈ S+q }
S+ diag(Ph, Pv)
the floor function defined by ⌊x⌋ = max{m ∈ Z | m ≤ x}
n Indicator function of the set B
The term induced by symmetry
D + ( p , q ) One-dimensional
n Two-dimensional
Two-dimentional singular Roesser systems

Ph ⊕ Pv Two-dimentional singular closed-loop systems

⌊x⌋ 5

IB

∗

1-D

2-D

2-DSSs

2-DSCS

LMIs Linear matrix inequalities
LKF Lyapunov–Krasovskii functional
LAS Lyapunov asymptotic stability
FRS Finite-region stability
FRB Finite-region boundedness
FTS Finite-time stability
FTB Finite-time boundedness
RS Reachable set
RSE Reachable set estimation
SFRS Singular finite-region stability
SFRB Singular finite-region boundedness
SFC State-feedback controller
TVDs Time-varying delays
✷ Completeness of a proof.

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INTRODUCTION

1. Literature review and motivations

Singular systems are used to describe dynamics of numerous practical models in
electrical circuit networks, power systems, multibody mechanics, aerospace engineering
and chemical and physical processes [8, 11, 79]. These systems are also known as de-
scriptor systems [15, 23, 65], implicit systems [5, 51] or differential/difference-algebraic
equations [16, 41]. In the state equation of a singular system, the state variables are
subject to both dynamical equations and algebraic constraints which result a number
of different features from classical systems such as impulsive behaviors in the state re-
sponse, non-properness of transfer matrix or non-causality between input/output and
states. Those characteristic properties make the study of singular systems much more
complicated and challenging than classical systems. On the other hand, as an inherent
characteristic, time-delay is ubiquitously encountered in engineering systems which has
various effects on the system performance [41,51]. Thus, the study of qualitative behav-
ior of time-delay systems plays an essential role in applied models which has received
significant research attention in the last two decades. We refer the reader to [16,57,64]
and the references therein for a few references. In particular, considerable effort from
researchers has been devoted to the analysis and control of singular systems with de-
lays and many results have been reported in the literature. We mention some existing
works [56, 61] for the stability analysis and [28, 74] for other control issues related to
singular delayed systems.

Two-dimensional (2-D) systems can be used to model many practical systems
where the information propagation occurs in each of the two independent directions [38,
59]. Recently, due to their widespread applications in circuit analysis, image processing,

seismographic data transmission or multi-dimensional digital filtering, the theory of 2-
D systems has attracted considerable research attention [1, 2, 6, 27]. There have been
a few papers concerning the problems of stability and stabilization of 2-D descriptor
systems. For example, in [91], the stability problem was studied for 2-D linear singular
systems in general model. Sufficient conditions were derived in a type of Lyapunov
matrix inequalities to ensure that a 2-D system is acceptable and asymptotically stable.
In [9], the problems of stability and stabilization via state feedback controllers were

7

investigated for a class of delay-free 2-D singular Roesser systems. By decomposing the
system into slow- and fast-subsystems, and based on the Lyapunov function method,
sufficient conditions in terms of linear matrix inequalities (LMIs) were derived to design
a stabilizing state feedback controller. The problem of H∞ control was also considered
in [40,80] for 2-D singular Roesser models with constant delays. By using the bounded
real lemma approach, delay-independent LMI-based conditions were derived for the
design of state feedback controllers that make the closed-loop system to be acceptable
and stable with a prescribed H∞ performance level. It is noted also that practical
models in real world applications produce different types of delays because of variable
networks transmission conditions. Since the properties of these delays may not be
identical, it is not reasonable to lump all the delays into one type [26]. Thus, it is
interesting and important to study 2-D singular models with different types of delays.
Besides, the use of Jensen-type inequalities usually produces undesired conservatism in
the derived stability conditions. Therefore, reducing the conservativeness of stability
conditions is always an important issue in applications of control engineering which
needs further investigation.

