DSpace at VNU: Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions

DSpace at VNU: Stability criteria for differential-algebraic equations with multiple delays and their numerical solution...

Stability criteria for differential-algebraic equations with multiple

delays and their numerical solutions

a

Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA

b

Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

a r t i c l e i n f o

Keywords:

Delay differential-algebraic equation

Multiple delays

Asymptotic stability

Regular pencil

Numerical solution

a b s t r a c t

This paper is concerned with the asymptotic stability of differential-algebraic equations with multiple delays and their numerical solutions First, we give a sufficient condition for delay-independent stability After characterizing the coefficient matrices that satisfy this stability condition, we propose some practical checkable criteria for asymptotic stabil-ity Then we investigate the stability of numerical solutions obtained by h-methods and BDF methods Finally, solvability and stability of a class of weakly regular delay differen-tial-algebraic equations are analyzed

Ó 2008 Elsevier Inc All rights reserved

1 Introduction

In this paper we consider the linear differential-algebraic equation with multiple delays

A _xðtÞ þ BxðtÞ þXM

i¼1

Ci_xðt siÞ þXM

i¼1

where A, B, Ci, Di(i = 1, 2, , M), are real (or complex) constant matrices of size m  m The time-delays are ordered increas-ingly, 0 <s1<s2<    <sM Matrix A is assumed to be singular with rank A = d < m We are also interested in a special subclass

of(1.1)in the form,

A _xðtÞ þ BxðtÞ þXM

i¼1

Ci_xðt  isÞ þXM

i¼1

That is bysi= is(i = 1, 2, , M), wheres> 0 is given From now on, if the unknown functions appear without argument and

no confusion arises, we mean that they are evaluated at the actual time t For example, we write x instead of x(t) and _x in-stead of _xðtÞ

Differential-algebraic equations (DAEs) play important roles in mathematical modeling of real-life problems arising in a wide range of applications, for example, multibody mechanics, prescribed path control, electrical design, chemically reacting systems, biology and biomedicine See[3,16]and references therein In many problems, the systems in consideration contain time-delays, see [2,5–7,10,19,20,22,24–26] While the theory and the numerics for delay ordinary differential equations (DODEs) have been well known and discussed for decades in a wide range of literature, see[12]and references therein, there are very few results for the theory of delay differential-algebraic equations (DDAEs) The main reason is that even for linear DDAEs, their dynamics have not been well understood yet, in particular when the pencil {A, B} in(1.1)is not regular The

0096-3003/$ - see front matter Ó 2008 Elsevier Inc All rights reserved.

* Corresponding author.

E-mail addresses: linhvh@vnu.edu.vn , vhlinh@hn.vnn.vn (V.H Linh).

Contents lists available atScienceDirect Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a m c

most difficult problem is that there exists no compressed form into which a tuple of more than two matrices can be simul-taneously transformed Most of the existing results are only for linear time-invariant regular DDAEs[10,24]or DDAEs of spe-cial form[2,19,25,26] Until now there have been only two papers concerning nonregular DAEs[7,20] A general result for DDAEs’ solvability and stability is still missing The following examples illustrate some significant differences between delay ODEs, DAEs without delays, and delay DAEs

Example 1 Consider the system

_x1ðtÞ þ x1ðtÞ  x1ðt  1Þ  x2ðt  1Þ ¼ 0

2x2ðtÞ þ x1ðt  1Þ þ x2ðt  1Þ ¼ 0



ðt P 0Þ;

where x1and x2are given by continuous functions on the initial interval (1, 0] The dynamics of x1is governed by a differ-ential operator and continuity of x1is expected The dynamics of x2is determined by a difference operator and unlike x1, this component is piecewise continuous, in general

Example 2 [7] Consider the following inhomogenous system:

_x1ðtÞ ¼ f ðtÞ

x1ðtÞ  x2ðt  1Þ ¼ gðtÞ



ðt P 0Þ:

The solution is given by

x1ðtÞ ¼

Z t

0

f ðsÞds þ c; x2ðtÞ ¼ gðt þ 1Þ þ

Z tþ1

0

f ðsÞds þ c ðt P 0Þ;

where c is a constant The system dynamics is not causal Not only is x2specified on (1, 0], but the solution depends on future integrals of the input f(t) This interesting phenomenon should be noted in addition to the well-known fact in the DAE theory that the solution may depend on derivatives of the input

