Stability and robust stability of singular linear difference equations

Thanh Nga STABILITY AND ROBUST STABILITY OF SINGULAR LINEAR DIFFERENCE EQUATIONS THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS... discrete-more general settings, for examp

VIETNAM NATIONAL UNIVERSITY, HANOI

VNU UNIVERSITY OF SCIENCE

Ngô Thi Thanh Nga

STABILITY AND ROBUST STABILITY

OF SINGULAR LINEAR DIFFERENCE EQUATIONS

THESIS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

VIETNAM NATIONAL UNIVERSITY, HANOI

VNU UNIVERSITY OF SCIENCE

Ngô Thi Thanh Nga

STABILITY AND ROBUST STABILITY

OF SINGULAR LINEAR DIFFERENCE EQUATIONS

Speciality: Dierential and Integral Equations Speciality Code: 62 46 01 03

THESIS FOR THE DEGREE OFDOCTOR OF PHYLOSOPHY IN MATHEMATICS

Supervisors: ASSOC PROF DR HABIL VŨ HOÀNG LINH

and PROF DR NGUYỄN HỮU DƯ

HANOI  2018

ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN

Ngô Thị Thanh Nga

TÍNH ỔN ĐỊNH VÀ ỔN ĐỊNH VỮNG CỦA

PHƯƠNG TRÌNH SAI PHÂN TUYẾN TÍNH SUY BIẾN

Chuyên ngành: Phương trình Vi phân và Tích phân

Mã số: 62 46 01 03

LUẬN ÁN TIẾN SĨ TOÁN HỌC

Người hướng dẫn khoa học:

PGS.TSKH VŨ HOÀNG LINH GS.TS NGUYỄN HỮU DƯ

HÀ NỘI – 2018

This work has been completed at the Faculty of Mathematics, Mechanics andInformatics, University of Science, Vietnam National University, Hanoi, underthe supervision of Assoc.Prof.Dr.habil Vu Hoang Linh and Prof.Dr NguyenHuu Du I hereby declare that the results presented in the thesis are new andhave never been published fully or partially in any other thesis/work

Author: Ngô Thị Thanh Nga

im-in research and study.

I would like to express sincere thanks to Assoc.Prof.Dr Lê Văn Hiện and Dr.Nguyễn Trung Hiếu for their useful comments and suggestions that led to theimprovement of the thesis I would also like to thank Dr Đỗ Đức Thuận for hiscollaboration in research My deepest appreciation goes to Prof Phạm Kỳ Anhand other members of "Seminar on Computational and Applied Mathematics",and also to the members of "Seminar on Differential Equations and DynamicalSystems" at the Faculty of Mathematics, Mechanics and Informatics, VNUUniversity of Science, Hanoi, for their valuable comments and discussions

I am grateful to my parents, brother, my beloved daughters, my husband andother members in my big family, who have provided me moral and emotionalsupport throughout my life

A very special gratitude goes to all Thang Long University, National dation for Science and Technology Development, the MOET project 911 forproviding the funding for me in the period of my study

Foun-Last but not least, I would like to thank my colleagues in Thang LongUniversity, the staffs of Vietnam Institute for Advanced Study in Mathematics,

my friends, and many other people beside me for their love, motivation andconstant guidance

Thanks all for your love and support!

This work is concerned with linear singular dierence equations (LSDEs) of

rst order and second order For LSDEs of rst order, by using the based approach we characterize the stability of the system under perturba-tions and establish the relation between the boundedness of solutions of non-homogeneous systems and the exponential/ uniform stability of the correspond-ing homogeneous systems We also extend the concept of Bohl exponent fromregular dierence equations to LSDEs and investigate its properties

projector-For LSDEs of second-order, we use the strangeness-index approach der the strangeness-free assumption we investigate the solvability of IVPs, theconsistency of initial conditions, and the relation between the solution sets

Un-of the systems and those Un-of the associated reduced regular systems By acomparison principle, some exponential stability criteria are obtained A Bohl-Perron-type theorem is also given to characterize the input-solution relation

of non-homogeneous equations Finally, the problem of robust stability underrestricted structured perturbations is investigated Also using the comparisonprinciple, an explicit bound for perturbations under which the systems preservetheir exponential stability is obtained

Tóm tắt

Trong công trình này chúng tôi nghiên cứu về phương trình sai phân suybiến tuyến tính cấp một và cấp hai Đối với phương trình sai phân suy biếntuyến tính cấp 1, chúng tôi sử dụng cách tiếp cận bằng phép chiếu và đưa rađược các kết quả như: đặc trưng hóa tính ổn định của hệ dưới tác động củanhiễu; thiết lập mối quan hệ giữa tính ổn định mũ/ ổn định đều của hệ thuầnnhất và tính chất nghiệm của hệ không thuần nhất; mở rộng khái niệm số mũBohl cho hệ sai phân suy biến và chỉ ra một số tính chất

Đối với phương trình sai phân suy biến cấp hai, chúng tôi sử dụng cách tiếpcận dùng chỉ số lạ Dưới giả thiết chỉ số lạ bằng không, chúng tôi nghiên cứutính giải được của bài toán giá trị ban đầu và các điều kiện đầu tương thích,mối quan hệ giữa tập nghiệm của hệ ban đầu và tập nghiệm của hệ được đưa

về dạng chính quy Bằng cách sử dụng nguyên lý so sánh, tiêu chuẩn cho sự ổnđịnh mũ được thiết lập Một định lý dạng Bohl-Perron được đưa ra nhằm đặctrưng mối quan hệ đầu vào-nghiệm của hệ không thuần nhất Cuối cùng, bàitoán về tính ổn định vững dưới tác động của nhiễu có cấu trúc được chỉ ra.Tiếp tục sử dụng nguyên lý so sánh một lần nữa, chúng tôi đưa ra được mộtchặn trên cho nhiễu để hệ bị nhiễu vẫn bảo toàn được chỉ số cũng như tính ổnđịnh mũ

