Luận văn stability and robust stability of singular linear difference equations

VIETNAM NATIONAL UNIVERSITY, HANOI VNU UNIVERSITY OF SCLENCE Ngô Thị Thanh Nga STABILITY AND ROBUST STABILITY OF SINGULAR LINEAR DIFFERENCE EQUATIONS Speciality: Differential and Inte

VIETNAM NATIONAL UNIVERSITY, HANOL VNU UNIVERSITY OF SCIENCE

Ngô Thị Thanh Nga

STAHBILITY AND ROBUST STABILTTY

OF SINGULAR, LINEAR DIFFERENCE EQUATIONS

THESIS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

HANOI - 2018

VIETNAM NATIONAL UNIVERSITY, HANOI

VNU UNIVERSITY OF SCLENCE

Ngô Thị Thanh Nga

STABILITY AND ROBUST STABILITY

OF SINGULAR LINEAR DIFFERENCE EQUATIONS

Speciality: Differential and Integral Equations

Speciality Code: 62 46 01 03

THESIS FOR THE DEGREE OF

DOCTOR OF PHYLOSOPITY IN MATHEMATICS

Supervisors: ASSOC PROF DR HABIL VU HOANG LINH

and PROF DR NGUYEN HUU DU

HANOI - 2018

DAI HOC QUOC GIA HA NOI

TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN

Ngõ Thị Thanh Nga

TINH ON BINH VA ON DINH VUNG CUA

PHƯƠNG TRÌNH SAI PHÂN TUYẾN TÍNH SUY HIẾN

Chuyên ngành: Phương trình Vi phãn và Tích phãn

Mã số: 62 46 01 03

LUẬN ÁN TIẾN SĨ TOÁN HỌC

Người hướng dẫn khoa, học:

PGS.TSKH VŨ HOÀNG LINH

GS.TS NGUYEN HUU DU

HÀ NỘI 2018

Declaration

‘Lhis work has beon completed at the Faculty of Mathematics, Mechanics and Trformaties, University of Science, Vietnam Natioual University, Hanoi, ander the supervision of Assoc.Prof,Dr.habil Vu Hoang Linh and Drof.Dr Nguyen Huu Du I hereby declare that the results presented in the thesis are new and have never been published fully or partially in any other thesis /work

Author; Ngo Thi Thanh Nga

in research and study

T would like to express sincere thanks to Assoc.Prof.Dr, Lé Van Hién and Dr Nenyén Trung Hiéy for their useful comments and suggestions that Ted to the improvement of the thesis 1 would also like to thank Dr D& Dite Thuan for his collaboration in research, My deepest appreciation goes to Prof Pham Ky Anh and other members of "Seminar on Computational and Applicd Mathematics",

and also to the members of "Sentinr on Differential Rquations and Dynamical Systems" at the Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, Hanoi, for their valuable comments and discussions

Tain grateful to my parents, brother, iny beloved daughters, my husband and other members in my hig family, who have provided me moral and emotional support throughout my life

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

Tast but nol Teast, T would Tike to thank any colleagues in ‘Thang Long University, the staffs of Vietnam Institute for Advanced Study in Mathematics,

Abstract

This work is concerned with lincar singular difference equations (L8DEs) of

first order anid second order Far LSDKs of firsk order, by using the projeclar-

based approach we characterize the stability of the system under perturba- tions and establish the relation between the boundedness of solutions of non- homogencous systems and the exponential, uniform stability of the correspond- ing homogeneous systems, We also extend Lhe concept of Bohl exponent from regular difference equations to LSDEs and investigate its properties

For LSDEs of second-order, we use the strangeness-index approach Un-

der the slrangenes+ free assumplion we investigate the solvability of [VPs, the

consistency of initial conditions, and the relation between the solution sets

of the systems and those of the associated reduced regular systems By a

comparison principle, some exponential stability criteria are obtained A Bohl-

Perrotelype Wieorem is also given lo characlerize Lhe itipul-salution relation

of non-hamogeneaus equations Finally, the problem of robust stability under

restricted structured perturbations is investigated Also using the comparison

principle, an explicit bound for perturbations under which the systems preserve their exponential stability is obtained.

công trinh nay chúng tôi nghiên cửu về phương trình sai phân suy

m kính cắp một và cấp hai Đối với phương trình sai phân suy biến

tuyến tính cấp 1, chúng tôi sứ dụng cách tiếp cận bằng phép chiếu và đưa ra

được các kết quả như: đặc trưng hóa tính ấn định của hệ dưới tác động của

m định đều của hệ thuần

: mở rộng khái niệm sổ mũ

nhiễu; thiết lặp mối quan hệ giữa tính ốn định mí

nhất và tính chắt nghiệm của hệ không thuần nhị

Bohl cho hệ sai phãn sny biến và chỉ ra một số tính chất

Đối với phương trình sai phản suy biển cắp hai, chúng töi sử dụng cách tiếp cần dùng chỉ số lạ Dưới giá khiết chỉ số lạ bằng không, phũng tối nghiên cửu tính giải được của bài toán giá trị ban đầu và các điều kiện đẫn tương thích,

mối quan hệ giữa tập nghiệm của hệ ban đầu và tập nghiệm của hệ được đưa

về dang chính quy Bằng cách sử đụng nguyên lý so sánh, tiếu chuẩn cho su én định mũ được thiết lập Một định lý dạng Bobl-Perron được đưa ra nhằm đặc

trưng mỗi quan hệ đầu vào-nghiệm cña hệ không thuần nhất Cuối cùng, bài toán về tính ốn định vững đưới tác động của nhiễu có cấu trúc được chỉ ra