A sufficient condition for the delay-independent asymptotic stability of DAEs with single delay is proposed in[25] Under this condition, the asymptotic stability of h-methods, BDF methods, general linear multistep methods, as well as implicit Runge–Kutta methods are analyzed Unfortunately, it is very difficult to verify this condition in practice The main aim of the present paper is to give a complement to this result in the stability theory for DDAEs Namely, we intend to derive de-lay-independent stability criteria for DDAEs of the form(1.1) and (1.2) We focus on practical stability criteria that are easily checkable Our results extend those obtained for neutral DODEs[13,14]to neutral DDAEs Under these criteria, we will show that numerical solutions obtained by the h-methods and BDF methods preserve the asymptotic stability of the DDAE This result includes the single delay DAEs result of[25]as a special case Further, we also investigate the solvability and the sta-bility of a special class of nonregular delay DAEs

The paper is organized as follows In the next section we review basic notions and results from the theory of DAEs and regular delay DAEs The main results of the paper lie in Section3, where we formulate sufficient conditions to provide the asymptotic stability of regular DDAEs We give a characterization of those coefficient matrices that satisfy the sufficient con-ditions We also propose some practical criteria for the asymptotic stability of DDAEs with multiple delays In Section4, we analyze the stability of numerical solutions to(1.1) and (1.2)using h-methods and BDF methods Finally, in the last section,

we discuss solvability and stability issues of a special class of weakly regular DDAEs

2 Preliminary

In this section, we give a brief summary of needed results on linear constant coefficient and delay DAEs We assume the reader is familiar with the basic theory of linear time invariant DAEs[3,11,16], such as

The matrix pencil {A, B} is said to be regular if there exists k 2 C such that the determinant of kA + B, denoted by det(kA + B), is nonzero The system(2.1)is solvable if and only if {A, B} is regular If detðkA þ BÞ ¼ 0 8k2 C, we say that {A, B} is irregular or non-regular If {A, B} is regular, then k is a (generalized finite) eigenvalue of {A, B} if det(kA + B) = 0 The set of all eigenvalues is called the spectrum of the pencil {A, B} and denoted byr{A, B} The maximum of the absolute values of the finite eigenvalues

is called the spectral radius of the pencil {A, B} and denoted byq(A, B) These concepts are also extended to the case of a given tuple of matrices fAigni¼0 (the generalized polynomial eigenvalue problem) by defining rðfAigni¼0Þ ¼ fk 2 C : det

ðPn

i¼0kniAiÞ ¼ 0g; and qðfAigni¼0Þ ¼ maxfjkj : k 2 Cand detðPn

i¼0kniAiÞ ¼ 0g: Thus, for a given matrix A 2 Cmm, the well-known spectrumr(A) and the spectral radiusq(A) arer(I, A) andq(I, A), respectively

Suppose that A is singular and pencil {A, B} is regular Then there exist nonsingular matrices W, T such that

WAT ¼ Id 0

0 N

; WBT ¼ B1 0

0 I

where N is nilpotent of index k[3,11,16] If N is a zero matrix, then k = 1 Furthermore, we may assume without loss of gen-erality, that N and B1are upper triangular If {A, B} is regular, the nilpotency index of N in(2.2)is called the index of matrix pencil {A, B} and we write index {A, B} = k If A is nonsingular, we set index {A, B} = 0

Definition 1 Suppose that {A, B} is regular Let Q be a projector onto the subspace of consistent initial conditions Let

P = I  Q We say that the zero solution of(2.1)is stable if, for anye> 0 there exists d > 0 such that for an arbitrary vector

x02 Cmsatisfying kx0k < d, the solution of the initial value problem

A _x þ Bx ¼ 0; t 2 ½0; 1Þ;

Pðxð0Þ  x0Þ ¼ 0



exists uniquely and the estimate kx(t)k <eholds for all t P 0 The zero solution is said to be asymptotically stable if it is stable and limt?1kx(t)k = 0 for solutions x of(2.1) If the zero solution of(2.1)is asymptotically stable, we say that system(2.1)is asymptotically stable

If index {A, B} = 1 one may choose Q as the projector onto (A)[11] A difference between ODE-s and DAE-s is that the equality x(0) = x0is not expected here, in general That is, for DAEs, we need consistent initial value x0such that(2.1)with the initial condition x(0) = x0holds for a smooth solution We do not consider impulsive solutions in this paper and for that reason will frequently make an index one assumption For linear time-invariant systems, the concepts of asymptotic stability and exponential stability are equivalent The system(2.1)is asymptotically stable if and only if the matrix pencil {A, B} is (asymptotically) stable, i.e.,rðA; BÞ  C;where Cdenotes the open left half complex plane[23] Clearlyr(RAS, RBS) =r(A, B) for nonsingular R, S