List of Notations

R (C) the set of real (complex) numbers

N the set of natural numbers

N(n 0 ) the set of integers that are greater than or equal to a given integer n 0

K R or C

Cd the d− dimensional complex vector space

Rd the d− dimensional Euclidean space

Cd,d the set of d × d matrices with entries in C

GL(Kd) the set of d × d invertible matrices with entries in K

kerE the kernel space of E

imE the image space of E

rank E the rank of matrix E

kxk norm of vector x

k∆k norm of matrix ∆

B(0, 1) unit disk on the complex plane

det A the determinant of matrix A

A the matrix tuple (A, B, C)

rK the (structured) stability radius

AH the conjugate transpose of matrix A

l p (n 0 ) the space of sequences {q n } n>n 0 ⊂ K d such that P

n>n 0

kq n k p < ∞, p > 1 ρ(A) spectral radius of matrix A

diag(σ 1 , · · · , σ p ) the matrix in C m,n whose ii entry is σ i for any i = 1, , p and the others are zero

u > 0 each component of vector u is positive

u > v means that u − v > 0

ond order 221.3 Further auxiliary results 25

Chapter 2 Singular systems of rst-order dierence equations 28

2.1 Stability notions for singular dierence equations 28

2.2 Stability of perturbed equations 32

2.2.1 The case of one-sided perturbation 33

2.2.2 The case of two-sided perturbation 38

2.3 Bohl-Perron-type stability theorems 41

2.3.1 Boundedness of solutions of nonhomogenous equations 41 2.3.2 Bohl-Perron-type theorems 46

2.4 Bohl exponents and exponential stability 55

2.4.1 Bohl exponents and their basic properties 55

2.4.2 Robustness of Bohl exponents 60

2.5 The case of unbounded canonical projector function 65

2.5.1 Uniform stability and exponential stability of perturbed equations 65

2.5.2 Bolh exponent of solutions and Bohl exponent of the system 67 2.6 Conclusion 71

Chapter 3 Singular systems of second-order dierence equations 72 3.1 Initial value problems 72

3.2 Exponential stability 83

3.2.1 Notion of exponential stability 83

3.2.2 Criteria for exponential stability 85

3.2.3 Bohl-Perron theorem 91

3.3 Robust stability 94

3.4 Conclusion 109

Conclusion 111

The author's publications related to the thesis 113

The evolution of certain phenomena in real-world over the course of time

is usually described by dierential and dierence equations In discrete-timescale, mathematical models lead to dierence equations Dierence equationsplay important roles in many areas such as control, biology, economics, Typ-ically, dierence equations can be described in the form

F (x(n + k), x(n + k − 1), · · · , x(n + 1), x(n)) = 0, (0.1)where k is a xed positive integer; n ∈ N; x : Z −→ Kd and F : Kd × Kd ×

· · · × Kd −→ Kd If k = 1, equation (0.1) is said to be of rst-order Otherwise,equation (0.1) is said to be of high-order If the highest order term x(n + k) issolved from (0.1) for each n, then we have explicit dierence equations, whichare given in the form

x(n + k) = f (x(n + k − 1), · · · , x(n + 1), x(n)) (0.2)Explicit dierence equations and their applications have been extensively in-vestigated in many papers and monographs; see [1], [32] and the referencestherein However, in many situations system (0.1) is not solvable for x(n + k).Then, we say (0.1) is an implicit dierence equation (IDE) or a singular dier-ence equation (SDE) The simplest case of SDEs is linear SDEs of rst-order,which are given by

Eny(n + 1) = Any(n) + qn, n ∈ N(n0), (0.3)where En, An ∈ Kd×d; y(n), qn ∈ Kd, N(n0) denotes the set of integers thatare greater than or equal to a given integer n0, and the leading matrix En may

be singular The homogeneous equation associated with (0.3) is

Enx(n + 1) = Anx(n), n ∈ N(n0) (0.4)

Unlike explicit dierence equations, the analysis of SDEs is more complicated.Even the solvability analysis is not trivial For example, consider the followingsystem

x(n + 1) = x(n) + fn,

0 = gn.Then, it is easy to see that solutions of the initial value problem (IVP) of SDE(0.5) either has innitely many solutions if gn ≡ 0 or otherwise no solution

We give here an example to illustrate applications of singular dierenceequations in practical areas (see Example 1-1.2 in [21])

Example 0.0.1 The fundamental dynamic Leontief model of economic tems is a singular system Its description model is ([52]):

sys-x(k) = Asys-x(k) + B[x(k + 1) − sys-x(k)] + d(k) (0.6)where x(k) is the n dimensional production vector of n sectors; A ∈ Rn×n

is an input-output (or production) matrix; Ax(k) stands for the fraction ofproduction required as input for the current production, B ∈ Rn×n is thecapital coecient matrix, and B[x(k + 1) − x(k)] is the amount for capacityexpansion, which often appears in the form of capital, d(k) is the vector thatincludes demand or consumption Equation (0.6) may be rewritten as

Bx(k + 1) = (I − A + B)x(k) − d(k)

In multisector economic systems, production augmention in one sector oftendoesn't need the investment from all other sectors, and moreover, in practicalcases only a few sectors can oer investment in capital to other sectors Thus,most of the elements in B are zero except for a few B is often singular Inthis sense the system (0.6) is a typical discrete-time singular system

The rst works on SDEs were done by S.L Campbell [17, 18], L Dai [21]and D.G Luenbeger [52, 54] in late 70's and 80's If En and An are constantmatrices, i.e En ≡ E, An ≡ A, then we have linear time-invariant (constant

coecient) SDEs Suppose that the pencil {E, A} is regular, i.e det(λE−A) 6≡

0, then there exists the so-called Weierstrass-Kronecker decomposition Thereare non-singular matrices T and W such that