The concept of Lyapunov stability, recognized as long-time behavior, has been
well investigated and developed over the decades. However, in many practical ap-
plications, it is only required that the system states do not exceed a certain bound

during a specified time interval for a given bound on initial states. This gives rise to
the use of finite-time stability (FTS) [24]. It is noted that a system may be finite-
time stable but not Lyapunov asymptotic stable and vice versa [24]. In the case of a
system with exogenous disturbance, we have the concept of finite-time boundedness
(FTB) [3]. Besides, when studying the qualitative properties of the singular systems,
the authors have introduced new concepts of singular finite-time stability (SFTS), sin-
gular finite-time boundedness (SFTB), and singular finite-time H∞ boundedness for
the first time [48–50, 86]. It is an extension of the concepts FTS and FTB for the
class of singular systems. Notice that SFTB implies that not only the dynamic part
of the system is finite-time bounded but also the algebraic part is finite-time bounded
since the static mode is regular and causal. Differ from 1-D systems, in 2-D systems,
the information propagation occurs in each of the two independent directions. In this
case, there is no explicit time concept, and it fails to extend the concepts of FTS and
FTB directly to such 2-D Roesser models. Therefore, it is expected that the transient
performance of a Roesser model is reflected by the behavior of the horizontal state
vector and vertical state vector which are required to be bounded over a given finite

8

region. Thus, the concepts of finite-region stability (FRS) and finite-region bounded-
ness (FRB) can be regarded as a natural extension of FTS and FTB. The FRS or FRB
implies that the state trajectory does not exceed a certain threshold in a pre-specified
finite region interval. It follows that FRB leads to FRS when the zero exogenous distur-
bance. The stability analysis problem under FRS and FRB concepts for 2-D systems
has received considerably less attention and only a few results for 2-D systems have
been reported [87–89]. However, up to the present, the analysis and control involving
FRS and FRB for 2-D singular systems have not been investigated in the literature.
Thus, it is natural and relevant to develop the problems of SFTS and SFTB for 2-D
singular systems with and without delays.

In addition to what has been stated, in the late of 1970s, the theory of dissipative
systems, first introduced by Willems to describe the system dissipativity [70], which
laid an important foundation for the dissipativity theory and is applied in many fields
such as system theory, circuit design, network synthesis, and control theory [?, 66, 90].
Dissipativity not only gives natural candidates for Lyapunov functions but also provides
a more useful method for the stability analysis and controller design of the dynamical
systems. In the perspective of energy, dissipativity was described by supply rates and
storage functions, which represent separately the energy supplied from outside the
system and the energy stored inside. The actual physical meaning of dissipativity is
that the one for which the energy dissipated inside the dynamical system is less than
the energy supplied from the external source of the system. Dissipativity is a more
general performance that can be reduced to H∞ performance, positive realness, and
passivity by choosing different dissipative coefficients. Thus, over the last few years,
dissipativity and its application have been extensively studied, and many significant
research results have been reported in the literature. There has been considerable
interest in the dissipativity analysis and control for various types of 1-D systems such as
nonlinear systems [60], singular systems [19, 74], stochastic systems [75, 85] or switched
systems etc. There are only a few results of research on the dissipation problem of 2-D
systems [1, 45]. Nevertheless, there are no studies on dissipative analysis for classes of
singular 2-D systems.