'Liếp tục sử đụng nguyên lý so sánh một lẳn nữa, chúng tôi đưa ra được một chặn trên cho nhiễu để hệ bị nhiễu vẫn bio toan được chỉ số cũng như tính ổn

định mũ.

the image space of £ the rank of matrix: F sora of veelor a norm of maciix &

unit disk on the complex plane

He ddermduaal ofimalzix 4 the matrix tuple (4, 8, ) the (structured) stability radius

1 conjupgie uranspone of ral vis A the space of sequences {gn fmany CK" such that 3° gx” < oc, p> 1 spectral radius of matrix A

1.1 Linear singular difference equations by tractability-index approach 11

11.1 Definition of index-7 sysleris and Lheir praperties 6 11

1.1.2 8olutions of Cauchy problem 18

1.2 Linear singular difference equations by strangeness-index approach 15

1.2.1, Definition of strangeness index and Briill’s results 1ã 1.2.2 The equivalence between two types of index definitions 21

1.2.3 Linear time-invariant singular difference equations af sec-

1.3 Further auxiliary resulis cuc ee 28

Chapter 2 Singular systems of first-order difference equations 28

2.1 Stability notions for singular difference equations 28

2.2 Stability of perturbed equations © 6 ee ee 3

2.2.2 The case of two-sided perturbation 38

2.8 Bobl-Perron-type stability theorems 2.6 ee eee eae 41

2.3.1 Boundedness of solutions of nonhomogenaus equations 41 2.9.2, Bolil-Perron-type theorems 6 ee 46

2.4 Bohl cxponents and exponential gability 55

2.4.1, Bohl exponents and their basie properties 55

2.4.2 Robusrness of Bohl cxponcnts 60

2.5 The case of unbounded canenieal projectorfunetion 65

2.5.1, Uniform stability and exponential stability of perturbed

3.3.1 Notion of eponential stability 8ä

3.2.2 Criteria for exponcntial stability

3.2.3 BohlL-Fcrron therem co 91

Introduction

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

is usually deseribed by differential and difference equations In discrete-time scale, mathematical modcls lead to difference equations Difference equations play imporlani roles in many areas such as conirel, biology, economies Typ ically, difference equations can be described in the form

Ban + By n(n + V)ye-+ a(n + 1), n(n) — 0 (0.1)

where & ix a fixed posilive inleger; a € Nj: % — Ke and Bs KY x KE +» x K¢ —+ K® If & = 1, equation (0.1) is said to be of first-order Otherwise, equation (0.1) is said to be of high-order If the highest order term x(n + È) is solved from (0.1) for cach n, then we have cxplicit differcnee cquations, which are given in the form

a(n bk) — f(xín | K 1), -,z(n 1 1) x(n)) (0.2)

Explieit difference equations ond their applications have been extensively in-

vestigated in many papers and monographs; see [1] [32] and the references Lherein However, in many siluations syslem (0.1) is not solvable far a(n + &) Then, we say (0.1) is an implicit difference equation (IDB) ara singular differ-

(SDE) The simplest case of SDEs is linear SDEs of first-order, which are given by

ence equation

Eny(n +1) — AnV(n) + an € Nữm), (0.3) where Ey, An € K¢4, y(n), gn € K¢, Ning) denotes the set of integers that are greater than or equal to a given integer np, and the leading matrix E, may

be singular The homogencous equation associated with (0.3) is

Unlike explicit difference equations, Lhe analysis of SDEs ix more complicated Even the solvability analysis is not trivial For example, consider the following

Then, it is casy to sce that solutions of the initial valuc problem (IVP) of SDE

(0.5) either has infinilely tmany solutions if g, = 0 or otherwise nọ golnliam

We give here an example to illustrate applications of singular difference equations in practical arcas (sec Example 1-1.2 in [21])

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

x(k) = Ax(k) Ez(kL1)- z(B| atk) (06)

where 2(k) is the n dimensional production vector of n sectors; A & Jin»?

is an input-output (or production) matrix; Az(k) stands for the fraction of production required as inpuk for Whe current production, Ho& i ia the capital coefficient matrix, and Bla(k — 1) — x(k)| is the amount for capacity expansion, which often appears in the form of capital, d(k) is the vector that

includes demand or consumption Equation (0.6) may be rewritten as

2

coefficient) SDF Suppose that the pencil {, A} is regular, ie, del(AB—A) #

0, then there exists the so-called Weierstrass-Kronecker decomposition There are non-singular matrices J and W' such that

where N is a nilpotent matrix of index rie NY — 0,N# 40,8 < 7 We also say that the index of matrix poncil (E, A) is r By introducing a variable change ity — Wa ancl multiplying both sides af (0.3) by 7, we oblain