"

A1 0

0 I

#,

where N is a nilpotent matrix of index r i.e Nr = 0, Nk 6= 0, k < r Wealso say that the index of matrix pencil (E, A) is r By introducing a variablechange xn = W yn and multiplying both sides of (0.3) by T , we obtain

y1(n + 1) = A1y1(n) + q1n,

N y2(n + 1) = y2(n) + q2n.The rst equation of the system is explicit If r = 1 then N ≡ 0, and the secondequation of the system is equivalent to y2(n) = −qn2, which is an algebraicequation In [17, 18], Campbell considered the following class of linear time-invariant SDEs

Ax(n + 1) = Bx(n) + fn, (0.7)where A, B ∈ Cm,m, A is singular For each consistent initial vector c, theIVP of the homogeneous equation, Ax(n + 1) = Bx(n), x(0) = c, n = 1, 2, has a unique solution if and only if there exists a λ ∈ C such that λA + B isnon-singular In this case the unique solution subject to x(0) = c is given by

where ˆfi = (λA−B)−1fi, ˆA = (λA−B)−1A, ˆB = (λA−B)−1A,ind( ˆA) = k,and

q ∈ Cm; ˆAD, ˆBD are Drazin inverses of ˆAand ˆB, respectively (for the denition

of Drazin inverse, see [17]) The existence of solutions was also established forthe discrete control problem

x(k + 1) = Ax(k) + Bu(k), k = 0, N − 1

Simultaneously, some results on linear time-invariant SDEs were applied to theLeontief dynamic model of multisector economy For non-autonomous discretesystem

Akx(k + 1) + Bkx(k) = fk, k > 0, Ak, Bk ∈ Cn×n, (0.8)

we assume that rank Ak ≡ r for all k Then, multiplying both sides of (0.8) by

an invertible matrix Pk, we obtain

"

Tk0

#x(k + 1) +

"

Ck

Dk

#x(k) =

invertible, the second one is the case where rank

"

Tk

Dk+1

#

= rank(Tk) for all

k The method used by Campbell is a slight extension of that employed byLuenberger [52, 54]

Recently, by adopting the projector-based analysis (developed by März et.al.since 80's) or the strangeness-free form formulation (Kunkel and Mehrmannsince 90's) for DAEs, some results on the existence and stability of solutionshave been obtained for SDEs by Ky Anh et.al [2, 3, 4, 5, 6] and by Brüll in[13, 14], respectively First, a denition of index-1 was extended to a class ofnonlinear SDEs in [5],

fn(x(n + 1), x(n)) = 0, where n > 0, x(n) ∈ Rd, fn : Rd× Rd −→ Rd

then the solvability of IVPs for nonlinear SDEs was investigated The Floquettheory for SDEs was developed in [6] It was proved that any index-1 SDEscould be transformed into its Kronecker normal form, then Floquet theorem

on the representation of the fundamental matrix of index-1 periodic SDEs hasbeen established As a consequence, the Lyapunov reduction theorem wasproved In [4], the Lyapunov function method was applied to study stability

of singular quasi-linear dierence equations of the following form

Anx(n + 1) + Bnx(n) = fn(x(n)), n > 0, x(n) ∈ Rd,where An, Bn ∈ Rd,d, fn(x) is a given Lipschitz continuous functions with asuciently small Lipschitz coecient, fn : Rd× Rd −→ Rd Recently, in [59],Nga derived conditions under which the asymptotic behavior of solutions ispreserved for perturbed linear SDEs This can be regarded as the rst attempt

of extending the stability results for regular dierence equations in [1, 32] toSDEs

In the last decade, interest in SDEs and their applications was continuedintensively by other research groups, as well There have been a number ofpapers that are closely related to the topic of this thesis, e.g., see [7, 33, 34, 35,

44, 45, 46, 57, 58, 63, 74, 75] Particular attention has been paid to stabilityand robust stability of singular discrete-time systems with or without delay,see [7, 33, 34, 44, 58, 63, 74, 75] For example, in [7] the authors investigatedthe stability of linear switched singular systems, which can be considered as aspecial class of linear time-varying singular systems

At the beginning of the 20th century, Bohl, and later Perron, proved thatthe bounded input - bounded state (also called Perron's property) of a non-homogeneous ordinary dierential equation (under some assumptions on thecoecient) implies the exponential stability of the corresponding homogeneousequation and vice versa More generally, instead of the boundedness property,

we can take a pair of appropriate Banach spaces B1 and B2 and consider the

B1-input - B2-state property (also called Perron's property), i.e., for any inputbelonging to B1, there exists a solution belonging to B2 This characterizationhas been generalized to various choices of space pairs and dierent kinds ofequations For example, Bohl-Perron-type stability theorems were formulatedfor ordinary dierential equations in Banach spaces in [22], for dierence equa-tions in Banach spaces in [8, 70], for dierence equations with delay in [9, 12]

In [8], Aulbach and Van Minh considered a dierence equation in a Banachspace

xn+1 = Anxn+ fn, (0.10)where sup

n

kAnk < ∞ It was proved that if the solution belongs to lp, 1 6 p 6

∞, for any sequence fn belonging to the same space, then all solutions of

xn+1 = Anxn (0.11)decay exponentially It was also proved that if for any sequence fn in l1 thecorresponding solution of (0.10) is bounded, then the zero solution of (0.11) isuniformly stable In [70], M Pituk improved the above results by showing that

if for any fn from lp, 1 < p < ∞, the corresponding solution of (0.10) belongs to

l∞ then the trivial solution of (0.11) is exponentially stable L Berezansky and

E Braverman proposed similar Bohl-Perron-type theorems for delay dierence

equations of the form

an analogous result for DAEs

Besides the problems of stability, researchers are interested in the robuststability problems The reason is that in modeling of real-life phenomena,uncertainties such as modeling errors due to simplifying assumptions, dataerror, etc arise Thus, the question whether a system preserves stability undersmall perturbations is very important for simulation and control problem Thedistance to instability can be characterized by the so-called stability radii,which was formulated in seminal works by Hinrichsen and Pitchard [40, 41].This problem is stated as follows Given a linear continuous-time system