On the other hand, in the systems and control theory, the state estimation prob-
lem plays an essential role. One of the widespread strategies for evaluation is the
reachable set estimation (RSE) for dynamical systems, which specifies a bounded set
in the state-space containing all system states starting from the origin in the effect of
bounded disturbances. It is worth noting that the object of the RSE is to find a small-

9

est possible compact set, typically in the form of an ellipsoid E = x ∈ Rn|x⊤P x ≤ r

with kernel matrix P = P ⊤ > 0 and radius r ≥ 0, to bound the reachable set of the
system [17, 42]. On the other hand, the insecurity of a system usually occurs as the
reachable set containing undesirable states. In practical engineering, the system is re-
garded safety if its reachable set can avoid an unsafe state in the state space. Therefore,
the problem of RSE raises an important research topic and has received great attention
from researchers during the past few decades. Numerous results involving this topic
for one-dimensional (1-D) systems have been reported in the literature. In particular,
Wang et al. [69] considered the problem of RSE for linear systems with time-varying
delays and polytopic uncertainties. The problem of RSE was also developed for sin-
gular continuous-time systems and switched discrete-time linear systems in [10, 18, 19].
The ellipsoid RS problem of linear continuous systems with time-varying delays and
linear discrete systems in control was considered in [7]. Improved results on RSE of
various classes of singular systems were formulated in [84] and [44, 46].

The RSE problem for 2-D systems with delays was recently developed in [32].
Based on an analysis scheme extended from the Lyapunov–Krasovskii functional (LKF)
method, and by utilizing some 2-D weighted summation inequalities, delay-dependent
conditions in terms of tractable LMIs were derived to ensure the existence of an ellipsoid
that attracts exponentially the system states in the presence of bounded disturbances.
However, to the best of authors knowledge, up to date, no result in the literature
focusing on RSE and controller design of 2-DSSs with time-varying delays has been
achieved. The aforementioned discussion strongly motivates us for the present study.

According to a comprehensive literature review, it is recognized that some impor-
tant problems in the systems and control theory have been successfully developed for
various classes of 2-D systems. However, such results have been less extended to 2-D
singular systems. It is important to note that, for 2-D singular systems, the existing
methodology for nominal systems is, in general, not directly adaptable. This urges
further development. From this observation, in this dissertation, we focus on the per-
formance analysis and controller design for 2-D singular Roesser systems with delays.

Specifically, the thesis focuses on the following aspects.

• Stability of 2-D singular Roesser systems with generalized directional delays.

• Finite-region stability and dissipative control of 2-D singular Roesser systems with
time-varying delays.

• Reachable set estimation and controller design for 2-D singular systems.

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2. Methodology

To deal with the aforementioned problems, in this thesis, we utilize the methods
of two time-scale mathematical induction and a slow- and fast-subsystem decomposi-
tion. Based on the 2-D Lyapunov–Krasovskii functional method combining with free
weighting matrix and zero-type equations techniques, analysis and design conditions
are formulated in terms of tractable linear matrix inequalities, which can be effectively
solved by various computational tools and convex optimization algorithms.

3. Objectives

The main purpose of this thesis is to study the stability and its applications in
control of some classes of discrete-time 2-D singular systems described by the Roesser
model with delays. The research includes the methodology development and estab-
lishment of new analysis and synthesis conditions. Specifically, this thesis is concerned
with the following problems.

3.1. Stability of 2-D singular Roesser systems with generalized delays

Consider a class of 2-D singular systems with mixed directional time-varying
delays described by the following Roesser model (2-D SRM)

Ehxh(i + 1, j) xh(i, j) xh(i − dh(i), j)
=A v + Ad v
Evxv(i, j + 1) x (i, j) x (i, j − dv(j))
τh(i) l=1 xh (1) (i − l, j) +
+ Aτ τv(j) l=1 xv(i, j − l) , i, j ∈ Z ,

where xh(i, j) ∈ Rnh and xv(i, j) ∈ Rnv are the horizontal state vector and the vertical
state vector, respectively, A, Ad, Aτ ∈ Rn×n (n = nh + nv) are given real matrices. The
system matrices Eh ∈ Rnh×nh and Ev ∈ Rnv×nv are singular with rank(Eh) = rh ≤ nh,
rank(Ev) = rv ≤ nv and r = rh + rv < nh + nv n. dh(i), τh(i) and dv(j), τv(j)
are respectively the directional time-varying delays along the horizontal and vertical

directions satisfying

dhm ≤ dh(i) ≤ dhM , dvm ≤ dv(j) ≤ dvM , (2a)

τhm ≤ τh(i) ≤ τhM , τvm ≤ τv(j) ≤ τvM , (2b)

11

where dhm, dhM , dvm, dvM , τhm, τhM , τvm, and τvM , are known nonnegative integers
involving the upper and the lower bounds of delays. Denote σh = max(dhM , τhM ) and
σv = max(dvM , τvM ).