2.1 Solvability of regular delay DAEs

The theory of delay ordinary differential equations (DODEs), when the leading matrix A in(2.3)is the identity matrix, has been widely discussed[12] These systems are classified by their type For a scalar DODE a _x þ bx þ c _xðt sÞþ dxðt sÞ ¼ f ðtÞ, the system is of retarded type if a – 0, c = 0, of neutral type if a – 0, c – 0, and of advanced type if a = 0, b – 0, and c – 0 One important attribute of the type is that it classifies how DODEs propagate discontinuities to future delay intervals (assuming

an initial value problem) Discontinuities in retarded systems become smoother in each successive interval, whereas discon-tinuities in advanced systems become less smooth in each successive interval Discondiscon-tinuities in neutral systems are carried into successive delay intervals with the same degree of smoothness Hence, we wish to study separately DDAEs which in-clude retarded and neutral DODEs, but to avoid altogether those which lead to DODEs of advanced type For some interesting examples of DDAEs and some DDAEs which ‘‘look like” they should be of retarded type but are actually neutral or advanced type, see[6,7]

In this section we consider DAEs with single delay

The delay DAE(2.3)is called regular[7]if the pencil {A, B} is regular and weakly regular if there exista;b;c2 C such that det(aA + bB +cD) – 0, i.e., the triplet {A, B, D} is regular We suppose initially that {A, B} is regular and has index k Note that neutral DAEs with single delay

A _x þ Bx þ C _xðt sÞ þ Dxðt sÞ ¼ 0 ð2:4Þ

can always be transformed to the form(2.3) Indeed, by defining a new variable y by y(t) = x(t s), we obtain a new delay DAE

with

~

x ¼ x

y

 

; eA ¼ A C

0 0

; eB ¼ B D

0 I

; D ¼e 0 0

I 0

:

However, this transformation can increase the index of the DAE system Further, the dimension of the new transformed sys-tem becomes 2m, which is less advantageous in practical computation

Proposition 1 The pencil feA; eBg is regular if and only if {A, B} is regular However, the index of feA; eBg is equal either k or k + 1, where k is the index of {A, B}

Proof The equivalence between the regularity of the two pencils is clear We verify the statement on the index of feA; eBg Without loss of generality, we assume that the pencil {A, B} is given in the Kronecker normal form(2.2)Correspondingly,

C and D are given in block form

C ¼ C1 C2

C C

; D ¼ D1 D2

D D

Thus feA; eBg can be assumed to be

eA ¼

I 0 C1 C2

0 N C3 C4

0 0 0 0

0 0 0 0

0

B

B

1 C

C; eB ¼

B1 0 D1 D2

0 I D3 D4

0 0 I 0

0 0 0 I

0 B B

1 C

C:

It is not difficult to verify that

indexfeA; eBg ¼ index

I 0 0 0

0 N C3 C4

0 0 0 0

0 0 0 0

0 B B

1 C

C;

B1 0 0 0

0 I 0 0

0 0 I 0

0 0 0 I

0 B B

1 C C

8

>

<

>

:

9

>

=

>

; :

Then indexfeA; eBg ¼ index eN where

e

N ¼

N C3 C4

0 0 0

0 0 0

0

B

1 C A:

Then

e

Nk

¼

Nk Nk1C3 Nk1C4

0

B

1 C A:

Hence, index eN ¼ k if Nk1C3= 0 and Nk1C4= 0 Otherwise index eN ¼ k þ 1 h

Corollary 1 Suppose that the pencil {A, B} is regular and has index-1 Further, suppose the matrices are in block form as in the proof ofProposition 1 Then the new pencil feA; eBg has index-1 if and only if C3= 0, C4= 0

Corollary 1means that the transformed system(2.5)has index-1 if and only if the pencil {A, B} has index-1 and the deriv-ative of x(t s) does not appear in the ‘‘algebraic part”

Now, we turn back to the regular delay DAE(2.3)with an initial condition x(t) = u(t), t 2 [s, 0], where u is a continuous function defined on [s, 0] The solvability of regular delay DAEs was discussed in detail in[5,6] Using appropriate constant coordinate changes, first we transform the matrix triplet A,B,D into the block form(2.2), (2.6) Then system(2.3)is decom-posed as follows:

z0þ B1z þ D1zðt sÞ þ D2wðt sÞ ¼ 0;

Nw0þ w þ D3zðt sÞ þ D4wðt sÞ ¼ 0; ð2:7Þ

where x is decomposed into ‘‘differential” variables z and ‘‘algebraic” variables w Using the nilpotency of N,

wðtÞ ¼ Xk1

i¼0

ðNÞi½D3zðiÞðt sÞ þ D4wðiÞðt sÞ: ð2:8Þ

Setting t = 0 in(2.8), we obtain the consistency condition for the initial condition The initial value problem for(2.3)with a consistent initial condition admits a unique solution, see[5,6,10]which can be obtained by solving the system(2.7)for z,w recursively on each interval ((l  1)s,ls], l = 1, 2, The definition of the asymptotic stability for DDAEs of the form(2.3)is similar to that for DODEs