˙x = Ax, A ∈ Km×m (0.13)Let us suppose that it is asymptotically stable Together with (0.13), consider

a perturbed system

˙x = (A + B∆C)x, (0.14)where ∆ is an uncertain perturbation, B and C are matrices that describe thestructure of the perturbation We dene the stability radius by

Finding an explicit formula for the real stability radius is more dicult In

1995, it was solved by Qiu et al [72] Stability radii for linear time systems are dened analogously Since 90's, the problem of stability radiihas been investigated for various systems, such as positive systems [64], delaysystems and higher-order systems [42, 65, 66] This problem was also stated in

discrete-more general settings, for example, Son and Thuan [67, 68] solved the problem

of a surjectivity radius for a linear operator, which implies stability radii results.Another extension is the robust stability of singular systems, i.e DAEs andSDEs Byers and Nichols [16], Qiu and Davison [71] proposed explicit formulasfor the complex stability radius of the DAEs of the form

E ˙x = Ax,where E is a singular matrix and the pencil {E, A} is supposed to has index-

1 Du considered higher-index DAEs and made some extensions to positivesystems [23] The behavior of complex stability radius for singularly perturbedDAEs was also analyzed by Du and Linh [24, 25] Generally, the problem ofrobust stability of DAEs is more complicated than that of ODEs due to thesingularity of E To this problem, the structure of the pencil {E, A} plays animportant role, which should be taken into account in the problem of stabilityanalysis

In [31], the authors proposed a formula for complex stability radius of lar systems of dierence equations As an application, they characterized theasymptotic behavior of stability radius of the system resulted by discretizing

singu-a DAE Extensions of these results to the higher-order singulsingu-ar systems singu-andsingular dynamic systems have been done recently in [37, 56] Extending suchresults for time-invariant systems to time-varying systems is much more dif-

cult Since asymptotic/exponential stability of a linear time-varying systemcannot be characterized by the spectrum of the coecient matrices, the ap-proaches developed for linear time-invariant systems are no longer applicable

By using novel results in the operator theory, Jacob succeeded in proving anexact formula for the stability radius of linear time-varying ODEs [43] In thespirit of Jacob's result, in [19, 26] Du and Linh extended the analysis to lineartime-varying DAEs An analogous result was obtained for singular systems

of linear time-varying dierence equations in [61] Recently, Mehrmann andThuan [56] characterized the stability radii of singular systems of higher-orderdierence equations by using an approach that was used in previous work fordelay DAEs in [29] To our knowledge, stability and robust stability of singularsystems of linear time-varying higher-order dierence equations have not beendiscussed in literature

The rst main aim of our work is to extend some existing stability results for

ODEs to linear SDEs of the forms (0.3) and (0.4) These results complementthose in [4, 5, 61] They can also be considered as the discrete-time analogues

of some recent results for DAEs, see [10, 11, 19, 49, 50] For the rigorousproofs of main results, we have to overcome the diculties that are causedsimultaneously by the singularity and the discrete-time nature of the systems

To the best of our knowledge, this research is the rst work that uses theconcept of Bohl exponent [15] to characterize exponential stability and robustexponential stability of singular discrete-time systems Furthermore, unlike theproblem formulation in [19, 61], here we consider a general class of allowablestructured perturbations arising in both the coecients of system (0.4) Wealso extend Bohl-Perron-type stability theorems in [8, 70] from regular explicitdierence equations to SDEs (0.3)

Another main aim of this thesis is to study solvability, exponential stability,and robust stability of linear time-varying SDEs of the form

Anx(n + 2) + Bnx(n + 1) + Cnx(n) = fn, n = n0, n0+ 1, , (0.15)where n0 ∈ N and coecients An, Bn, Cn ∈ Cd,d

, n > n0 The leading cient Anis supposed to be singular with rank An ≡ d1 < dfor all n > n0 Underthe strangeness-free assumption, we propose an explicit construction of the so-called consistent initial conditions by which the IVP admits a unique solution

coe-We also show a relation between SDE (0.15) and a uniquely determined explicitdierence equation Combining this characterization and a comparison tech-nique, which is similar to that of [60], we obtain exponential stability criteriafor SDEs (0.15) Exponential stability is also characterized by a Bohl-Perron-type theorem Next, we make use of a recent result for linear time-invariantSDEs in [56] and obtain bounds for robust stability of SDEs (0.15) when thecoecients are subject to structured perturbations The approach proposed

in this thesis can be extended to higher-order SDEs and similar results can

be obtained However, for the sake of simplicity, in this work, we restrict theinvestigation to second-order SDEs The transformation into strangeness-freeform based on matrix decompositions plays a key role in our analysis Up to ourknowledge, this is the rst work addressing the stability and robust stability

of linear time-varying singular dierence equations of higher-order

The thesis is organized as follows

• In the rst chapter, we recall the tractability-index notion and a decoupling

technique for linear singular systems of index-1 by using projectors Then,the strangeness-free index notion is introduced and a relation between twoindex notions is discussed We also mention some results on stability andstability radii in [56] for second-order linear time-invariant systems.