Initial condition of (1) is defined by

xh(k, j) = φ(k, j), k ∈ Z[−σh, 0], j ∈ Z+,

(3)

xv(i, l) = ψ(i, l), l ∈ Z[−σv, 0], i ∈ Z+,

where φ(k, .) ∈ l2(Z+), k ∈ Z[−σh, 0] and ψ(., l) ∈ l2(Z+), l ∈ Z[−σv, 0].

The pair (E, A) with E = diag(Eh, Ev) is said to be regular if det[EI(z, w) −A] is
not identically zero, where I(z, w) = diag(zInh , wInv ), and causal if deg(det(sE −A)) =
rank(E). If the pair (E, A) is regular and causal then the 2-D SRM (1) is said to be
acceptable. In addition, the acceptable 2-D SRM (1) is internally stable if, for any
initial condition (3), it holds that

lim sup xh(i, j) : i + j = q = 0.
xv(i, j)
q→∞

System (1) is admissible if it is acceptable and internally stable.

Regularity and causality of the pair (E, A) guarantee system (1) to have a de-
composition into slow- and fast-subsystems which deduce the global existence and
uniqueness of solution of system (1). Our aim here is to establish conditions for 2-D
systems as (1) to be admissible, that is, system (1) is regular, causal, and internally
stable. This topic will be studied and presented in Chapter 2.

3.2. Finite-region stability and dissipativity of 2-D singular systems with
time-varying delays

3.2.1. Finite-region stability of 2-D singular systems with directional delays

Consider the following 2-D singular system with delays

Ehxh(i + 1, j) xh(i, j) xh(i − dh, j)
= A v + Ad v , (4)
Evxv(i, j + 1) x (i, j) x (i, j − dv)

where xh(i, j) ∈ Rnh and xv(i, j) ∈ Rnv are the horizontal and vertical state vectors,

Ahh Ahv Ahh Ahv
respectively. A = Avh Avv and Ad = d d Avh Avv are given real matrices. The
d d
system matrices Eh ∈ Rnh×nh and Ev ∈ Rnv×nv are singular with rank(Eh) = rh ≤ nh,

12

rank(Ev) = rv ≤ nv, r = rh + rv < nh + nv n and dh, dv are positive integers
representing directional delays. Initial conditions of (4) are specified as

xh(k, j) = φ(k, j), k ∈ Z[−dh, 0], j ≥ 0,
(5)

xv(i, l) = ψ(i, l), l ∈ Z[−dv, 0], i ≥ 0.

In addition, for given positive integers D1 ∈ N+ and D2 ∈ N+, we define the rectangle
finite-region

D1 × D2 = (i, j) ∈ N20|0 ≤ i ≤ D1, 0 ≤ j ≤ D2 .

For a given matrix W ≥ 0, we denote the weighted norm of φ and ψ as

φ W = sup φ⊤(k, j)W φ(k, j) : −dh ≤ k ≤ 0, j ≥ 0 ,

l∞

ψ W = sup ψ⊤(i, l)W ψ(i, l) : i ≥ 0, −dv ≤ l ≤ 0 .
l∞

We define the regularity and causality of system (4) by the regularity and causality

of the matrix pair (E, A). System (4) is said to be singularly finite-region stable (SFRS)
with respect to (c, ch, cv, D1 × D2, Γ) if for any initial sequences φ, ψ that satisfy

max Eh⊤ΓhEh 2 Γh < ch
φ l∞ , Eh φ l∞

and

Ev⊤Γv Ev 2 Γv
max ψ l∞ , Ev ψ l∞ < cv,

it holds that

x⊤(i, j)E⊤ΓEx(i, j) < c (6)

for all (i, j) ∈ D1 × D2, where c, ch, cv is positive scalars, E = diag(Eh, Ev) and