Definition 2 [10,24,25]The trivial solution of the DDAE(2.3)is said to be stable if for anye> 0 theres exists d = d(e) such that for all contiuous functions u satisfying the consistent condition and supt2[s,0]ku(t)k < d, the solution x = x(t, u) of the initial value problem for (2.3) satifies kx(t, u)k <e for all t P 0 The trivial solution of the DDAE (2.3) is said to be asymptotically stable if it is stable and furthermore limt?1kx(t, u)k = 0

For higher-index problems (k > 1), the formula for w involves derivatives of the solution taken in the past It was shown in [5]that solutions for(2.3)can be continuous on only finite intervals even if the initial function u (or the input, if there is an input function) is infinitely differentiable Further, discontinuities do not necessarily get smoothed out as with the nonsin-gular problem

Example 3 [6] Consider a two dimensional delay DAE system

0 1

0 0

x0ðtÞ þ 1 0

0 1

xðtÞ þ 0 0

1 0

xðt  1Þ ¼ 0;

with x = (x1, x2)T This system has index-2 It is easy to see that x1satisfies an advanced type equation x1ðtÞ ¼ x0

1ðt  1Þ, so that

x1ðtÞ ¼ xðmÞ1 ðt  mÞ That is, x1(and x2, too) becomes progressively less smooth The system behaves like those of advanced type

For the simplest case k = 1, the situation is somewhat better The evolution of z is given by a delay differential equation, meanwhile a difference operator defines the dynamics of w If a continuous initial function u is given, then z is continuous and w is piecewise continuous in general Furthermore, z is differentiable and w is continuous except possibly at integer mul-tiples ofs The system(2.3)behaves like a neutral delay system

Extending all the results in this section to multiple-delay DAEs of the form(1.1) or (1.2)is straightforward We note that the smoothness of solutions now may be even worse Even in index-1 problems, the distance between the jump (or break) points can become arbitrarily small as t is increasing except for the case when all the ratiossi/sj, i – j are rational numbers This fact gives rise to practical difficulties for numerical methods

2.2 Delay DAEs of Hessenberg form

Delay DAEs arising in applications frequently have special structure One of the most important class of systems is that of Hessenberg forms which generalizes non-delay DAEs of Hessenberg form[3]

Definition 3 Linear delay DAEs of the form

_x1þ B1x1þ B2x2þ D1x1ðt sÞ þ D2x2ðt sÞ ¼ 0; ð2:9Þ

B3x1þ B4x2þ D3x1ðt sÞ ¼ 0; ð2:10Þ

where B4is nonsingular, is called semi-explicit index-1 linear DDAEs or index-1 linear DDAEs in Hessenberg form Note {A, B}

is an index one pencil and D4= 0

Linear delay DAEs of the form

_x1þ B1x1þ B2x2þ D1x1ðt sÞ ¼ 0; ð2:11Þ

where B3B2is nonsingular, is called semi-explicit index-2 linear DDAEs or index-2 linear DDAEs in Hessenberg form Here {A, B} is an index two Hessenberg pencil and D2= 0, D3= 0, and D4= 0

Delay DAEs of the form(2.9)-(2.10)come from the linearization of index-1 nonlinear DDAEs in Hessenberg form

f ðt; _x1ðtÞ; x1ðtÞ; x1ðt sÞ; x2ðtÞ; x2ðt sÞÞ ¼ 0; ð2:13Þ gðt; x1ðtÞ; x1ðt sÞ; x2ðtÞÞ ¼ 0 ð2:14Þ

along a particular solution, where the Jacobian gx2is assumed nonsingular Similarly, by linearizing index-2 nonlinear DDAEs

f ðt; _x1ðtÞ; x1ðtÞ; x1ðt sÞ; x2ðtÞ; x2ðt sÞÞ ¼ 0; ð2:15Þ

where gx

1fx2is assumed nonsingular, one obtains DDAEs of the form(2.11) and (2.12), see[2,26]

The derivative of the unknown function at a delayed time may appear in(2.9), (2.10) and (2.11), (2.12), as well Namely, DDAEs of the form

_x1þ B1x1þ B2x2þ C1_x1ðt sÞ þ C2_x2ðt sÞ þ D1x1ðt sÞ þ D2x2ðt sÞ ¼ 0; ð2:17Þ

B3x1þ B4x2þ D3x1ðt sÞ ¼ 0; ð2:18Þ

where B4is nonsingular, are called index-1 linear neutral DDAEs in Hessenberg form Further, DDAEs of the form

_x1þ B1x1þ B2x2þ C1_x1ðt sÞ þ D1x1ðt sÞ ¼ 0; ð2:19Þ

where B3B2is nonsingular, are called index-2 linear neutral DDAEs in Hessenberg form