• In the second chapter, we study the preservation of uniform stability andexponential stability when the coecients of system (1.2) are subject toperturbations We also present Bohl-Perron-type theorems that establish

a relation between exponential/uniform stability of homogeneous system(0.4) and the boundedness of solutions to non-homogeneous system (0.3).Next, we give the notion of Bohl exponents for linear singular systems (1.2)and analyze its properties including sensitivity to perturbations occurring

in the system coecients We also discuss the case of unbounded canonicalprojector

• In the third chapter, we investigate the solvability of SDE (0.15) Then,

we introduce the notion of exponential stability for homogeneous SDEsassociated with (0.15) and establish some criteria for exponential stability.Next, we consider SDE (0.15) subject to structured perturbations andobtain a bound for the perturbations such that the exponential stability

2 Vu Hoang Linh, Ngo Thi Thanh Nga, BohlPerron Type Stability rems for Linear Singular Dierence Equations, Vietnam J Math (2017).https://doi.org/10.1007/s10013-017-0245-z (ESCI, Scopus)

Theo-3 Vu Hoang Linh, Ngo Thi Thanh Nga, Do Duc Thuan, Exponential stabilityand robust stability for linear time-varying singular systems of second-order dierence equations, SIAM J Matrix Anal Appl., 39-1 (2018), 204-233.(SCI)

and also presented at the following conferences and seminars

1 The 2nd PPICTA, Dynamical Systems Session, November 13-17, 2017,Busan, Korea

2 "Vietnam-Korea Joint Conference on Selected Topics in Mathematics",February 20-24, 2017, Da Nang, Vietnam

3 "Vietnam-Korea Joint Workshop on Dynamical Systems and Related ics", March, 2016, Vietnam Institute for Advanced Study in Mathematics

Top-4 Conference on Mathematics, Mechanics and Informatics, VNU University

of Science, Hanoi, 2014 and 2016

5 Seminar on Computational and Applied Mathematics, Faculty of ematics, Mechanics and Informatics, VNU University of Science, Hanoi,2014-2017

Math-6 Seminar on Dierential Equations and Dynamical Systems, Faculty ofMathematics, Mechanics and Informatics, VNU University of Science,Hanoi, 2014-2017

Chapter 1

Preliminaries

In this chapter, we introduce some fundamental notions on singular lineardierence equations and other auxiliary results which will be used in nextchapters

1.1 Linear singular dierence equations by tractability-index

approach

In this section, we recall the index notion and a decoupling technique forlinear singular systems of index-1 by using appropriate projectors This can beconsidered as a discrete analogue of projector approach for DAEs [48] Considerthe following linear singular dierence equation (LSDE)

Eny(n + 1) = Any(n) + qn, n ∈ N(n0), (1.1)where En, An ∈ Kd,d and qn ∈ Kd The homogeneous system associated with(1.1) is given by

Enx(n + 1) = Anx(n), n ∈ N(n0) (1.2)

1.1.1 Denition of index-1 systems and their properties

Consider singular system (1.1) Denote Nn := kerEn and let Qn be a jection onto Nn Put Pn := I − Qn Let Tn ∈ GL(Kd) (n ≥ n0+ 1) be suchthat Tn|Nn is an isomorphism between Nn and Nn−1 We introduce following

pro-matrices and subspaces associating with (1.1)

Gn := En−AnTnQn (n ≥ n0+1), Sn := {z ∈ Kd : Anz ∈ ImEn} (n ≥ n0)

We have the following lemma

Lemma 1.1.1 ([3, Lemma 2.3]) The following assertions are equivalent forany n ∈ N(n0):

(i) The matrix Gn := En − AnTnQn is nonsingular;

(ii) Nn−1⊕ Sn = Kd;

(iii) Nn−1∩ Sn = {0}

Proof of Lemma 1.1.1 is available in Appendix

By virtue of Lemma 1.1.1, we can dene LSDEs of tractability index-1 (see[3, Denition 2.2])

Denition 1.1.2 The LSDE (1.1) is said to be of tractability index-1 (index-1for short) if for all n ∈ N(n0+ 1), the following two conditions hold

(i) rankEn = r(constant);

(ii) Nn−1∩ Sn = {0}

Hereafter, we always assume that dim Sn 0 = r and let En 0 −1 ∈ Kd,d be a

xed matrix satisfying the relation Kd = Sn0 ⊕kerEn 0 −1 Thus, condition (ii)

in Denition 1.1.2 holds for all n ∈ N(n0) and the operators Tn as well asmatrices Gn are dened for all n ∈ N(n0)

Lemma 1.1.3 ([3]) Suppose the LSDE (1.1) is of index-1 and Qn are arbitraryprojections onto Nn, n ≥ n0 Then, the following relations hold:

(i) Pn = G−1n En, where Pn := I − Qn; (1.3)(ii) PnG−1n An = PnG−1n AnPn−1; QnG−1n An = QnG−1n AnPn−1− Tn−1Qn−1;

(1.4)(iii) Qen−1 := −TnQnG−1n An is the projector onto Nn−1 along Sn (1.5)Proof of Lemma 1.1.3 is available in Appendix

Due to (iii), the projector Qen−1 dened in Lemma 1.1.3 is uniquely mined, i.e., it does not depend on the choice of Qn and Tn Thus, the corre-sponding projectorPen := I − eQn is also unique They together form a canonicalprojector pair We have some properties involving the canonical projector pair

deter-as follows

Lemma 1.1.4 The matrices PenG−1n and TnQnG−1n are independent of thechoice of Tn and Qn

Proof of Lemma 1.1.4 is available in Appendix

As a consequence of Lemma 1.1.4, it follows immediately that the matricese

PnGe−1n and TnQenGe−1n are independent of the choice of Tn Here, the sponding scaling matrix Gen := En − AnTnQen is set

corre-1.1.2 Solutions of Cauchy problem

In this section, we briey present a decoupling technique for index-1 LSDEs

By virtue of Lemma 1.1.1, we see that the matrices Gn are nonsingular forall n ≥ n0 Hence, multiplying (1.1) by PnG−1n and QnG−1n , respectively, andapplying formulas(1.3)-(1.4) in Lemma 1.1.3 we decouple the index-1 LSDEinto the following system