Γ ∈ S+ ⊕ S+ is a given positive definite matrix. In the first part of Chapter 3, we

nh nv

will derive tractable LMI-based conditions by which a 2-D singular system in the form

of (4) is regular, causal and SFRS.

3.2.2. Finite-region dissipativity analysis for 2-D singular systems with mixed delays

Consider a class of 2-D singular Roesser systems with mixed time-varying delays

Ehxh(i + 1, j) xh(i, j) xh(i − dh(i), j)
=A v + Ad v
Evxv(i, j + 1) x (i, j) x (i, j − dv(j))

+ Aτ k=1 τh(i) xh(i − k, j) + Bww(i, j), (7a)
τv(j) v
l=1 x (i, j − l)

13

xh(i, j) xh(i − dh(i), j)
z(i, j) = C v + Cd v
x (i, j) x (i, j − dv(j))

+ Cτ k=1 τh(i) xh(i − k, j) + Dww(i, j), (7b)
l=1 τv(j) xv(i, j − l)

where xh(i, j) ∈ Rnh and xv(i, j) ∈ Rnv are the horizontal and vertical state vectors,
w(i, j) ∈ Rm is the exogenous disturbance, z(i, j) ∈ Rs is the regulated output. The
system matrices A, Ad, Aτ ∈ Rn×n, Bw ∈ Rn×m, C, Cd, Cτ ∈ Rs×n and Dw ∈
Rs×m are known with appropriate dimensions. The functions dh, dv, τh, τv : N0 → N0
represent directional time-varying delays along the horizontal and vertical directions,

which satisfy

dhm ≤ dh(i) ≤ dhM , dvm ≤ dv(j) ≤ dvM ,

τhm ≤ τh(i) ≤ τhM , τvm ≤ τv(j) ≤ τvM , (8)

where dhm, dvm, τhm, τvm and dhM , dvM , τhM , τvM are given positive integers involving
the upper and the lower bounds of delays. The initial condition of (7) is specified as

xh(k, j) = φ(k, j), k ∈ Z[−σh, 0], 0 ≤ j ≤ D2,

xh(k, j) = 0, k ∈ Z[−σh, 0], j > D2,

xv(i, l) = ψ(i, l), l ∈ Z[−σv, 0], 0 ≤ i ≤ D1,

xv(i, l) = 0, l ∈ Z[−σv, 0], i > D1, (9)

where σh = max{dhM , τhM }, σv = max{dvM , τvM } and D1, D2 ∈ Z+ are some given

positive integers. We define the space 
 
 w⊤(i, j)w(i, j) ≤ ρ2 .

W2ρ = w : D1 × D2 → Rm (i,j)∈D1×D2

The exogenous disturbance w(i, j) is assumed to belong to W2ρ for a given ρ > 0. In ad-
dition, for given positive scalars ch, cv, and a positive definite matrix Γ = diag(Γh, Γv),
we define the following regions referred to starting regions of initial states

Eh = x ∈ Rnh x⊤Eh⊤ΓhEhx < c2 ∩ x ∈ Rnh x⊤Γhx < c2 ,
Ev =

h h

x ∈ Rnv x⊤ Ev⊤Γv Ev x < c2 ∩ x ∈ Rnv x⊤ Γv x < c2 .

v v

Given scalars c, ch, cv and integers D1, D2, which are positive, and a matrix
Γ = diag(Γh, Γv) ∈ S+n , system (7) is said to be singularly finite-region bounded (SFRB)
with respect to (c, ch, cv, D1, D2, Γ) if it is regular, causal and for any disturbance input