Neutral delay DAEs of the forms(2.17), (2.18) and (2.19), (2.20)can be transformed to delay DAEs of the forms(2.9), (2.10) and (2.11), (2.12)by introducing new auxiliary variables as discussed in the previous section.Proposition 1shows the index of the transformed delay DAEs have the same index as the original neutral delay DAEs

One of the most important features of delay DAEs in Hessenberg form is that one can easily get the so-called underlying DODEs For example, for the system2.9,2.10, one can solve x2from(2.10), then insert into(2.9), and get the underlying DODE

_x1þ ðB1 B2B1

4 B3Þx1þ ðD1 D2B1

4 B3 B2B1

4 D3Þx1ðt sÞ  D2B1

4 D3x1ðt  2sÞ ¼ 0: ð2:21Þ

Note that the UDODE(2.21)now has double delays while the original DDAE(2.9) and (2.10)has a single delay Further, it is easy to see that the index-1 DDAE(2.9) and (2.10)is asymptotically stable if and only if its UDODE(2.21)is asymptotically stable Similarly, we can derive the underlying neutral DODE for the index-1 neutral DDAE of the form(2.17) and (2.18)

Obtaining the underlying DODE for semi-explicit index-2 DDAE of the form(2.11) and (2.12)is a little bit more compli-cated than the index-1 case First, observe that by differentiating(2.12)and inserting the result into(2.11), we obtain a hid-den constraint

B3B1x1þ B3B2x2þ B3D1x1ðt sÞ ¼ 0: ð2:22Þ

Since B3B2is invertible, one can calculate the index-2 algebraic variable x2 from x1 Next, we proceed as follows (see [1,8]) Denote the row number and the column number of B2by m1and m2, respectively Take a matrix R 2 Rðm 1 m 2 Þm 1 whose linearly independent normalized rows form a basis for the null space of BT

2 Then RB2= 0 and the matrix R

B3

is invertible

Defining new variables u = Rx1, we can calculate x1from u by

x1¼ R

B3

 1

u 0

 

where S is defined by RS = I, B3S = 0 The underlying DODE is

From(2.22) and (2.23), it is clearly seen that the semi-explicit index-2 DDAE(2.11) and (2.12)is asymptotically stable if and only if the UDODE(2.24)is We obtain analogously the underlying neutral DODE for the index-2 neutral DDAE of the form (2.19) and (2.20)

From the above introduction of DDAE in Hessenberg form, we conclude that if one wants to investigate the stability of DDAEs in Hessenberg form, it makes sense to consider their underlying DODEs

3 Stability criteria

Now consider the DAE of multiple delays of the form(1.1) or (1.2) The characteristic equation for(1.1)is defined by

PðsÞ ¼ det sA þ B þ sXM

i¼1

CiessiþXM

i¼1

Diessi

!

For a given s 2 C, we denote its real and imaginary parts by Re(s) and Im(s), respectively It is well known, see[25], that the system(1.1)is asymptotically stable if all the roots of(3.1)have negative real part and they are bounded away from the imaginary axis, i.e., for all root kiof(3.1)(i = 1, 2, ) and for some positivel, the inequalities

hold Note that(3.1)may have infinitely roots and they may accumulate at a finite point on the complex plane or at infinity

In this section, we will derive some sufficient conditions for(3.2) We will need the following definition and an auxiliary re-sult, which are well known in the theory of nonnegative matrices[17]

Definition 4 Let W 2 Cnnwith elements wijand jWj denote the nonnegative matrix in Rnn with element jwijj For two matrices U; V 2 Rnn, we write U 6 V if and only if uij6vijfor each i,j 2 {1, 2, , n} In particular,q(W) 6q(jWj)

Lemma 1 Let W 2 Cnnand V 2 Rnn If jWj 6 V, thenq(W) 6q(V)

3.1 Delay-independent asymptotic stability

We introduce the following two-variable polynomials:

Qðs; zÞ ¼ det sA þ B þXM

i¼1

ðsCiþ DiÞz

!

Rðs; zÞ ¼ det sA þ B þXM

i¼1

ðsCiþ DiÞzi

!

Lemma 2 Suppose that

ðiiÞ sup

ReðsÞP0

q XM i¼1

jðsA þ BÞ1ðsCiþ DiÞj

!

Then Q(s,z)–0 for all s; z 2 C such that Re(s) P 0, jzj 6 1

Proof Suppose that the contrary happens, i.e., there exist s, Re(s) P 0 and z, jzj 6 1 such that Q(s,z) = 0 This implies that there exist a vectorv–0 such that

sA þ B þXM

i¼1

ðsCiþ DiÞz

or equivalently (since (sA + B) is invertible)

ðsA þ BÞ1XM

i¼1

ðsCiþ DiÞzv¼ v:

This means that 1 is an eigenvalue of ðsA þ BÞ1PM

i¼1ðsCiþ DiÞz, which implies

q ðsA þ BÞ1XM

i¼1

ðsCiþ DiÞz

!