Pny(n + 1) = PnG−1n AnPn−1y(n) + PnG−1n qn, (1.6)

0 = QnG−1n Any(n) + QnG−1n qn (1.7)Multiplying both sides of equation (1.7) by Tn and using the second equality

in (1.4), this equation is rewritten as

Qn−1y(n) = − eQn−1Pn−1y(n) + TnQnG−1n qn (1.8)Thus, solution y(n) is decomposed as a sum of two components Pn−1y(n) and

Qn−1y(n), where the dynamic component Pn−1y(n) is governed by equation(1.6), while the algebraic component is determined by algebraic equation(1.8) Inspired by this decoupling procedure, we formulate the correctly statedinitial condition for index-1 LSDE (1.1) as

Pn0−1(y(n0) − y0) = 0, y0 ∈ Kd is arbitrarily given (1.9)Therefore, the Cauchy problem (1.1)-(1.9) has a unique solution dened onN(n0) [3]

Remark 1.1.5 The initial condition (1.9) is actually independent of the choice

of Pn 0 −1 A given initial vector y0 is said to be consistent with LSDE (1.1) ife

Qn0−1y0 = Tn0Qen0Ge−1n0qn0 Then, the Cauchy problem for (1.1) with consistentinitial condition y(n0) = y0 admits a unique solution

Next, we consider homogeneous equation (1.2), where qn ≡ 0, n ∈ N(n0).Let us dene z(n) = Pn−1x(n) The regular ordinary dierence equation

e

Pny(n + 1) = PenGe−1n AnPen−1y(n) + ePnGe−1n qn for all n ∈ N(n0),

e

Qn−1y(n) = TnQenGe−1n qn (1.11)From (1.11), it is easy to see that (1.2) is equivalent to

By using (1.11) and the constant-variation formula for inhomogeneous ular dierence equations, any solution y(·) of (1.1) can be expressed by

1.2.1 Denition of strangeness index and Brüll's results

Now we introduce the denition of strangeness index for discrete-time casewhich was rst constructed by T Brüll's in [13]

Consider a linear time-varying discrete-time descriptor system

Ekx(k + 1) = Akx(k) + fk, x(n0) = ˆx, k ∈ N(n0), (1.15)

where Ek, Ak ∈ Cm,n for k ∈ N(n0), x(k) ∈ Cn for k ∈ N(n0) are state vectors,

fk ∈ Cm are given vectors and ˆx ∈ Cn is an initial condition given at the point

k = n0

Denition 1.2.1 ([13]) Let Ek, Ak, eEk, eAk ∈ Cm,n for k ∈ N(n0) Then, twosequences of matrix pairs {(Ek, Ak)}k∈N(n0) and {(Eek, eAk)}k∈N(n0) are said to beglobally equivalent (on N(n0)) if there exist two pointwise nonsingular matrixsequences {Pk}k∈N(n0) with Pk ∈ Cm,m and {Qk}k∈N(n0) with Qk ∈ Cn,n suchthat PkEkQk+1 = eEk and PkAkQk = eAk for all k ∈ N(n0) We denote thisequivalence by {(Ek, Ak)}k∈N(n0) ∼ {( eEk, eAk)}k∈N(n0)

If {(Ek, Ak)}k∈N(n0) ∼ {( eEk, eAk)}k∈N(n0), we multiply both sides of equation(1.15) by Pk and change variable x(k) = Qkx(k)e Then, we obtain an equivalentequation of (1.15) as

rf = rank(E) (corresponds to forward direction)

rb = rank(A) (corresponds to backward direction)

hf = rank(ZHA) (rank of ZHA; forward)

hb = rank(YHE) (rank of YHE; backward)

= rf + hf − rb,

c = rb− hf (common part)

a = min(hf, n − rf) (algebraic part)

where the last block column has u columns and the last block row has v rows.

We have that either s = 0, u = 0 or s = u = 0 The quantities dened aboveare called local characteristics or local invariants of the matrix pair (E, A).Lemma 1.2.4 ([13]) Consider system (1.15) and introduce the matrix se-quence {Zk}k>n0, where Zk is a basis of corange(Ek) = ker(EkH) for k ∈ N(n0).Let

where all matrices [E(1)

E(2)] have full-row rank of rf

Equation from (1.15) associated with (1.17) can be written as

Ek(1)x(1)(k + 1) + Ek(2)x(2)(k + 1) = A(1)k x(1)(k) + fk(1),

0 = x(k)(2)+ fk(2),

0 = fk(3)for k ∈ N(n0) This system is equivalent to the system given by

Ek(1)x(1)(k + 1) = A(1)k x(1)(k) + efk(1),

0 = x(2)(k) + fk(2),

0 = fk(3)(where efk(1) = fk(1)+ Ek(2)fk+1(2) ) which is connected with the sequence of matrix

We have another important property presented in the following theorem

Theorem 1.2.5 ([13, Theorem 6]) Suppose that the sequences of matrix pairs

Now, returning to equation (1.15) with the sequence of matrix pairs {(Ek, Ak)}k>n0,

we dene a sequence (of sequences of matrix pairs) {{(Ek,i, Ak,i)}k>n0}i∈N by

the following backtrack procedure For the beginner {(Ek,0, Ak,0)}k>n0 :=

{(Ek, Ak)}k>n0 For i ∈ N,

• First, assume that the local invariants of {(Ek,i, Ak,i)}k>n0 are constant

rf,i := rf,ik , hf,i := hkf,i (1.19)

We reduce this sequence by Lemma 1.2.4 to the form (1.17)

{(Ek,i, Ak,i)}k>n0 ∼

char-Lemma 1.2.6 ([13, char-Lemma 7] ) Let the sequences {(rf,i, hf,i)}i∈N and

{{(Ek,i, Ak,i)}k>n0}i∈N be dened as in (1.19) In particular, let the constantrank assumptions (1.16) hold for every {(Ek,i, Ak,i)}k>n0 For all i ∈ N, den-ing the quantities

hf,−1 := 0, si:= rf,i− rf,i+1, vi := m − rf,i− hf,i (1.21)

µ = min{i ∈ N | si = 0} (1.22)the strangeness index of the sequence of matrix pairs {(Ek,i, Ak,i)}k>n0 and

of the associated descriptor system (1.15) In the case that µ = 0, we say{(Ek,i, Ak,i)}k>n0 and (1.15) are strangeness-free

Remark 1.2.8 Consider the sequence of matrix pairs {(Ek,i, Ak,i)}k>n0 withstrangeness index µ We can see that after µ + 1 reduction steps from (1.17)

to (1.18) and µ + 1 equivalence transformations, the sequence of matrix pairs{(Ek,0, Ak,0)}k>n0 can be transformed to a sequence of the form

x(1)(k + 1) = A(1)k x(1)(k) + A(3)k x(3)(k) + fk(1), rf,µ

0 = x(2)(k) + fk(2), hf,µ

where with uµ := n − rf,µ − hf,µ we have x(3)

(k) ∈ Cuµ and each in geneous term f(1)

homo-k , fk(2), fk(3) is determined by the original homogeneous terms

fk, , fk+µ+1 as in (1.15) for all k ∈ N(n0).For the associated forward problem

0 = ˆx(2) = −fn(2)0 are satised.The corresponding initial value problem is uniquely solvable if and only if, inaddition, uµ = 0 holds

In order to get the existence and uniqueness of solutions of initial valueproblems, we give here a denition of strangeness-free systems in the strictsense

Denition 1.2.10 The system

Eny(n + 1) = Any(n) + qn, n ∈ N(n0), (1.1)where En, An ∈ Kd,d, qn ∈ Kd, rank En ≡ r < d, is called a strangeness-freesystem if

{(Ek, Ak)}k>n0 ∼

( "

Ek10

#,

is invertible for all k ∈ N(n0)

1.2.2 The equivalence between two types of index denitions

Consider system (1.1) and suppose that rank En ≡ r < d We explain arelation between two types of index denitions for the sequence {(Ek, Ak)}k>n0(index-1 tractable and strangeness-free) Note that, both index-1 and strangeness-free properties are invariant under global equivalence Thus, in order to get asimpler form, we transform the sequence {(Ek, Ak)}k>n0 as follows

we can conclude that system (1.1) is index-1 tractable if and only if

1.2.3 Linear time-invariant singular dierence equations of second order

In this subsection, we briey review recent results on stability and stabilityradii in [56] for SDEs of second order

Consider the following linear time-invariant implicit dierence equation ofsecond order

Ax(n + 2) + Bx(n + 1) + Cx(n) = fn, n ∈ N(n0) (1.24)where n0 ∈ N and A, B, C ∈ Cd,d

, n > n0 The leading coecient A issupposed to be singular with rank A ≡ d1 < d for all n > n0 We also assumethat the initial conditions

x(n0) = x0, x(n0+ 1) = x1, (1.25)with x0, x1 ∈ Cd, are given The homogeneous equation of (1.24) is

Ax(n + 2) + Bx(n + 1) + Cx(n) = 0, n ∈ N(n0) (1.26)Denition 1.2.11 • A pair of initial conditions x0, x1 is said to be consis-tent with (1.24) if the associated initial value problem (1.24) has at leastone solution

• Equation (1.24) is said to be regular if for any consistent pair of initialconditions, the associated initial value problem (1.24) has a unique solu-tion

• A solution vector xe ∈ Cn is said to be an asymptotic equilibrium of (1.24)

Note that, for time-invariant equations, these two types of stability totic stability and exponential stability) are equivalent

(asymp-Set

P (λ) := λ2A + λB + C

and denote the set of nite roots of P by

σ(P ) = {λ ∈ C | det(P (λ)) = 0}

Theorem 1.2.13 ([56, Theorem 2.4]) Assume that (1.26) is regular and thatthe initial conditions are consistent Then, the following statements are equiv-alent

(i) Equation (1.26) is asymptotically stable

(ii) σ(P ) ⊂ B(0, 1), where B(0, 1) is the unit disk on the complex plane

According to [55], for any matrix tuple (A, B, C), there exists a non-singularmatrix W ∈ Cd,d such that

C(3) has full-row rank

Denition 1.2.14 Equation (1.26) is said to be strangeness-free if there exists

a non-singular matrix W ∈ Cd,d, that transforms the matrix tuple (A, B, C) tothe form (1.27), such that ˆA in (1.28) is invertible

Suppose that system (1.26) is asymptotically stable and consider a perturbedequation of (1.26) described by

eAx(n + 2) + eBx(n + 1) + eCx(n) = 0, n > n0, (1.29)where

e

A = A + D1∆1E, eB = B + D2∆2E, eC = C + D3∆3E,

∆i ∈ Cli,q, i = 1, 2, 3, are perturbations and Di ∈ Cd,li

, E ∈ Cq,d, i = 1, 2, 3,are matrices that restrict the structure of the perturbations

l = l1+ l2+ l3 and consider the set of destabilizing perturbations

VK= {∆ ∈ Kl,q|(1.29) is nonregular or not asymptotically stable }

We dene the structured stability radius of (1.26) subject to structured bations (1.29) as

pertur-rK(A; D, E) = inf{k∆k : ∆ ∈ VK}, (1.31)where A refers to the matrix tuple (A, B, C)

Denition 1.2.15 Consider a strangeness-free equation (1.26) and let W ∈

Cd,d be such that (1.27) holds A structured perturbation as in (1.29) is said to

be allowable if (1.29) is still strangeness-free with the same triple (d1, d2, d3),i.e., there exists a sequence of unitary matrices fW ∈ Cd,d such that