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w ∈ W2ρ, it holds that

φ(k, j) ∈ Eh, k ∈ Z[−σh, 0], 0 ≤ j ≤ D1,
ψ(i, l) ∈ Ev l ∈ Z[−σv, 0], 0 ≤ i ≤ D2,

x⊤(i, j)E⊤ΓEx(i, j) < c2 (10)

for all (i, j) ∈ D1 × D2. Moreover, for given real matrices Q, S , R with Q = Q⊤ and
R = R⊤, system (7) is said to be finite-region (Q, S , R)-dissipative with respect to

(c, ch, cv, D1, D2, Γ) if it is SFRB with respect to (c, ch, cv, D1, D2, Γ) and the following
dissipation inequality is satisfied

Tv −1 Th−1

V h(xh(Th, j)) − V h(xh(0, j)) + [V v(xv(i, Tv)) − V v(xv(i, 0))]

j=0 i=0

Th−1 Tv−1

≤ SE(z(i, j), w(i, j)). (11)

i=0 j=0

where SE(z, w) = z⊤Qz + 2z⊤S w + w⊤Rw is the supplied rate function.

In Chapter 3, we first derive tractable LMI-based conditions by which 2-D singular
systems in the form of (7) are regular, causal, and SFRB. Then we formulate design
conditions to ensure system (7) is (Q, S , R)-dissipative.

3.3. Reachable set estimation and controller design for 2-D singular
Roesser with time-varying delays

In Chapter 4, we consider the problem of RSE for 2-DSSs with time-varying
delays described by the following Roesser model

E xh(i + 1, j) = A xh(i, j) + A xh(i − d d h(i), j) + Bu(i, j) + Dw(i, j), (12)vvv
x (i, j + 1) x (i, j) x (i, j − dv(j))

where E = diag(Eh, Ev) and given system matrices

Ahh Ahv Ahh Ahv B = B1 , D = D1 .
A = Avh Avv , Ad = d d Avh Avv , B2 D2
d d

xh(i, j) ∈ Rnh and xv(i, j) ∈ Rnv are state vectors, u(i, j) ∈ Rs is the control input

and w(i, j) ∈ Rs is the exogenous disturbance vector. dh(i) and dv(j) are directional

TVDs along the horizontal and vertical directions which satisfy

dhm ≤ dh(i) ≤ dhM , dvm ≤ dv(j) ≤ dvM , (13)

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with known nonnegative integers corresponding to the upper and lower bounds of delays
dhm, dhM , dvm and dvM . Initial condition of (12) is specified as

xh(k, j) = φh(k, j), k = −dhM , . . . , −1, 0, j ≥ 0
(14)

xv(i, l) = φv(i, l), l = −dvM , . . . , −1, 0, i ≥ 0,

where φh(., .) and φv(., .) are given two-parameter sequences.
Moreover, w(i, j) represents the disturbance which belongs to ℓ2(N2). In other

words, there exists a scalar ρ > 0 such that

w⊤(i, j)w(i, j) ≤ ρ2. (15)

i+j=p

For system (12), an SFC will be designed in the form

xh(i, j)
u(i, j) = Kx(i, j) = K1 K2 xv(i, j) , (16)

where K is the controller gain matrix which will be determined. The 2-D singular
closed-loop system of (12) can be expressed as

xh(i + 1, j) xh(i, j) xh(i − dh(i), j) + Dw(i, j), (17)
Ev = Ac v + Ad v
x (i, j + 1) x (i, j) x (i, j − dv(j))

where

Ahh Ahv c c
Ac = A + BK = Avh Avv ,
c c

hh = Ahh + B1K1, hv = Ahv + B1K2,

Ac Ac

vh = Avh + B2K1, vv = Avv + B2K2.