#(n 11) = Aw'(n) bon

Nin | Y= wn) |g

The first cquation of the system is explicit fr — 1 then N = 0, and the second

equation of the system is equivalent to y(n) = —y2, which is am algebraic

equation In [17,18], Campbell considered the following class of linear Lime: invariant SDEs

where A,K € 0") A is singular For each consistent inilial veelor ¢, (he

IVP of the homogeneous equation, Ax(n — 1} = Bz{n), z(0) = e,n = 1,3,

has a unique solution if and only if there cxists a A.C C such that AA : B is

tran-dingnlar In thấy case the unique soluion subjech ta z(0) = ¢ is given by

g€ 0" AP BP? are Drain inverses of A and A, respec

of Dravin inverse, see [17]] The exislence of soluli

ively (or the definition

s was alse established for

we assume that ranh Ay =r for all k Then, muliiplying both sides of (0.8) by

an invertible matrix F,, we obtain

where rank ?¿ = r,⁄¡ € €"*", 'here are four possibilities but solutions could

Th only be shown in two cases, The frst one is the ease where | ,) * | is

Fi{a(n | 1),2(0)) — 0, where n > 0,2(n) € RY fy REx Rs RE

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

on the representation af the fundamental matrix of index-1 periodic SDFs has been established As a consequence, the Lyapunov reduction theorem was proved, In [4], the Lyapunov function method was applied to study stability

of singular quasi-linear difference cquations of the following form

Auzn 1) Bzr(m = fa@(n)), 2B 0, 2(n) c RY,

Tm the last decade, inlerest in SDEs and their applications was continued intensively by other research groups, as well There have been a number of papers that are closely related to the topic of this thesis, c.g., see [T, 33, 34, 35,

44, 45, 46, 57, 58, 63, 74, 75] Particular attention has becn paid to stability and tobust slabilily of singular discrelelime systems with or wilhouk delay, see [7, 33, 34, 44, 58, 63, 74, 75] For example, in [7] the authors investigated the stability of lincar switched singular systems, which can be considered as a special class of lincar time-varying singular systems

At the beginning of the 20th century, Bohl, and later Perron, proved that the bounded input - bounded state (also called Perron’s property) of a non- homogencous ordinary differential equation [under some assumptions on the coefficient) implies the exponential slabilily of Lhe corresponding hamageneoms equation and vice versa More generally, instead of the boundedness property,

we can take a pair of appropriate Banach spaces By and Bz and consider the

By-input - By-state property (also called Perron’s property}, i.c., for any input

belonging lo By, Uhere exisly a solulion belonging lo Ba ‘Thi charac erizalion has been generalized to various choices of space pairs and different kinds of equations Tor example, Bohl-Perron-type stability theorems were formulated for ordinary differential equations in Banach spaces in [22], for difference cqua- tions in Banach spaces in [8, 70], for difference equations with delay in [9, 12]

Tn |8[, Aulbach and Van Minh considered a difference equation ina Banach space

where sup |A,|| < oc It was proved that if the solution belongs to ),1 <p <

se, for any sequence f,, belonging to the same space, then all solutions af

decay exponentially Tl was also proved thal if for any sequence fi, in dy the corresponding solution of (0.10) is bounded, then the vero solution af (0.11) is uniformly stable In [70], M Pituk improved the above results by showing that

if for any fy from Ip 1 <p < 00, the corresponding solution of (0.10) belongs to

fog then the trivial solution of (0.14) is exponentially slahle 1 Bere:

sky an

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

equations of the form

a

ant 1) = 3) A(m,k}r(k) + Ƒín) (0.12)

& -đ

Annalagons RohlEPerron bheorems were also formulated for differenliabalgebraic

equations in [10, 19] The approach used in those papers is to decouple the system into an ordinary differential equation (ODE) and an algebraic equation, then to apply the conventional Bohl-Perron thcorem for ODEs in order to get

an analogous resull for DAK

Besides the problems of stability, researchers are interested in the robust stability problems The reason is thab in modeling of real-life phenomena, uncerlainties such as modeling error due to simplifying assumptions, dala error, etc arise, Thus, the question whether a system preserves stability under small perturbations is very important for simulation and control problem The distance to instability ean be characterized by the so-called stability radii, which was formulated in seminal works by Hinriehsen and Pilchard [40, 41]

This problem is stated as follows Given a linear continuous-time system

Lel us suppose (hal il is asymptotically slable ‘Together with (0.13), consider

a perlurbed syslem

where A is an uncertain perturbation, B and C are matrices that describe the

slrueture af the perkurbation We define the siabilily radius by

re — inf] All, (0.14) is unstable}

Here || - || is a matrix norm and if K = C(Rres.) thon rg is called the complex

Finding an explicit formula for the real slabilily radius is more diffecull, Tu

4995, it was solved by Qiu et al [72] Stability radii for linear discrete- time systems are defined analogously Since 9's, the problem of stability radii

has been investigated for various systems, such as positive systoms [64], delay

systema and higher-order aysierms [42, 65, 66] ‘Thix problem wax also slated in

§

tore general sellingy, for example, San 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, ic DAEs and SDEs Byers and Nichols [16], Qiu and Davison [71] proposed explicit formulas for the complex slabilily radins of the DAKs of the form

bet — Ax,

where E is a singular matrix and the peneil {E,.A} is supposed to has index-

1 Du considered higher-index DAKs and made some exletisions la positive

systems [23] The behavior of complex stability radius for singularly perturbed DAEs was also analyzed by Du and Linh [24, 25] Generally, the problem of robust stability of DAEs is more complicated than that of ODEs duc to the singularity of # ‘To this problem, the slruelure of the pencil {4, A} plays au important role, which should be taken into account in the problem of stability

analysis

in 81], the authors proposed a formula far complex stabilily radius of singu- lar systems of difference equations As an application, they characterized the asymptotic behavior of stability radius of the system resulted by discretizing

a DAE Extensions of these results to the higher-order singular systems and

singular dynamic systems have been deme recer

in (37, 58] Extending such results for time-invariant systems ta time-varying systems is much more dif ficult Since asymptotic/exponential stability of a linear time-varying system

cannot be characterized by the spectrum of the caefficient matrices, the ap- proaches developed for linear time-invariant systems are no longer applicable

By using novel resulls in the aperalor Leary, Jacob succeeded in proving an

exact formula for the stability radius of linear time-varying ODEs [43] In the spirit of Jacob's result, in [19, 26] Du and Linh extended the analysis to linear time-varying DAEs An analogous result was obtained for singular systems

of linear Lime-varying dilference equations in [61] Recently, Mehrmarm and

Thuan |56| characterized the stability radii of singular systems of higher-order difference equations by using an approach that was used in previous work for delay DAEs in [29] To our knowledge, stability and robust stability of singular systems of linear time-varying higher-order difference equations have not been

discussed in literature

‘The first main aim of our work is to extond some oxisting stability results for

GDEs to linear SEs of the forms (0.3) and (0.4) These resulis complement

those in [4, 5, 61] They can also he considered as the discrete-time analogues

of some recent results for DAEs, sce [10, 11, 19, 49, 50], For the rigorous

proofs of main results, we have to overeome the diffeultics that are eauscd

simullaneously by the singularily and the diserele-Limse nalure of the systems

To the best of our knowledge, this research is the first work that uses the

concept of Bohl cxponent [15] to characterize exponential stability and robust

exponential stability of singular discrete-time systems Furthermore, unlike the

problem formulation in [19, 61], here we consider a general class of allowable

structured perturbations arising in both the coefficients of system (0.4) We

also extend Bohl-Perron-type stability theorems in [8 70] from regular explicit

difference cquations to SDEs (0

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

Ay n+ 2) + Byrle +t Cyrla) = far =a te tdy , (015)

where ny € Nand cocfficients An, Ba Cn € C™, n> ny The leading cocfi- cient Ay is supposed Lo be singular wilh rank Ap = d; < dfor all n> ro Under

the strangeness-free assumption, we propose an explicit construction of the so-

called consistent initial conditions by which the IVP admits a unique solution

Yo also show a relation betwoon SDE (0.15) and a uniquely determined explicit difference equation, Combining this charactorization and a comparison tech- nique, which is similar lo thal of [60|, we oblain exponential slabilily criteria for SDEs (0.15) Exponential stability is also characterized by a Bohl-Perron- type theorem Next, we make usc of a recent result for lincar time-invariant SDEs in [56] and obtain bounds for robust stability of SDEs (0.15) when the

coelliciertss are subjes

lo slruelured 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 the investigation to second-order SDKs 'lhe transformation into strangeness-free form based on matrix decompositions plays a key role in our analysis Up to our

knowledge, lis is Uhe first work addressing Lhe stabilily and robust stabilily

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

The thesis is organized as follows

# In the first chapter, we recall lhe Uraclabilily-index noblion and a decoupling

8

lechnique for linear singular syslems of indexe1 by using projeclars Then,

the strangeness-free index notion is introduced and a relation between two index notions is discussed We also mention some results on stability and

stability radii in [6] for second-order linear time-invariant systems

In the sccond chapter, we study the prescrvation of uniform stability and exponential stability when the coefficients of system (1.2) are subject to perlurbations We alse present, BohlPerronelype Uneoremy thal 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 [he syslem coefficients, We alsa discuss the case of unbounded canonical

projector

In the Uhird chapter, we investigale the solvability of SK (0.15) Then,

we introduce the notion of exponential stability for homogeneous SDEs associated with (0.15) and cstablish some critoria for exponential stability Nexi, we consider SDK (0.15) subject to slruchured perturbations and obtain a bound for the perturbations auch Uhal Uhe exponential stabilily

is preserved

© Conelusions and discussion of fulure works are drawn in the Tash section

Parts of this thesis have been published in

1 Nguyen Hun Du, Vu Hoang Linh and Ngo Thi Thanh Nga (2016), Ow stability and Bohl exponent of linear singular systems of difference equa- tions with variable cocfiicients, J Differ Equations Appl., 22, 1450 1377, (SCIB)

2 Vu Hoang Linh, Ngo Thi Thanh Nga, Bohl Perron Type Stability Theo-

remiss for Linear Singular Difference Kquations, Vietnam J Math (2017)

hblpse {/deviorg/10.1007/s1 0013-01 7-0245-r (KSCI, Scopus)

4 Vụ Hoang Linh, Ngo Thì 'Lhanh Nga, Do Due 'lhuan, Exponential stability and robusi xiabilily for near lime-varying singular systers of second=

order difference equations, STAM J Mairiz: Anal Apph, 39-1 (2018), 204- 233.(8CD)

and also presented al the following conference

"Vietnam-Korea Joint Conference on Selected Topics in Mathematics",

February 20-24, 2017, Da Nang, Vietnam

Gysters and Relaled Top: ics", March, 2016, Vietnam Institute for Advanced Study in Mathematics

Conference on Malhemalics, Mechanics and Informaties, VNU Universily

of Science, Hanoi, 2014 and 2016

| Seminar on Computational and Applied Mathemraties, Facutly of Malth-

ematics, Mechanics and Informatics, VNU University of Science, Hanoi,

where £i,.4, € K and g, C K4 The homogencous system associated with (1.1) ix given by

1.1.1 Dafinition of index-1 systems and thair properties

Consider singular system (1.1) Denote Ny, :— kerE,, and let Q,, be a pro-

jection onto Ny Put Py :-— F—@Qp Let T, € GL(K%) (n > ng + 1) be such

that J4|, is an isomorphism between N, and Ny We introduce following

1

inalrices ard subspaces associaling with (1.1)

Gaim Fy AnTiQn (mờ ng+), Sym fe Ks Ans 6 TU} (a 2 ao)

We have the following lemma

Lemma 1.1.1 ({3, Lemma 2.3]) The following assertions are equivalent for

any n C N(ng):

(i) The mulvie Cy = Ey — AnlnQn is nousingular;

(ii) Nai & S, — Kt;

{ili} NaN Sp — {0}

Proof of Lemma 1.1.1 is available in Appendix

By virtue of Lemma 1.1.1, we can define LSDEs of traetabilily index-1 (sea [3, Definition 2.2])

Definition 1.1.2 ‘The LSM (4.1) is said ba be of traelabilily index-1 (index-1

for short) if for all n € N(no +1), the following two conditions hald

(i) rank, — r{constant);

(lì PGyLAu = PaGRLAuPnSui On@y'An = QnGz*AnPaa- Ta Qn:

(149 (ii) Ont — —TaQn@a'An is the projector onto Np_1 along Sp (1.5)

Proof of Lemma 1.1.3 is available in Appendix

12

Due to (iii), the projector ,1 defined in Lomma 1.1.8 is uniquely deter-

mined, ic it docs not depend on the choice of Q, and ?„ ‘hus, the corre- sponding projector ,

1—§, is also unique They together forma canonical

projector pair We have some properties involving the canonical projector pair

as follows

Lemma 1.1.4 The matrices P,Gz) and In@aC,' are independent of the

choice of Ty 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 matrices

PG) and Tr@,Gz! arc independent of the choice of Ty Here, the corre-

sporiding scaling malrix Gy = Bn — An‘TnGn is sel

1.1.2 Salutians of Cauchy problem

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

By virtue of Lemma 1.1.1, we see that the matrices Gy are nonsingular for all n > no Hence, multiplying (1.1) by HGy! and QaC@,!, respectively, and applying, formulas(1.)-(1.4) in Lemma 1.1.3 we decouple the index-1 L8DE inla the following system

Ly

Balu 1) — PG! Au Pa ual) = Bly lưu, (18)

(Ì = Quan 'Án(n) + QuỚn an al

Multiplying both sides of equation (1.7) by 7 and using the second equality

in (1.4), this equation is rewritten as

Qn-yln) = -On1Paryl2) | TrQnGa an (1.8)

Thus, solution y(n} is decomposed as a sum of two components Py_yy(n) and Qp-ry(n), where the “dynamic” component P,—1y(n) is governed by equation (1.6), while Ghe “algebraic” component is determined by algebraic equation (1.8) Inspired by this decoupling procedure, we farmulate the correctly stated initial condition for index-1 LSDE (1.1) as

Py, 1(g(H0) — m0) — 0, yo € KY is arbitrarily given (1.9)

Therefore, the Cauchy problom (1.1)-(1.9) has a unique solution defined on N(ap) [8]

13

Remark 1.1.5, The iwitind condition (1.9) is actully indepenent of the choiex

of Pag1 A given initial vector yy is said to be consistent with LSDE (1.1) if

Qno-190 = Tno@noGnd Inq» Then, the Cauchy problem for (1.1) with consistent

initial condition y(no) = yo admits a unique solution

Nexl, we cunsider homogeneous equation (1.2), where gp = 0, m € N(ue) Let us define 2(n) — P,12(n) The regular ordinary difference equation

ones By Lemma 1.1.3, equation (1.1) is decoupled as

Py(n tt) — P.GyAnPriy(n) + PrGalgn for all n € N(x), E : an (1.11)

Trom (1.11), ít is easy tơ see that (1.2) is equivalent o

Pttn +l) — B.Gz1A,Paax(n) for all n € N{nv) (142)