B(2)0

C(2)e

B(2)e

D(s) :=h s2D1 sD2 D3 i, H(s) := EP (s)−1D(s),

we have the following proposition

Proposition 1.2.16 ([56, Proposition 4.4]) Consider an asymptotically stablesystem of the form (1.26) If the system is strangeness-free and subjected tostructured perturbation as in (1.29) with structure matrices Di satisfying (1.34)and if the perturbation ∆ satises

k∆k < kH(∞)k−1 = ( lim

s→∞kH(s)k)−1,then the structured perturbation is allowable, i.e., the perturbed equation (1.29)

is strangeness-free with the same block sizes d1, d2, d3

Let

b

A−1 =h M1 M2 M3 i

We have the following result for the case of positive equations

Theorem 1.2.17 ([56, Theorem 4.6]) Let (1.26) be strangeness-free and itive Assume that (1.26) is asymptotically stable and subjected to structuredperturbations as in (1.29) with E > 0 and MiD(i)j > 0 for all i, j = 1, 2, 3.Then, we have

pos-rC(A; D, E) = rR(A; D, E) = 1

max{kH(1)k, kH(∞)k}.Furthermore, if the structure matrices Disatisfy (1.34) and k∆k < rC(A; D, E),then (1.29) is strangeness-free with the same block sizes d1, d2, d3 as for (1.26).1.3 Further auxiliary results

The following auxiliary lemma (also known as the discrete Gronwall lemma)will be useful in the estimation of solutions of SDEs

Lemma 1.3.1 ([1, Corollary 4.1.2]) Let p, q be non-negative real numbers and{u(n)} and {f(n)} be non-negative sequences for all n ∈ N(n0), where n0 ∈ N

is given Suppose that

u(n) 6 p + q

n−1

X

l=n 0

f (l)u(l)for all n ∈ N(n0)

Then, we have the estimate

(I + F N )−1 = I − F (I + N F )−1N

Proof of Lemma 1.3.2 is available in Appendix

We give here the principle that will be used several times in two next ters It is a special case of the uniformly boundedness principle (see [62]).Theorem 1.3.3 Let X and Y be Banach spaces Let Γ be a collection ofcontinuous linear mappings from X into Y and suppose that

p (1.35)

Singular value decomposition (SVD) of matrices is a well-known tool innumerical linear algebra, see [69].

Theorem 1.3.5 Let A ∈ Cm,n Then, there exist a unitary matrix U ∈ Cm,m

and a unitary matrix V ∈ Cn,n such that

A = Udiag(σ1, · · · , σp)V,where σ1 > · · · > σp > 0, p = min(m, n) If A is real, then U and V may betaken to be real orthogonal matrices Such values σ1, · · · , σp are called singularvalues

Note that diag(σ1, · · · , σp) is a matrix in Cm,n whose ii entry is σi for any

i = 1, , p and the others are zero

is introduced and we characterize the relation between the exponential ity and Bohl exponent Finally, robustness of Bohl exponent with respect toallowable perturbations is investigated This chapter is written on the basis oftwo papers [1] and [2] in the list of the publications used in this thesis.

stabil-2.1 Stability notions for singular dierence equations

From now on, we always suppose that the linear SDE (1.2) has index-1 andits Cauchy operator Φ(n, m) is dened in Subsection 1.1.2 The followingstability notions generalize those for ODEs See also [4, 6]

Denition 2.1.1 The zero solution of equation (1.2) is said to be stable if for

any ε > 0 and n1 ∈ N(n0) there exists a positive constant δ = δ(ε, n1) suchthat the inequality kPen1−1x1k < δ implies kx(k)k < ε for all k ∈ N(n1), wherex(·) is the solution of (1.2) satisfying Pen1−1(x(n1) − x1) = 0.

Denition 2.1.2 The zero solution of equation (1.2) is said to be uniformlystable if it is stable and the constant δ mentioned in Denition 2.1.1 is inde-pendent of n1

Denition 2.1.3 The zero solution of equation (1.2) is said to be cally stable if it is stable and limk→∞kx(k)k = 0, where x(·) is the solution of(1.2) with Pen1−1(x(n1) − x1) = 0

asymptoti-If the zero solution of equation (1.2) is stable (resp uniformly stable, totically stable), then we say the equation (1.2) is stable (resp uniformlystable, asymptotically stable)

asymp-Denition 2.1.4 Equation (1.2) is said to be exponentially stable if there existconstants K > 0 and 0 < ω < 1 such that kx(n)k 6 Kωn−mk ePm−1x(m)k =

Kωn−mkx(m)k, n, m ∈ N(n0), n > m, for every solution x(·) of (1.2)

The following characterizations of uniform stability and exponential ity are straightforward generalizations of the well-known results for ordinarydierence equations, see [1, 32]

stabil-Theorem 2.1.5 Suppose that the projector function {Pem} is bounded Then,equation (1.2) is uniformly stable if and only if there exists a constant C > 0such that

kΦ(n, m)k 6 C for all n, m ∈ N(n0), n > m (2.1)Proof The proof contains two parts

Necessity Since equation (1.2) is uniformly stable, there exists a positiveconstant δ such that, for any n1 ∈ N(n0), if kPen1−1x1k < δ, then kx(k)k < 1 forall k ∈ N(n1), where x(k) is the solution of equation (1.2) with Pen1−1(x(n1) −

x1) = 0(we have chosen ε = 1 ) We note that Φ(k, n1) = Φ(k, n1) ePn1−1 then,x(k) = Φ(k, n1)x(n1) = Φ(k, n1) ePn1−1x(n1) = Φ(k, n1) ePn1−1x1 = Φ(k, n1)x1

On the other hand, for an arbitrary vector a ∈ Kd, kak = 1, it is easy to seethat if

x1 = δ

2 sup

m∈N(n )

k ePmka