Ac Ac

The reachable set of the singular closed-loop system (17) under zero initial condition
is defined as

Rx = x(i, j) ∈ Rn x(i, j) satisfies (17) with φh = 0, φv = 0

for any w(i, j) satisfies (15), where x(i, j) = xh(i, j)
the existence of an ellipsoid xv(i, j) . We will derive conditions for

E(P, r) = x ∈ Rn x⊤P x ≤ r, P ⊤ = P > 0, r ≥ 0

that bounds the reachable set of singular closed-loop systems (17) under zero initial
condition. The objective of this chapter is to

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• Design an SFC in the form of (16) such that, in the effect of exogenous distur-
bances, the reachable set of (17) is confined in a certain ellipsoid.

• For a given symmetric positive definite matrix Φ0 and positive scalars β > 0,
0 < α < 1, design an SFC in the form of (16) such that the reachable set of (17)
is bounded within the given ellipsoid E(Φ0, 1−α βρ2 ).

4. Summary of the results

The problems of stability, dissipativity analysis and reachable set estimation are
considered for discrete-time 2-D singular Roesser systems with delays. Main results
presented in this thesis can be summarized as follows.

1. By constructing an improved 2-D LKF comprising quadratic terms and summation
terms in single, double, and triple forms combining with the technique of zero-
type matrix equations, LMI-based conditions are established to guarantee the
regularity, causality, and stability of 2-D descriptor systems with interval discrete
and distributed time-varying delays.

2. The concepts of SFRS and SFRB are suitably developed for 2-D singular systems.
By using a 2-D Lyapunov-like functional and a novel multiple Lyapunov-function
that contains the time-varying delays, we derive conditions that guarantee the
properties of SFRS and SFRB of a class of 2-D singular Roesser systems with
delays. LMI-based conditions which ensure that a 2-D singular system with mixed

time-varying delays is strictly µ-(Q, R, S)-dissipative are also formulated.

3. The RSE problem is first addressed for 2-DSSs with time-varying delays. Tractable
conditions are derived to get an ellipsoidal estimation of the reachable set of the
system. Then, an SFC is designed to make the reachable set of the closed-loop
system be contained within a given ellipsoid.

The obtained results have been published in 03 papers and 01 preprint, and have
been presented at

• The weekly seminar on Differential and Integral Equations, Division of Mathemat-
ical Analysis, Faculty of Mathematics and Informatics, Hanoi National University
of Education.

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• Workshop Dynamical Systems and Related Topics, Vietnam Institute for Ad-
vanced Study in Mathematics, Hanoi, December 23-25, 2019.

• PhD Annual Conferences, Faculty of Mathematics and Informatics, Hanoi Na-
tional University of Education, 2020-2022.

• Workshop Selected Research Topics in Modern Mathematical Analysis, Hue Uni-
versity, Hue, August 18-21, 2022.

• The 10th Vietnam Mathematics Congress, Da Nang, August 08-12, 2023.

5. Thesis structure

The rest of this thesis is organized as follows.

Chapter 1 presents preliminary results on the stability and stabilization of discrete-

time 1-D singular systems. Basic concepts and related results about finite-time
stability, stabilization, and useful technical lemmas are recalled and presented.
Chapter 2 is devoted to the problems of regularity, causality, and stability of 2-D
singular (descriptor) systems with generalized directional delays.
Chapter 3 deals with the problems of SFRS and SFRB for 2-D singular systems
described by the Roesser model. Dissipativity of 2-D singular systems with mixed
time-varying delays is also discussed in this chapter.
Chapter 4 investigates the problems of reachable set estimation and controller design
for a class of 2-D singular Roesser systems with time-varying delays. Based on
a 2-D LKF scheme developed in this chapter, LMI-based conditions are derived
to establish an ellipsoid estimation for the reachable set of the system subject
to bounded disturbances. Controller design problem to ensure the reachable set
of the resulting closed-loop system confined in a prescribed ellipsoidal set is also
presented.

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