Qrizin) = 0

We now construct the Cauchy operator for homogeneous equation (1.2) There

exists a unique matrix function denoted by (B(n,m)})azm satisfying

E,B(n — Lm) — An8(n,m), - P„_¡(8(m,m) — ï) — 0

The matrix funetian (®{n,7n));>m is called the Cauchy operator associated

with LSDE (1.2)

with canonical projector pair P,, Qx, we obtain

By using the decoupled system (1.12), that is constructed

Bin.m)— J] Gy Ag n> m > nạ, and B(n,m) — Pri

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

no

ain) — Bln, m) Fray) 1 Pb | DAG so 1 Tu G14, (118)

for a % m2 ap For this formula, we note thải 3Ÿ — ơn, he second (ern of

the right hand side is equal to zero

Particularly, in the homogeneous case, any salubion t(.) of (1.2) can be

expressed by x(n} — O{n,m)P,,12(m) for n Bm ¥ no

Remark 1.1.6 /f we use an arbitrary non-canonical projector pair Pa, Qn for decoupling purpose, then the Cauchy operator (®(n,m))nz2n Can be alternatively constructed as follows First, the Cauchy operator associated with inherent difference equation (1.10) is defined by

for alin & m % no Due to the fact that B(n,m) = B(n,m)Py1 and the

results of Lemma 1.1.4, formulas (1.13) and {1.14} obviously coincide

1.2 Linear singular difference equations by strangeness-index

approach

1.2.1 Definition of strangeness index and Hriill's results

Now we introduce the definition of strangeness index for discrete-time case which was first constructed by ‘I’ Briill’s in [13]

Consider a linear time-varying discrete-time descriptor system

EyzΠ+ 1) = Antik) + fe, 2(ro) = k & Ning), (4.18)

15

where Fi, Ag € C7" for k € Nững),z(k) € CP for k N(vg) are slale vectors,

fe © C™ are given vectors and # € C” is an initial condition given at the point

k=

Definition 1.2.1 ([13]) Let Bs, Ae, Be, dy € C™ for b E N(ny) Then, two

scquencos of matrix pairs {(Ex, Ax)) ecrtaa) and {(Ex, Ax)} crsoo) are said to be

globally equivalent (on N(no)) if there exist two pointwise nonsingular matrix

sequences {Ph}ecrtun) With By € CM!" anid {Okfeergany Wilh Qk © CM sách

that P,EyQbii — Ey and Py ARQk — Ất for all k € Nína) We denate this

equivalence by {(, Áe)}eesino ~~ {(Ee, As} }eertino)-

Hf ((Ey Ag) Jeening) ~ (CB Ae) }xexing): We multiply both sides of equation

(1.15) by Py, and change variable x(k) = Qj@(A) Then, we obtain an equivalent

equation of (1.18) as

Definition 1.3.3 ([14]} Two pairs of matrices (B, A) (E, A) ¢ C™ are called

locally equivalent if Lhere exish nonsingular malrices PEC" and Q, A Ee Cet

rp = rank(#} (corresponds to forward direction}

rụ = rank(4) (corresponds to backward direction}

hy = rank(Y # E) (ranh dƒ VTE; backuard)

—rÿ | hy Tụ,

16

s=hị cứ (strangeness)

vom=rp—hy (vanishing equations)

are invariant under local equivalence, and (E, A) is locally equivalent to the canonical form

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

We have that cither s = 0,u =0 ors =u=0 The quantities defined above are called local characteristics or local invariants of the matriz pair (Hi, A)

Lemma 1.2.4 ([13|] Consider system (1.15) and introduce the matrix se-

quence {Zi }ign, where Zy is a basis of corange( Ej) — ker(E#) for k € N(no) Let

* = rank(Ek) & © Nino)

Fquation from (1.15) associated with (1.17) can be wrillen as

EOaO + 1) + BOR e+ 1) = AMZ ay 4 £4,

0=z(#)® + J8, 0-72

for k € N(ng) This system is equivalent to the system given by

EQ ROE 1) — APES + RY,

We have another imporlant property presented in Lhe fallowing theorem,

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

the following backtrack procedure For the beginner {(Exo,.4kn)}}g>ạ :—

{(Ze, Ag) }eony Por ac K,

We reduce ibis sequence by Lemma 1.24 ta the form (1.17)

al, (2! 1) Bel we Al iv

{ (Fats Ana) bez ~ 0 0 |,| 0 hạ,

Lemma 1.246 ([18, Lennna 7] ) Let the sequences (Crp hepa psen anil

{{ (Beis Ae} bean hes be defined as in (1.19) In particular, let the constant rank assumptions (1.16) hold for every {(Exi Ani) }icony For all i CN, defin- ing the quantities

We have that

Tịi Đau hye Shp, tar 2 ti, 82 snr, 5% BO for dlieN

Furthermore, there evists an a € N such that 8j4q —0 for alli € N

Definition 1.2.7 ([13, Detinition 8] } Let {(L:, At) }izny be a sequence of max trix pairs and let the corresponding sequence of characteristic valnes {(ryi, ha} bien

as in (1.19) be well-defined In particular, let (1.16) hold for every entry

{Bais Ana) }icong of the sequence (of sequences of matrix pairs) Then, ac- cording to (1.21) we call

Uhe sérungeness inden of Lhe sequence of matrix pains {( Fine: Aggibeeng and

of the associated descriptor system (1.15) In the case that z — 0, we say {(Bkis Ana) }iony and (1.15) are strangeness-free

with stiow steps from (1.17) Remark 1.2.8 Consider the sequence of matrix pairs {(Exs, An,

lo (1.18) oved jo-$ 1 equinulence trmformations, the sequence of tmutein pairs { (Ego, Anu) eons can be transformed to a sequence of the form

with all Ex) having full-row rank reay = yy (since s, = 0) We can further

reduce the last sequence of matrix pairs to a canonical form

deseriplor systern (1.18) is equimnient (in the sense thut there is an one-to- one correspondence between the solution/sequence spaces) to a discrete-time descriptor systcm of the form

Definition 1.2.10 The system

Fuy( + \) — Auylnt) +e, 1 € Nino), (1)

where Hy, An € K™, gn € lK“, rank F„, = r < d, is called a strangeness-free

.2.2 The equivalence between twa types of index definitions

Consider system (1.1) and suppose that rank Z, =r < d We cxplain a relation between two types of index definitions for the sequence {(Bk Az) }uzna (index-1 Uraclable and strangeness free) Note lal, both index-1 and strangeness free properties are invariant under global equivalence Thus, in order to get a simpler form, we transform the sequence {(Zx, Ax) }eony a8 follows

{ (Ex, Ae) Hoa {(| 0 "| › E ae bom

where Ey! is an invertible matrix in K"" for all k € Nino) From Definition

1.2.10 system, (1.1) is strangeness free Hf and only if Ayr ix invertible for all

Qe olay and Tr = ta

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

Tạ

Ti, — Fa — A ¬ i

as invertible for all & @ Ning) It means that Aj’ is invertible far all k € N(g)

Henee, the two types of index are cquivalont execpt for the condition on A,,,

2

1.2.3 Linear time-invariant singular difference equations of second order

In this subsection, we briefly review recent results on stability and stability radii in [56] for SDEs of sccond order

Consider the following linear time-invariant implicit difference equation of second order

Ax(n +2) + đen + 1) + Caín) — fa, n © N(r9) (1.24)

where 9 € Nand ABC € CH in Bom ‘Vhe leading coelficient A is

supposed to be singular with rank A = d; < d for all n > nọ We also assume

that the initial conditions

#Íne) — #o, {ng + 1] — 21, (1.25)

wilh #o, zị € Cổ, ane given The homogeneons equation of (1.24) ix

Ac(n 2) | Bain 1) Cz(@nm=0, nc Nữ) (1.26)

Definition 1.2.11 A pair of initial conditions x9, x; is said to be consis-

end wilh (1.24) if the associated inilial value problem (1.24) has ab least

one solution

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

A solution vector 2, € C” is said to be an asymptotic equilibrium of (1.24)

if Jim fe — (A+ B- Che yee fe

Definition 1.2.12 Equation (1.26) is said to be asymptotically stable if it is

regular and unique solution x(k) satisfies

Jim |kŒ)JJ—0

for all consistent initial conditions zy z1 such that max{|zo|| |£i|l} <7 for

some 7) > 0

Note thal, for bime-tvariant equations, Lhese lve lypes of slabilily (asymp:

totic stability and exponential stability) are equivalent

Set

ĐỘ) —A?A+AH+C

22

and denote the sel of finike tools af P by

ø(P) —{A 6©| det(P(À)) — 0}

Theorem 1.2.1 ([ñ6, 'heorem 2.4|) Assume that (1.26) és regular and that

the initial conditions are consis

(ij Equation (1.26) is asymptotically stable

(ii) ø(P) C B(0, 1), where B(O, 1) is the unit disk on the complex plane

According to [58], for any matrix tuple (A, B, C), there exists a non-singular

matrix W € C43 such that

where AQ), AG CO € Cd, BO CC e Chet, COE ChE dy 4 dg tdg—a

and Al), Bp!) have full-row rank Furthermore, if (1

C® has full-row rank

Definition 1.2.14, Equation (1.26) is said ta be strangeness-free if there exists

a non-singular matrix W € C44, that transforms the matrix tuple (A, B,C’) to

the form (1.27), such that A in (1.28) is invertible

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

Axin 12) 1 Ben 1) x(n) = 0, n > ap, (1.29)

where

A= A+ DAH, B= B+ Dyodgk, G

Ai; 6#, 2 — 1,2,3, are perturbations and D; € C44, Be C84 4 —1,2,3,

are matrices that restrict the structure of the perturbations

C+ Dadak,

23

Ai

A=las|,Ð=[i Mm Da], (1.30)

As

Ll +ly4ly and consider the set of destabilizing perturbations

Vc — {A € K'4|(1.29) is nonregular or not asymptotically stable }

We define the structured stability radius of (1.26) subject to structured pertur- bations (1.29) as

re(A: D,E} —inf{JAl : A € Ved, (1.31) where A refers lo (he malrix Luple (A, B,C)

Definition 1.2.15 Gonsider a slrangenesefree equation (1.26) and let W&

C4 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 (dy, đa, dạ),

1.e., there exists a sequence of unitary matrices P< C4 such that

where DY € O%°, ý 7 — 1,9,1 According lo [16, Lemma 3.3], if Lhe girue tured perturbation is allowable, then DM A,E = 0 for all 1 <i <j <3 This can be achioved by requiring that

DP =0,1gi<j<3 (1.34)

Defining

Dis) — [ sD so Ds | Hs) — EP{s)"D(s),

we have Lhe following proposition

Proposition 1.2.16 {[56, Proposition 4.4]) Consider an asymptotically stable system of the form (1.26) if the system iv strungenessfren und subjecten bo

structured perturbation as in (1.29) with structure matrices D; satisfying (1.34)

and if the perturbation A satisfies

WA] < || foo}? = Cin (5) 7

then the struckuved perturution ix allowable, Le., Ue perturbed equation (1.29)

We have the following result for the case of positive equations

Theorem 1.2.17 ([56, Theorem 4.6]) Let (1.26) be wtrunyeessefier: und pos itive, Assume that (1.28) is asymptotically stable and subjected to structured perturbations as in (1.20) with E > 0 and MiD™ > 0 for all i,j — 1.2.3

1.3 Further auxiliary results

‘The following auxiliary lemma {also known as the discrete Gronwall lemma) will be nseful in the estimation of solutions of SDEs

25

Lemma 1.3.1 ([1, Corollary 4.1.2)) Let p,q be non-negative real nambers end {u(n)} and {f(n)} be non-negative sequences for all n € Ning), where ny EN

is given Suppose that

Lemma 1.3.2 Let N,F be constant matrices, NC KM F cK! (TK — B or

Cj) YLANE is awertible then 1+ 4N is also invertible and

URN) 11 - FU ENE IN Proof of Lemma 1.3.2 is available in Appendix

We give here the prineiple tad will be used several Limes in two next chap

ters It is a special case of the uniformly boundedness principle (see |62])

Theorem 1.3.3 Let X ond ¥ be Banach spaces Let T be a collection of continuous linear mappings from X into ¥ and suppose that

sup [Az|| < % for every x © X

Then, there exists an M < 00 such that

| Az|| < Aljz|| for 2 © X and ACT

In seckion 2.3, we will use (he inequalily regarded as a dixcrele Lype Young's inequality for convolution

Theorem 1.3.4 Let {an}, {bn} 2%» be sequences of nonnegative real num- bers, 1 p< có Then

(= (dy uh) C52) 5 9” (1.38)

ix a well-known lool in

Singular value decomposition (SVD) af matric

numerical linear algebra, see [69]

Theorem 1.3.5 Let AEC" Then, there exis a unitary matrix U € CP and @ unitary matrix V € C™" such that

A-Unhinglo:, + pV, where 0) B+ Boyd Op — minfran) HỆ Ain vent, then U and Vo may be taken to be real orthogonal matrices Such unlues ơi, ‹‹- dp are called singular

values

Note that diag(Øi, - ,đy) is a matrix in C"™” whose di entry is oj for any

i =1 p and the others are zero

a7

jecled io allowable perlurbalians We also investigale cerlain inpul-slale ree

lations and the exponential stability for singular systems of linear difference

equations For the treatment of singular systems, we use the projecter-based approach Based on the result of decoupling, we construct admissible spaces for the inhomogeneity part of the singular systems ‘Ihree so-called Bohl-Perron-

lype stability Uheorems, which are known in lhe Titeralure of regular explicit

difference equations, are extended ta SDRs Next the nation af Rah! exponent

is introduced and we characterize the relation between the exponential stabil- ity and Bohl exponent Finally, robustness of Bohl exponent with respect to

allowable perturbations is irr igaled ‘This chapter is wrillen on Lhe basis of

lwo papers [1] and [2| in the Tist of Lhe publications: used in thủ Lheni,

2.1 Stability notions for singular difference equations

From new on, we always suppose (hat Lhe linear SIK (1.2) has index-1 and

ils Cauchy operalar ®(n mr) is defined in Subsechion 1.1.2 ‘The lattowing

stability notions generalize those for ODEs See also [4, 6]

Definition 2.1.1 The zero solution af equation (1.2) is said lo be stable if for

28

any © > (and m1 € N(ng) bhere exists a posilive constant 6 — d(e,ra) such

that the inequality ||P,,-12'l] <6 implies ||x(k) | < £ for all k e Nứn,), where

z(-) is the solution of (1.2) satisfying ,_¡(z(m) z!) =0,

Definition 2.1.2 The zero solution of cquation (1.2) is said to be uniformly

stable if it is stable and the constant 6 mentioned in Definition 2.1.1 is inde

pendent of 74

Definition 2.1.8 The vere solution af equation (1.2) is said bo be aymploti-

cally stable if it is stable and limp |x(k)|| = 0, where x{-) is the solution of

(1-2) with P„„_i(z[m)- zD — 0

If the zero solution of equation (1.2) is stable (resp uniformly stable, asymp- lolically slable), (hen we say the equation (1.9) is stable (resp uniforraly stable, asymptotically stable)

Definition 2.1.4 Equation (1.2) is said to be exponentially stable if there exist

constants K > 0 and 0 < w <1 such that |[x(n)|| < Re” " | iz(m)|| —

Ke™ ™lle(m)]|.n,m © N(no).2 > m, for every solution «(-) of (1.2)

The following characterizations of uniform stability and exponential stabil- ity are straightforward generalizations of the well-known results for ordinary difference cquations, sce [1, 32)

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

| đín,m)|| < C for all nym Ning), n > m (2.4)

Proof The proof contains lwo paris

Necessity Since equation (1.2) is uniformly stable, there exists a positive

couslanl 6 such Uhab, for any ny) € N(via), if |Pay—sa? | < ổ, khien ||e(#)|| < 1 for

all k €N(m), where 2(k) is the solution of equation (1.2) with P,,1(¢(m) —

z!) —0 (we have chosen ¢— 1) We note that @(k, 1) — #&(k, n.)Õ, _: then,

#Œ) — (bs rey )arlrtn) — ®(Mu mà) Pịy trữ) — (KE gan! — (kom at