P1:

But

q XM

i¼1

jðsA þ BÞ1ðsCiþ DiÞj

!

Pq ðsA þ BÞ1XM

i¼1

ðsCiþ DiÞz

!

;

which contradicts(3.6) h

Note that in the single delay case, i.e M = 1, the statement holds without the need of taking the absolute value in(3.6), see also[25] Further, due to the maximum principle in complex analysis, it suffices to take the supremum on the imaginary axis Re(s) = 0 in the assumption(3.6)

Theorem 1 Suppose that the assumptions(3.5) and (3.6)inLemma 2hold Then the system(1.1)is asymptotically stable for all sets of the delays fsigMi¼1, i.e., the asymptotic stability of(1.1)is delay-independent

Proof Similarly to the proof ofLemma 2, it is not difficult to show that the equation P(s) = 0 has only roots with negative real part Next, we prove that the real parts of the roots are bounded away from 0 Suppose that this statement is not true Then, there exists a sequence {sn} such that limn?1Re(sn) = 0 meanwhile P(sn) = 0 Choose a positive numberesuch that

e<1  sup

ReðsÞ¼0

q XM i¼1

jðsA þ BÞ1ðsCiþ DiÞj

! :

It is obvious that there exists a sufficiently large N0 such that for n P N0, we have ReðsnÞ P1l, where



l¼ maxfReðkÞ; k 2rðA; BÞg < 0 is the spectral abscissa of the pencil {A, B}, and je sMsnj 6 ð1 e=2Þ1 For each sn, there exists

a vectorvn–0 such that

snA þ B þXM

i¼1

ðsnCiþ DiÞesis n

which implies

q XM

i¼1

jðsnA þ BÞ1ðsnCiþ DiÞj

!

P1 e=2:

Now we observe that the entries of the matrix functions

ðsA þ BÞ1ðsCiþ DiÞ; i ¼ 1; 2; ; M

are rational function of s Only the finite eigenvalues of {A, B} may be poles of these functions Thus, each element of the (non-negative) matrix function

XM

i¼1

jðsA þ BÞ1ðsCiþ DiÞj

has the form jsjapq

ðapqþ Oð1=jsjÞ, whereapqare some integers and apqare nonnegative numbers (1 6 p; q 6 m) Hence, for an arbitrarily small>0, there exists a bound s1>0 such that

AðsÞð1 Þ 6XM

i¼1

jðsA þ BÞ1ðsCiþ DiÞj 6 AðsÞð1 þÞ ð3:7Þ

for all jsj P s , where the elements of the matrix function AðsÞ are defined by

ApqðsÞ ¼ jsjapqapq:

Leta¼ maxp;qapqand the matrix function AðsÞ be decomposed such that

AðsÞ ¼ jsjaAð0Þþ Að1ÞðsÞ

;

where Að0Þis a nonnegative constant matrix and each entry of Að1ÞðsÞ is either zero or negative power of jsj Next, we inves-tigate the asymptotic behavior of the spectral radius of AðsÞ as jsj tends to infinity The following cases are possible:

 Ifa60, thenqðAðsÞÞ has a finite limit as jsj ? 1;

 Ifa> 0 andqðAð0ÞÞ > 0, thenqðAðsÞÞ tends to infinity as jsj ?1;

 Ifa> 0 andqðAð0ÞÞ ¼ 0, then due to the Puisseux series of the eigenvalues of (Að0Þþ Að1ÞðsÞÞ, see[15], we have

qðAð0Þþ Að1ÞðsÞÞ ¼ jsjbðc þ oð1ÞÞ;

where c > 0 is a constant and b is a negative fractional number In other words, we use the fact that the eigenvalues can be expanded into fractional power series of 1/jsj Depending on the sign ofa+ b, the spectral radius of AðsÞ either converges to a finite number or tends to infinity as jsj ? 1

Summarizing the above cases, the spectral radius of AðsÞ either converges to a finite number or tends to infinity as jsj ? 1 Sincein(3.7)is arbitrarily chosen, the same statement holds for the spectral radius of

XM

i¼1

jðsA þ BÞ1ðsCiþ DiÞj;

which is a function of s The assumption(3.6)implies that the latter function must converge to a finite limit as jsj ? 1 On the other hand, this function is continuous in the domain fs 2 C; ReðsÞ P1lg Consequently, it is uniformly continuous in the considered domain

Finally, due to the verified uniform continuity, there exists snsufficiently close to the imaginary axis such that

q XM

i¼1

jðsnA þ BÞ1ðsnCiþ DiÞj

!

q XM i¼1

jðImðsnÞA þ BÞ1ðImðsnÞCiþ DiÞj

!











6e=2:

We obtain

q XM

i¼1

jðsnA þ BÞ1

ðsnCiþ DiÞj

!

6q XM i¼1

jðImðsnÞA þ BÞ1ðImðsnÞCiþ DiÞj

!

þe=2

6 sup

ReðsÞ¼0

q XM i¼1

jðsA þ BÞ1ðsCiþ DiÞj

!

þe=2 < 1 e=2;

which yields contradiction The proof is complete h

Assumptions(3.5) and (3.6)come from the straightforward generalization of the corresponding stability conditions for neutral DAEs with single delay given in[25] In that paper, a third assumption juTAuj > juTCuj;8u 2 Cm, was needed to ensure that all the roots of the characteristic equations are bounded away from the imaginary axis From the proof ofTheorem 1, we see that such an assumption is redundant and can be ignored

Sometimes it is more convenient to check the assumptions by using an operator norm instead of the spectral radius Corollary 2 Suppose that the assumption(3.5)holds and

sup

ReðsÞ¼0

XM

i¼1

jðsA þ BÞ1ðsCiþ DiÞj











<1:

Then the system(1.1)is delay-independently asymptotically stable

We have similar statements for the system(1.2)

Lemma 3 Suppose that

ðiiÞ sup

ReðsÞP0

q fsCiþ DigMi¼0

where C :¼ A, D :¼ B Then R(s,z) – 0 for all s; z 2 C such that Re(s) P 0, jzj 6 1

Proof Recall that

q fsCiþ DigMi¼0

¼ max jkj; det XM

i¼0

ðsCiþ DiÞkMi

!

¼ 0

for a given fixed s:

Hence, for a given s, Re(s) P 0, if z 2 C is such that R(s, z) = 0, then 1/z is an eigenvalue of the polynomial eigenvalue problem with data fsCiþ DigMi¼0 Note that z cannot be zero because det(sA + B) – 0 for all s, Re(s) P 0 Therefore, assumption(3.9) im-plies that for a given s, Re(s) P 0, if R(s, z) = 0, then jzj > 1 h

Theorem 2 Suppose that the assumptions(3.8) and (3.9)inLemma 3hold Then the system(1.2)is asymptotically stable for all

sP0, i.e., the asymptotic stability of(1.2)is delay-independent

Proof Similar to the proof ofTheorem 1 h

Next, we attempt to characterize the set of admissible coefficient matrices which satisfy the assumptions ofTheorem 1 and analyze the effect of the index of the pencil {A, B} with second assumption(3.6) For sake of simplicity, and due to Prop-osition 1, we consider the single delay Eq.(2.3) The assumption(3.6)now becomes supRe(s)=0q((sA + B)1D) < 1

Assume the coefficient matrices are transformed again in block form We have

ðsA þ BÞ1D ¼ ðsI þ B1Þ

1

D1 ðsI þ B1Þ1D2

ðsN þ IÞ1D3 ðsN þ IÞ1D4

! :

Using the nilpotency of N, it is obvious that the spectral radius of the matrix

ðsI þ B1Þ1D1 ðsI þ B1Þ1D2

Pk1

i¼0ðNÞisiD3 Pk1

i¼0ðNÞisiD4

!

is necessarily bounded for s, Re(s) = 0 Since all the entries of the first block row tend to 0 as jsj tends to infinity, we get some consequences on D4 Namely, if k = 1, thenq(D4) < 1 must be satisfied For higher index cases, we haveq(D4) < 1 andq(Ni D

4) = 0 for i = 1, 2, , k  1, otherwise the spectral radius in question is unbounded That is, for higher index pencil {A, B}, the block D4(and D3as well) must be of special structure Taking into account the result ofProposition 1, the same statement holds for higher index neutral delay DAEs(2.4)and for higher index DAEs systems with multiple delays of the form(1.1) and (1.2) Note that these necessary conditions on D4are trivially satisfied by the delay and neutral delay DAEs of Hessenberg forms

With the assumption(3.5), the problem of finding sufficient condition for asymptotic stability for delay DAEs is closely related to the robust stability question of DAE, see[4,21,9,18] We have a nominal DAE system without delays which is as-sumed to be asymptotically stable The delay terms can be considered uncertain perturbations From this point of view, a somewhat simpler condition can be given instead of(3.6)

Proposition 2 Consider the delay DAE of the form(1.1) Suppose that Ci= 0, for i = 1, 2, , M and assumption(3.5)holds Then if

k Dð 1 D2    DMÞk < sup

ReðsÞ¼0

kðsA þ BÞ1k

!1

the delay DAE system is asymptotically stable

Proof Eq.(3.10)implies(3.6) See also[18] Note that in(3.10)we can take any matrix norm induced by a vector norm h Unfortunately, if the index of {A, B} is greater than 1, then the right hand side of(3.10)is simply zero, and the proposition does not apply This once again confirms that for higher-index problems, the coefficient matrices Ci, Di, (i = 1, 2, , M) must be highly structured so that the asymptotic stability would be preserved

3.2 Practical algebraic stability criteria

Theorems 1 and 2give us sufficient conditions for the asymptotic stability of delay DAEs of the form(1.1) and (1.2), respectively Unfortunately, checking the conditions(3.6) or (3.9)is rather difficult because computing the supremum of the spectral radius of a matrix function or a polynomial matrix function over an unbounded domain is very costly In this section, we propose some checkable algebraic criteria for the asymptotic stability Our results extend some recent results for neutral delay ODEs, see[13,14], to neutral delay DAEs

(a) The index-1 case We first restrict the investigation to index-1 problems Since the matrices A and B can easily be transformed to the upper block-triangular form using QZ or QR decompositions, we assume that

A ¼ A1 A2

0 0

; B ¼ B1 B2

0 B

:

The index-1 assumption on the pencil {A, B} implies that the submatrix B4is invertible Furthermore, due toCorollary 1, we assume that

Ci¼ Ci1 Ci2

0 0

; Di¼ Di1 Di2

Di3 Di4

; i ¼ 1; 2; ; M:

We introduce some auxiliary matrix sequences

Li¼ ðA þ BÞ1ðDiþ CiÞ; Mi¼ ðA þ BÞ1ðDi CiÞ; E ¼ ðA þ BÞ1ðA  BÞ ð3:11Þ

for i = 1, 2, , M

Lemma 4 Let the assumption(3.5)hold Then

ðsA þ BÞ1ðsCiþ DiÞ ¼ ðI  zEÞ1ðzMiþ LiÞ

for all Re(s) P 0, where z ¼1s (which implies jzj 6 1, z – 1)

Proof The proof is similar to the proof of Theorem 2.2[13] It is easy to derive

I  zE ¼ I 1  s

1 þ sðA þ BÞ

1

ðA  BÞ

¼ ðA þ BÞ1½ðA þ BÞð1 þ sÞ  ð1  sÞðA  BÞð1 þ sÞ1

¼ 2ðA þ BÞ1ðsA þ BÞð1 þ sÞ1:

In the same way, we get

zMiþ Li¼ 2ðA þ BÞ1ðsCiþ DiÞð1 þ sÞ1 for all i ¼ 1; 2; ;

which yields the statement h

Now, let S = (A1+ B1)1 By direct calculations, we have

E ¼ SðA1 B1Þ 2SA2

0 I

¼: E1 E2

0 I

;

Li¼ S½Di1þ Ci1 ðA2þ B2ÞB

1

4 Di3 S½Di2þ Ci2 ðA2þ B2ÞB1

4 Di4

B14 Di3 B14 Di4

!

¼: Li1 Li2

Li3 Li4

;

Mi¼ S½Di1 Ci1 ðA2þ B2ÞB

1

4 Di3 S½Di2 Ci2 ðA2þ B2ÞB1

4 Di4

B1

4 Di4

!

¼: Mi1 Mi2

Mi3 Mi4

Matrix E always has an eigenvalue k = 1, which makes the straightforward extension of the results in[13,14]impossible sinceq(jEj) < 1 would be required We can still give estimates for the left hand-side of(3.6) and (3.9)by estimating sepa-rately the ‘‘differential” part and the ‘‘algebraic” one Furthermore, in order to ease the matrix calculations, we may trans-form A1, B1, and B4into upper triangular form prior to the calculations

We have

I  zE ¼ I  zE1 zE2

0 ð1 þ zÞI

; zMiþ Li¼ zMi1þ Li1 zMi2þ Li2

ð1 þ zÞLi3 ð1 þ zÞLi4

Note that Li3= Mi3, Li4= Mi4

We introduce the following auxiliary matrices

Definition 5 Assumeq(jE1j) < 1 For an integer l P 0, and for i = 1, , M; j = 1, 2, let

GijðlÞ ¼Xl

m¼0

fjEm1Lijj þ jEm1Meijjg þ ðI  jE1jÞ1

ðjElþ11 Lijj þ jElþ11 MeijjÞ; ð3:13Þ

where eMi1¼ Mi1þ E2Li3; eMi2¼ Mi2þ E2Li4; i ¼ 1; 2; ; M Further, let

GiðlÞ ¼ Gi1ðlÞ Gi2ðlÞ

jLi3j jLi4j

The following estimate will be very useful (see also[13, Theorem 3.1])

Proposition 3 Assume that the assumption(3.5)holds and the pencil {A, B} has index 1 Further, assumeq(jE1j) < 1 Then for any

z satisfying jzj 6 1, we have

jðI  zEÞ1ðL þ z eM Þj 6 GðlÞ 6 Gð0Þ: