DSpace at VNU: On the robust stability of implicit linear systems containing a small parameter in the leading term

IMA Journal of Mathematical Control and Information (2006) 23, 67–84
doi:10.1093/imamci/dni044
Advance Access publication on July 27, 2005

On the robust stability of implicit linear systems containing a small
parameter in the leading term
N GUYEN H UU D U AND V U H OANG L INH†
Faculty of Mathematics, Mechanics, and Informatics, University of Natural Sciences,
Vietnam National University, 334, Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam

This paper deals with the robust stability of implicit linear systems containing a small parameter in the
leading term. Based on possible changes in the algebraic structure of the matrix pencils, a classification of
such systems is given. The main attention is paid to the cases when the appearance of the small parameter
causes some structure change in the matrix pencil. First, we give sufficient conditions providing the
asymptotic stability of the parameterized system. Then, we give a formula for the complex stability radius
and characterize its asymptotic behaviour as the parameter tends to zero. The structure-invariant cases are
discussed, too. A conclusion concerning the parameter dependence of the robust stability is obtained.
Keywords: robust stability; stability radius; singular perturbation; implicit systems; dependence on
parameter.

1. Introduction
Recently, the system robustness analysis occurring in a wide range of applications has attracted considerable attention of researchers from mathematical and engineering communities. The concept of stability
radii introduced by Hinrichsen & Pritchard (1986a,b) has been applied and analysed for different classes
of systems in a large number of research papers, e.g. see a fairly up-to-date list of references in Bracke
(2000). A lot of problems arising in applied fields, such as modelling electrical circuits, multi-body
mechanics, optimal control, etc., can be described by differential–algebraic equations (for short, we
write DAEs; other frequently used names are implicit systems, singular systems, generalized state-space
systems and descriptor systems) which may contain one or several small parameters as well, see Brenan
et al. (1989), Section 1.3 and Kurina (1993). In this paper, the robust stability and the sensitivity depending on data for implicit linear time-invariant systems containing a small parameter will be analysed.
E XAMPLE 1 Consider the classical singular perturbation problem
y1 (x) = A11 y1 (x) + A12 y2 (x),

εy2 (x) = A21 y1 (x) + A22 y2 (x),
where yi (x) ∈ Cni, Ai j ∈ Cni ×n j, i, j ∈ {1, 2}, and ε is a parameter (0 < ε
1). This is the simplest
example for implicit systems containing a small parameter in the leading term and it has been discussed
widely in the literature of control theory, e.g. see Kokotovic et al. (1986), Dragan & Halanay (1999).
Note that the leading matrix is non-singular for ε > 0 and singular for ε = 0. A concrete numerical
example modelling a voltage regulator controlled by a so-called corrected near-optimal state feedback
† Email:

c The author 2005. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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[Received on 27 July 2004; revised on 9 January 2005; accepted on 12 January 2005]


68

N. H. DU AND V. H. LINH

law can be found in Kokotovic et al. (1986) and Qiu & Davison (1992). For examples in optimal control
problems described by implicit systems of more general form, see Kurina (1993) and references cited
therein.

z = (0

1

0)y.

Here, G
0, g > 0, C > 0 are assumed. In general, there are five small parasitic parameters
G 1 , G 2 , G 3 , C1 , C2 . We may be interested in the situation when either C1 or C2 is varying and very
close to zero and all the others are fixed for simplicity. Note that in this problem the leading matrix is
always singular. Furthermore, its rank is equal to 1 if C2 = 0 and equal to 2 otherwise.
We consider the implicit parameterized system of differential equations
(E + εF)y (x) = Ay(x),

(1.1)

where y(·): R → Cn is a vector function, E, F, A are constant matrices in Cn×n and ε is a small
positive parameter. The matrix E may be singular, but the pencil {E, A} is assumed to be regular (see
Section 2 of this paper). The matrix F describes the parameter-perturbation direction in the leading
term. In the main part of this paper, we suppose the following assumption.
A SSUMPTION A1 Pencil {E, A} is (asymptotically) stable and index{E, A} = 1.
The asymptotic stability of {E, A} means exactly that all finite generalized eigenvalues of the pencil
lie in the open left half C− of the complex plane. The almost trivial subcase when E is non-singular
is treated separately in Section 5.1. It is well known that in the case of a singular E, the asymptotic
stability of the system may be lost under an arbitrarily small perturbation affecting the leading term,
see Qiu & Davison (1992), Byers & Nichols (1993) and Du et al. (2003). The first question is how the
direction F should be to ensure the asymptotic stability of (1.1) for all sufficiently small ε?
In lots of applications, the coefficient matrix A is under the effect of an uncertain perturbation. Let
an appropriate matrix F be chosen, we consider the perturbed system
(E + ε F)y (x) = (A + B∆C)y(x),

(1.2)

and determine the complex structured stability radius defined by
r (E + εF, A; B, C) = inf{ ∆ , ∆ ∈ C p×q and (1.2) is not asymptotically stable}.

(1.3)

Here, matrix ∆ is an uncertain perturbation, matrices B ∈ Cn× p , C ∈ Cq×n specify the structure of
perturbation. The norm used here is an arbitrary matrix norm induced by vector norms. A formula of the
complex stability radius for index-1 systems was proposed in Qiu & Davison (1992), Byers & Nichols
(1993), and for general implicit systems in Du (1999), Du et al. (2003). It is worth mentioning that the

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E XAMPLE 2 In general, engineering applications may contain several small parameters and the original
equations may be DAEs as well (i.e. the leading matrix is singular). Consider the circuit known as a
loaded degree-one Hazony section under small loading, see Brenan et al. (1989). This circuit has timeinvariant linear resistors, capacitors, a current source and a gyrator. The resistances are large and the
capacitances are small. The problem is described by the linear time-invariant DAE
⎛
⎞
⎞
⎛ ⎞
⎛
0
0
g
1
G1
1
C1
⎝0
C2 0⎠ y + ⎝ g G + G 2 −1⎠ y = ⎝0⎠ u,
C −C2 0
G3
−G 3

−1
0


ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS

69

2. Preliminary
Consider a general implicit system of linear differential equations
E y (x) = Ay(x),
where E and A are given constant matrices in
may be non-singular or singular.

Kn×n ,

(2.1)

K = C or R. The leading coefficient matrix E

D EFINITION 1 The matrix pencil {E, A} is said to be regular if there exists λ ∈ C such that the
determinant of (λE − A), denoted by det(λE − A), is different from zero. Otherwise, if det(λE − A) =
0, ∀ λ ∈ C, we say that {E, A} is irregular.

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stability radius with respect to the so-called admissible perturbations occurring in both the coefficient
matrices was examined first by Byers & Nichols (1993). In Hinrichsen & Pritchard (1990), it is shown
that the complex structured stability radius of an explicit system, i.e. when E = I is set, depends
continuously on the data triplet {A, B, C}. The second question arising here is how the stability radius

of an implicit system depends on the leading term. Does r (E + εF, A; B, C) tend to r (E, A; B, C) as
ε tends to zero? If it is not true, what is the asymptotic behaviour of r (E + εF, A; B, C) for small ε?
Since ε is small, the computation of r (E + εF, A; B, C) may lead to an ill-posed problem, in general.
Therefore, an answer of the latter question seems to be of interest from both the theoretical and numerical
point of view.
Dragan (1998) considered the classical singular perturbation problem, i.e. a special case of (1.1)
with E = diag(In 1 , 0), F = diag(0, In 2 ), where In 1 , In 2 are identity matrices of indicated sizes. Based
on some results in the control theory and the asymptotic theory for singularly perturbed differential
equations, he has shown it may happen that the stability radius r (E + εF, A; B, C) does not tend to that
of the reduced system as ε tends to zero. Moreover, the exact limit was obtained. A result closely related
to Dragan (1998) was obtained in Tuan & Hosoe (1997), where the asymptotic behaviour of the H∞
norm of the transfer function for the classical singularly perturbed control problem was characterized.
We should also mention the reference lists in these papers, Kokotovic et al. (1986), and Dragan &
Halanay (1999) for more results on the classical singular perturbation problem.
Recently, by developing a new approach which is direct and completely different from that in Dragan
(1998), the authors of this paper have extended Dragan’s result to a more general class of singularly perturbed systems, see Du & Linh (2005). Based on the idea presented there, here we aim to give a more
complete analysis for the robust stability of the system of quite general form (1.1). Our approach is based
rather on some basic results in linear algebra and classical analysis. We will show that the uniform convergence of associated artificial transfer functions on the imaginary axis as the parameter tends to zero
plays an important role in the question whether the operations of taking limit and supremum commute.
The paper is organized as follows. In Section 2, we recall some basic notions and introduce a computable formula of the complex stability radius for implicit systems. In Section 3, based on the structure
properties of the matrix pencil related to (1.1), we classify the parameter perturbations and for the sake
of a convenient treatment, the system will be transformed into a block form. Some auxiliary statements
are given, too. Section 4 focuses on the main case when the perturbation in the leading term causes
structure changes in the matrix pencil. Sufficient conditions are given for providing the asymptotic stability of (1.1) and the asymptotic behaviour of r (E + εF, A; B, C) is characterized as ε tends to zero.
Section 5 discusses the problem with regular perturbation, i.e. the case when the appearance of the small
parameter ε does not cause changes in the structure of the matrix pencil. The paper is closed with some
conclusions.


70

N. H. DU AND V. H. LINH

If {E, A} is regular, then a complex number λ is called a (generalized finite) eigenvalue of {E, A} if
det(λE − A) = 0. The set of all eigenvalues is called the spectrum of the pencil {E, A} and denoted by
σ {E, A}.
Suppose that the pencil {E, A} is regular. If the matrix E is singular, then there exist non-singular
matrices W, T ∈ Cn×n such that
E=W

Ir
0

0
T −1 ,
N

A=W

H
0

0
T −1 ,
In−r

(2.2)

D EFINITION 2 Suppose that {E, A} is regular. The nilpotency index of N in the Weierstrass–Kronecker
form (2.2) is called the index of matrix pencil {E, A} and we write index {E, A} = k. If E is nonsingular, we set index {E, A} = 0.

From (2.2), it is easy to verify that for a regular matrix pencil {E, A}
deg{λ → det(λE − A)} = rank E if and only if index {E, A}

1.

(2.3)

D EFINITION 3 Suppose that {E, A} is regular. We say that the trivial zero solution of (2.1) is asymptotically (and also exponentially) stable if, for an arbitrary vector y0 ∈ Kn , there are positive constants
c, α such that the solution of the initial-value problem
E y (x) = Ay(x),
P(y(0) − y0 ) = 0

x ∈ [0, ∞),

exists uniquely and the estimate y(x)
c P y0 e−αx holds for all x 0. Here, P is an appropriately
n×n
chosen projector in K .
If the zero solution of (2.1) is asymptotically stable, we then also say that the system (2.1) is asymptotically stable.
For instance, in case index {E, A} = 1, one may choose P = I − Q where Q is the projector onto
ker (E) along S = {z ∈ Cn , Az ∈ im E}, see Griepentrog & M¨arz (1986). A difference between ordinary differential equations and DAEs is that the equality y(0) = y0 is not expected here, in general. We
also remark that, for linear time-invariant systems, the concepts of asymptotic stability and exponential
stability are equivalent. One may easily verify the following statement.
P ROPOSITION 1 The system (2.1) is asymptotically stable if and only if the matrix pencil {E, A} is
(asymptotically) stable, i.e.
σ (E, A) ⊂ C− ,
where C− denotes the open left half complex plane.
We refer to Brenan et al. (1989) and Griepentrog & M¨arz (1986) for more details on the theory of DAEs.
Now, let us suppose that system (2.1) is asymptotically stable and consider the disturbed system
E y = (A + B∆C)y,

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where Ir , In−r are identity matrices of indicated size, H ∈ Cr ×r , and N ∈ C(n−r )×(n−r ) is a matrix of
nilpotency index k, k ∈ N = {1, 2, . . .}, i.e. N k = 0, N i = 0 for i = 1, 2, . . . , k − 1. Formula (2.2) is
well known to be the canonical Weierstrass–Kronecker form of pencil {E, A}, see Brenan et al. (1989)
and Griepentrog & M¨arz (1986). If N is a zero matrix, then k = 1 holds.


71

ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS

where B ∈ Kn× p , C ∈ Kq×n are given matrices and ∆ ∈ K p×q is an uncertain disturbance. The matrix
B∆C is called a structured perturbation. We define
V K (E, A; B, C) = {∆ ∈ K p×q , {E, A + B∆C} is either unstable or irregular}.
In Du (1999) (see also Du et al., 2003), the structured stability radii of (2.1) are defined as
rK (E, A; B, C) = inf{ ∆ , ∆ ∈ VK (E, A; B, C)},

P ROPOSITION 2 (Du, 1999, see also Qiu & Davison, 1992; Byers & Nichols, 1993) Suppose that the
matrix pencil {E, A} is regular and asymptotically stable. Then the complex stability radius of (2.1) has
the representation
sup C(s E − A)−1 B

r (E, A; B, C) =

−1

s∈iR

.

(2.4)

Here, iR denotes the imaginary axis of the complex plane. Note that the above formula of the stability
radius holds not only for the Frobenius norm (see Byers & Nichols, 1993) and the Euclidean norm (see
Qiu & Davison, 1992) but also for any matrix norm induced by vector norms.
R EMARK 1 The matrix function G(s) = C(s E − A)−1 B is called the associated transfer matrix, see
Hinrichsen & Pritchard (1986b). For a non-singular matrix E, it is easy to verify that
lim

|s|→∞

G(s) = lim

|s|→∞

C(s E − A)−1 B = 0.

In the case of a singular E, by using the canonical Weierstrass–Kronecker form (2.2), we write
G(s) = C(s E − A)−1 B = C T

(s Ir − H )−1
0

0
−

k−1
i

i=0 (s N )

W −1 B

(2.5)

and easily deduce that G(s) tends to either a finite number (e.g. in case k = 1) or infinity when
|s| → +∞. Hence, it is clear that the stability radius for a system of index less than or equal to 1
is strictly positive. This fact does not hold for a higher index system with respect to an arbitrary perturbation structure. For example, if k > 1 and B = C = I , one may find that
lim

|s|→∞

G(s) = +∞,

which implies r (E, A; B, C) = 0.
R EMARK 2 It is not difficult to show that in the case of index-1 systems, the stability radius introduced
here has also the structure-preserving property, see Qiu & Davison (1992) and Byers & Nichols (1993).
That is, under any perturbation of the norm less than the value of the stability radius, not only the
asymptotic stability is preserved, but also the index of the perturbed matrix pencil. The latter property
also implies the degree invariance for the generalized characteristic polynomial.

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where · is an arbitrary matrix norm induced by vector norms. Depending on K = C or K = R,
we talk about the complex or real stability radius, respectively. In this paper, we focus on the complex
one, only. Thus, if we do not mention explicitly, the stability radius will mean the structured complex
one. Furthermore, the subscript C in the notation of the stability radius will be omitted for brevity. The
following result is analogous to that for explicit linear systems, see Hinrichsen & Pritchard (1986b).

72

N. H. DU AND V. H. LINH

M=

M11
M21

M12
M22

be given with appropriate block sizes. Suppose that M22 is non-singular. Then, the decomposition
M=

In 1
0

−1
M12 M22
In 2

−1
M21
M11 − M12 M22
0

0
M22

0

In 1

−1
M21
M22

In 2

(2.6)

holds. A similar decomposition holds true in the case of a non-singular M11 , as well.
3. Classification and the simplified block form of the problem
Now, we return to the parameterized system (1.1). In the theory of DAEs, relevant structure properties
of matrix pencils play important roles. In addition to the index assumption on the matrix pencil {E, A},
we require, and later give sufficient conditions, that the index of the parameterized matrix pencil be a
fixed constant less than or equal to 1 and the rank of the leading term be invariant for all sufficiently
small positive ε. We are interested in the following cases:
C1. Index-change case: The index of {E + εF, A} changes from 0 to 1 when ε becomes 0, i.e. there
exists ε0 > 0 such that index{E + εF, A} = 0 for all ε, 0 < ε ε0 , but index{E, A} = 1. Of course,
the index change implies the rank change of the leading term.
C2. Rank-change case: The index of {E +εF, A} is equal to 1 for all sufficiently small ε and ε = 0;
the rank of (E + εF) is constant for all sufficiently small ε but changes when ε becomes 0. This means
exactly that some generalized finite eigenvalues may be lost as ε reaches 0.
C3. Structure-invariance case: The index of the parameterized matrix pencil {E + εF, A} as well
as the rank of the leading term do not change for all sufficiently small ε, i.e. there exists ε0 > 0 such that
index {E + ε F, A} = index{E, A} and rank(E + εF) = rankE for all ε, 0 < ε ε0 . This parameter
perturbation ε F belongs to the class of admissible perturbations defined in Byers & Nichols (1993).

If Case C3 occurs, we refer to (1.1) as a regular perturbation problem since the appearance of ε does
not cause changes of the relevant structure properties of the matrix pencil, while Cases C1–C2 are called
singularly perturbed problems. In general, degenerate cases also occur, e.g. when—for ε in an arbitrary
small right neighbourhood of 0—matrix pencil {E + εF, A} has a varying index, has a constant higher
index or is irregular. These cases are excluded from our consideration.
For the sake of simplicity, from now on the problem (1.1) is set in a block form. Without loss of
generality, we suppose that the triplet {E, F, A} has the form
E=

E 11
0

0
,
0

F=

F11
F21

F12
,
F22

A=

A11
A21

A12
,
A22

(3.1)

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R EMARK 3 The concept of the stability radius can be extended in a more general sense as follows.
Suppose that all the eigenvalues of the undisturbed matrix pencil lie in a prescribed open subset Cg
of the complex plane. We want to determine how large perturbations the system can tolerate without
losing the property that its spectrum remains in Cg . In the asymptotic stability analysis of differential
equations, the open subset Cg is chosen to be C− . It is trivial to obtain a formula of a Cg -stability
radius analogously to that in Proposition 2. In fact, we should simply replace iR by the boundary set of
Cb = C\Cg . As a consequence of the definition, positivity of a Cg -stability radius with an appropriately
chosen subset Cg implies the continuity of the pencil spectrum with respect to data.
In the next sections, we need a well-known decomposition formula of block matrices, see, e.g.
Gantmacher (1960), Section 2.5. Let an arbitrary matrix


ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS

73

where E 11 is non-singular. The condition index{E, A} = 1 is equivalent to the non-singularity of A22 .
Furthermore, the matrices B, C are decomposed into block forms as
B=

B1
,

B2

C2 ).

C = (C1

(3.2)

Here, all the submatrices are supposed to have appropriate sizes, e.g. Ai j has the size n i ×n j , i, j = 1, 2,
where n 1 + n 2 = n. If the original matrix E is not of the sparse and block form as in (3.1), one may use,
e.g. a singular-value decomposition
E = U ∗ Σ V,

E new = Σ,

Fnew = U F V ∗ ,

Anew = U AV ∗ .

(3.3)

Accordingly, the perturbation structure is determined by
Bnew = U B,

Cnew = C V ∗ .

Finally, the new matrices should be decomposed appropriately to have the form (3.1). It is obvious that
the new system and the original one are equivalent in the sense that they possess the same robust-stability
properties (asymptotic stability, the complex structured stability radius).
R EMARK 4 As a matter of fact, it is sufficient to use any decomposition of the form

E=P

E 11
0

0
Q,
0

where matrices E 11 , P, Q are non-singular, in order to transform the problem (1.1) into the form (3.1).
Now, we present some auxiliary statements which will be useful in the next sections. The first one
can be checked easily. Hence, it is stated without proof.
L EMMA 1 Suppose that P, Q, R are arbitrary matrices in Cn×n , ε is a real parameter and s is a complex
variable. If P is a non-singular matrix, then there exist a sufficiently small ε0 > 0 such that
(a) Matrix (P + εQ) is invertible for all ε, 0 ε ε0 . Furthermore, the norm of the inverse matrix
(P + εQ)−1 is bounded on [0, ε0 ].
(b) The norm of [s(P + εQ) − R]−1 converges uniformly to 0 w.r.t. ε on [0, ε0 ] when |s| tends
to +∞.
The latter statement remains valid in the case when R is a bounded matrix function of variables ε, s.
The next statement deals with the possible order change of operations of taking limit and supremum for
a two-variable continuous function in a non-compact domain.
L EMMA 2 Suppose that f (x, t) is a continuous real function in domain D = [a, b] × [0, ∞), where
a, b are finite numbers. Furthermore, f (x, t) converges to a continuous function g(x) uniformly with
respect to x ∈ [a, b] as t tends to infinity. Then,
(a) f (x, t) is uniformly continuous on D.
(b) limx→x0 supt∈[0,∞) f (x, t) = supt∈[0,∞) f (x0 , t), for all x0 ∈ [a, b].

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n

where U, V are unitary matrices, Σ = diag(σ1 , σ2 , . . . , σn ), {σi }i=1
are the singular values of E in a
non-increasing order. Then, one obtains an equivalent problem with the new data set


74

N. H. DU AND V. H. LINH

Proof. Let be an arbitrarily small positive number.
(a) Due to the uniform convergence, there exists a number T such that
| f (x, t) − g(x)|

/3,

∀ x ∈ [a, b], t

T.

Let us fix such a number T . Since g(x) is continuous on [a, b], thus uniformly continuous, there
exists a number δ1 > 0 such that
|g(x1 ) − g(x2 )|

/3,

∀ |x1 − x2 |

δ1 .

| f (x1 , t1 ) − f (x2 , t2 )|

| f (x1 , t1 ) − g(x1 )| + | f (x2 , t2 ) − g(x2 )| + |g(x1 ) − g(x2 )|

holds for all t1 , t2 T, |x1 − x2 | δ1 . On the other hand, since f (x, t) is uniformly continuous
on the compact subdomain D1 = [a, b] × [0, T ], there exists a number δ2 > 0 such that
| f (x1 , t1 ) − f (x2 , t2 )|

,

∀ (x1 , t1 ), (x2 , t2 ) ∈ D1 ,

(x1 , t1 ) − (x2 , t2 )

δ2 .

Since f (x, t) is continuous and T is fixed, it is also clear that there exists a number δ3 > 0
such that
| f (x1 , t1 ) − f (x2 , t2 )|

,

∀ x1 , x2 ∈ [a, b], t1 < T, t2 > T,

(x1 , t1 ) − (x2 , t2 )

δ3 .

Set δ = min{δ1 , δ2 , δ3 } and obtain
| f (x1 , t1 ) − f (x2 , t2 )|

,

∀ (x1 , t1 ), (x2 , t2 ) ∈ D,

(x1 , t1 ) − (x2 , t2 )

δ.

This proves the uniform continuity of f (x, t) on D.
(b) As a consequence of the first statement, there exist a number κ (independent of x0 , t) such that
f (x0 , t) −

f (x, t)

f (x0 , t) + ,

∀ x, t : |x − x0 |

κ, t

0.

Taking the supremum with t varying on [0, ∞), one obtains
sup

f (x0 , t) −

t∈[0,∞)

f (x, t)

sup
t∈[0,∞)

sup

f (x0 , t) + ,

∀ x : |x − x0 |

κ.

t∈[0,∞)

The proof of the second statement is complete.
P ROPOSITION 3 Suppose that E ∈ Cn×n is a constant matrix, A(ε): [0, ε0 ] → Cn×n , B(ε): [0, ε0 ] →
Cn× p , C(ε): [0, ε0 ] → Cq×n are continuous matrix functions, the pencil {E, A(0)} is of index-1 and
stable. Then, the relation
lim sup G(ε, t) = sup G(0, t) ,
(3.4)
ε→+0 t∈iR

where G(ε, t) := C(ε)(tE

− A(ε))−1 B(ε),

t∈iR

holds true.

Proof. First, we note that there obviously exists a number ε1 ε0 such that the pencil {E, A(ε)} is of
index-1 and stable for all ε
ε1 , too. Therefore, the function G(ε, t) is well defined (and continuous)
when ε ∈ [0, ε1 ], t ∈ iR. Without loss of generality, we assume that the matrices E, A(ε), B(ε), C(ε)

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Hence, the inequality


ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS

75

are in block form (3.1)–(3.2). The index of {E, A(ε)} implies that A22 (ε) is invertible for all ε ∈ [0, ε1 ].
Using formula (2.6) for computing the inverse of block matrices, after some matrix calculations, we
arrive at
G(ε, t) = D(ε) + C(ε)(tE11 − A(ε))−1 B(ε), with
A(ε) = A11 (ε) − A12 (ε)A22 (ε)−1 A21 (ε),
C(ε) = C1 (ε) − C2 (ε)A22 (ε)−1 A21 (ε),

B(ε) = B1 (ε) − A12 (ε)A22 (ε)−1 B2 (ε),

(3.5)

D(ε) = −C2 (ε)A22 (ε)−1 B2 (ε).

C OROLLARY 1 Under the same assumptions as in Proposition 3, we have
lim r (E, A(ε); B(ε), C(ε)) = r (E, A(0); B(0), C(0)).

ε→+0

We conclude that the complex stability radius of an index-1 system depends continuously on the second
term as well as the perturbation structure.
4. The singular perturbation problem
4.1

The index-change case

In this section and the next one, we suppose that the problem is set of form (3.1), (3.2). First, we
characterize the class of direction matrices F inducing the index change (Case C1).
P ROPOSITION 4 The parameterized leading term (E + εF) is non-singular for all sufficiently small ε
if and only if the matrix pencil {E, F} is regular.
From now on, we are interested in an extension of the robust-stability result on the classical singular
perturbation problem (Dragan, 1998). Techniques used in Du & Linh (2005) can be applied with a slight
modification. In this subsection, in addition to the assumptions in Section 1, we suppose the following
conditions.
A SSUMPTION A2 Matrix F22 is non-singular.
A SSUMPTION A3 Matrix pencils {F22 , A22 } and {E 11 , A11 − A12 A−1
22 A21 } are (asymptotically) stable,
i.e. their spectra belong to the open left half plane C− .
It is trivial to check the equality σ {E, A} = σ {E 11 , A11 − A12 A−1
22 A21 }, i.e. the second stability
condition in Assumption A3 has already been provided in fact by the stability of pencil {E, A}, see
Assumption A1. First, we establish a statement on the asymptotic stability of (1.1).
P ROPOSITION 5 Assume that Assumptions A1–A3 hold true. Then the system (1.1), (3.1) is asymptotically stable for all sufficiently small ε, i.e. there exists a positive number ε such that σ {E + εF, A} ⊂ C−
for each parameter ε ∈ [0, ε].

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The matrix functions A(ε), B(ε), C(ε), D(ε) are continuous and since E11 is non-singular, the expression C(ε)(tE11 − A(ε))−1 B(ε) converges to zero uniformly with respect to ε ∈ [0, ε1 ] as |t| tends
to infinity. Thus, the function G(ε, t) fulfils the conditions of Lemma 2. Therefore, the equality
(3.4) holds.
We obtain an extension of Hinrichsen & Pritchard (1990), Proposition 2.2, to index-1 DAEs.


76

N. H. DU AND V. H. LINH

Proof. The proposition means exactly that the parameterized system is asymptotically stable for sufficiently small ε if the reduced slow system with pencil {E, A} and the so-called boundary-layer fast
system with pencil {εF22 , A22 } are simultaneously asymptotically stable.
It is clear that the pencil {E + εF, A} has index-0 for sufficiently small ε and consequently, it
has exactly n finite (non-zero) eigenvalues. Denote the eigenvalues (counted with multiplicity) of the
following matrix pencils for ε > 0, respectively,
σ {E 11 , A11 − A12 A−1
22 A21 } = µ1 , . . . , µn 1 ,
σ {F22 , A22 } = ν1 , . . . , νn 2 ,
We will show that, by ordering the eigenvalues of {E + ε F, A} appropriately, we have
ελ j (ε) = ν j + o(1),
λn 2 +i (ε) = µi + o(1),

j = 1, 2, . . . , n 2 ,
i = 1, 2, . . . , n 1 , ε → +0,

i.e. the first n 1 eigenvalues of {E + εF, A} are asymptotically equal to those of the reduced pencil
{E, A}, while the other n 2 eigenvalues have asymptotic behaviour like those of pencil {ε F22 , A22 } as ε
tends to zero.
On one hand, we observe that λ is an eigenvalue of {E + εF, A} if and only if ελ is an eigenvalue
of the pencil

ε A11 ε A12
E 11 + εF11 εF12
,
.
(4.1)
F21
F22
A21
A22
The matrix pencil (4.1) contains small parameter perturbations in both terms. Since the reduced leading
term (when ε = 0 is set) is non-singular, so is the parameterized leading term for sufficiently small ε,
see Lemma 1, part (a). Then, the eigenvalue problem of the matrix pair (4.1) may be reduced to the
classical eigenvalue problem of a perturbed matrix. Invoking the continuity of the spectrum of matrices,
there are n 2 eigenvalues of (4.1) (ordered appropriately) such that
ελ j (ε) = ν j + o(1),

j = 1, 2, . . . , n 2 , ε → +0.

Consequently, there exists a number ε1 such that the pencil {E + ε F, A} has n 2 eigenvalues in C− for
ε ε 1 . Furthermore, those tend to infinity as ε tends to zero.
On the other hand, if λ is a non-zero eigenvalue of pencil {E + ε F, A}, then the value λ−1 is a
non-zero eigenvalue of pencil {A, E + εF} and vice versa. Since A is non-singular (otherwise 0 would
be an eigenvalue of {E, A} which contradicts the stability assumption), by a similar argument as above,
we state that the spectrum of {A, E + εF} tends to that of {A, E} as ε tends to zero. Taking into account
that µi−1 , i = 1, 2, . . . , n 1 , are non-zero eigenvalues of {A, E}, there are (other) n 1 eigenvalues of
{E + ε F, A} (ordered appropriately) such that
λn 2 +i (ε)−1 = µi−1 + o(1),

i = 1, 2, . . . , n 1 , ε → +0.

Hence, there exists a number ε 2 such that the pencil {E + ε F, A} has n 1 bounded eigenvalues in C− for
ε ε2 .
By setting ε = min{ε1 , ε 2 }, the above arguments imply that the parameterized pencil {E + ε F, A}
should have exactly n = n 1 + n 2 eigenvalues in C− for each ε ∈ (0, ε]. The proof of Proposition 4 is
completed.

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σ {E + ε F, A} = λ1 (ε), λ2 (ε), . . . , λn 1 +n 2 (ε) .


77

ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS

r (E, A; B, C) =

sup G s (t)

−1

t∈iR

,

A = A11 − A12 A−1
22 A21 ,
C = C1 − C2 A−1
22 A21 ,

G S (t) = D + C(t E 11 − A)−1 B,
B = B1 − A12 A−1
22 B2 ,

(4.2)

D = −C2 A−1
22 B2 ;

and
r (F22 , A22 ; B2 , C2 ) =

sup G F (s)

s∈iR

−1

,

where G F (s) = C2 (s F22 − A22 )−1 B2 .

(4.3)

We note that the stability radius corresponding to the quadruplet {F22 , A22 ; B2 , C2 } and that for {ε F22 ,
A22 ; B2 , C2 } are the same for all positive ε. Similarly to the results in Dragan (1998) and Du & Linh
(2005), we state the following theorem.
T HEOREM 1 Let Assumptions A1–A3 hold. Then the stability radius of the singular perturbation problem (1.1), (3.1) is asymptotically equal to the minimum of the stability radii of the reduced slow system
and of the fast boundary-layer system for small ε, i.e.
lim r (E + εF, A; B, C) = min{r (E, A; B, C), r (F22 , A22 ; B2 , C2 )}.

ε→+0

Proof. Let us fix a number ε provided by Proposition 5. For ε ∈ (0, ε], a formula similar to (4.2) can be
obtained for r (E + εF, A; B, C) as well. However, it is too complicated and inappropriate for further
examination. It is more advantageous, first, to transform the system (1.1), (3.1) into an equivalent one
with a block-diagonal leading term. Applying the decomposition formula (2.6) to (E + εF), we have
E + εF =

In 1
0

−1
F12 F22
In 2

−1
F21 ]
0
E 11 + ε[F11 − F12 F22
0
ε F22

0

In 1
−1
F22 F21

In 2

.

Let us denote the upper and lower triangular matrices in the above formula by P and Q, respectively,
and introduce the new equivalent system with the modified coefficient matrices
Eε =
A = P −1 AQ −1 =

−1
F21 ]
E 11 + ε[F11 − F12 F22
0

In 1
0

−1
−F12 F22
In 2

0
,
ε F22
(4.4)

A11
A21

A12
A22

In 1

−1
F21
−F22

0
In 2

.

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R EMARK 5 A similar idea, combined with the continuity of roots of polynomials, was used in the proof
of Theorem 1 in Du & Linh (2005). The argument used here is closely related to the positivity of the
stability radius defined and analysed in Byers & Nichols (1993), see Remark 3 of this paper. Another
proof of Proposition 5 is also possible, if one transforms the matrix E + ε F into the diagonal form and
then applies Proposition 3.1.1 in Dragan & Halanay (1999) to the transformed eigenvalue problem.
Let us denote the stability radius of the reduced slow system (corresponding to the quadruplet
{E, A, B, C}) and that of the boundary-layer system (corresponding to the quadruplet {F22 , A22 ; B2 ,
C2 }) by r (E, A; B, C) and r (F22 , A22 ; B2 , C2 ), respectively. Using Proposition 2 and formula (2.6) for
computing the inverses of block matrices, we obtain explicit expressions for r (E, A; B, C), r (F22 , A22 ;
B2 , C2 ) as follows


78

N. H. DU AND V. H. LINH

The perturbation structure is described by the new matrix pair
B = P −1 B =

−1
B1 − F12 F22
B2

B2

−1
C = C Q −1 = (C1 − C2 F22
F21

,

C2 ).

We will refer to the blocks of A, B, C marked with bar. Accordingly, we have
−1
F21 ).
E ε,11 = E 11 + ε(F11 − F12 F22

−1

−1

A = A11 − A12 A22 A21 ,

B = B 1 − A12 A22 B 2 ,

−1

−1

C = C 1 − C 2 A22 A21 ,

D = −C 2 A22 B 2 ,

where A, B, C, D were defined in (4.2). Applying Proposition 2 to the new system and taking into
account that the stability radius corresponding to the quadruplet {(E + εF), A, B, C} and that corresponding to {E ε , A, B, C} are the same. For ε ∈ [0, ε], the formula
r (E + εF, A; B, C) = r (E ε , A, B, C) =

sup G ε (t)

−1

t∈iR

,

where G ε (t) = Dε (t) + Cε (t)(t E ε,11 − Aε (t))−1 Bε (t),
Aε (t) = A11 + A12 (tεF22 − A22

)−1 A

21 ,

Cε (t) = C 1 + C 2 (tεF22 − A22 )−1 A21 ,

Bε (t) = B 1 + A12 (tεF22 − A22

(4.5)
)−1 B

2,

Dε (t) = C 2 (tεF22 − A22 )−1 B 2

is valid. As ε tends to zero, the function G ε (t) converges point-wise to G S (t) for t ∈ iR. However,
taking limit and supremum do not commute, i.e. it may happen that limε→+0 r (E + εF, A; B, C) =
r (E, A; B, C).
From now on, the proof advances quite analogously to that of Theorem 2 in Du & Linh (2005). The
key idea is that, on one hand, for sufficiently large |t|, the second expression in the formula of G ε (t) is
arbitrarily small; therefore, the function G ε (t) is close to G F (s) with s = εt. On the other hand, in a
given bounded domain, the function G ε (t) can be approximated by G S (t) for small ε. Indeed, since F22
is non-singular and A22 = A22 holds, we have
sup (tεF22 − A22 )−1 = sup (s F22 − A22 )−1 < +∞.

t∈iR

s∈iR

Hence, the functions
Aε (t) , Bε (t) , Cε (t) and Dε (t)
are bounded on iR and the bounds are independent of positive ε. Due to Lemma 1, part (b), the function
Cε (t)(t E ε,11 − Aε (t))−1 Bε (t) converges to zero uniformly w.r.t. ε ∈ [0, ε] as |t| → ∞.

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Now, we arrive at a robust-stability problem with data {E ε , A, B, C} which has the form very close to

that analysed in Du & Linh (2005). By elementary calculations, it is easy to verify that


79

ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS

Let us choose an arbitrary δ > 0. We show that the inequalities
max sup G S (t) , sup G F (s)
t∈iR

− 2δ

s∈iR

sup G ε (t)

t∈iR

max sup G S (t) , sup G F (s)
t∈iR

(4.6)

+δ

s∈iR

hold for sufficiently small ε. We recall that the variables t and s are considered on the line iR, only.

Cε (t)(t E ε,11 − Aε (t))−1 Bε (t)
Therefore, for t with |t|

δ,

|t|

T.

T , we have
G ε (t)

C 2 (tεF22 − A22 )−1 B 2 + δ.

Hence, we obtain
sup C 2 (tεF22 − A22 )−1 B 2 + δ

sup G ε (t)

|t| T

|t| T

G F (s) + δ

= sup

|s| εT

sup G F (s) + δ.

s∈iR

(4.7)

On the other hand, on the compact domain {(t, ε), |t| T, 0 ε ε}, G ε (t) is continuous
as a two-variable function, hence uniformly continuous, too. Therefore, there exists a sufficiently
small ε1 = ε1 (δ) such that for ε ε1 , we have
sup G ε (t)

|t| T

Thus, for ε

sup G S (t) + δ

|t| T

sup G S (t) + δ.

t∈iR

ε1 , we obtain
sup G ε (t)

t∈iR

max sup G S (t) , sup G F (s)
t∈iR

s∈iR

+ δ.

(b) Now, we prove the first inequality in (4.6).
Analogously to (4.7), we have
sup G ε (t)

|t| T

sup

|s| εT

G F (s) − δ.

Since G F (s) is continuous w.r.t. s ∈ iR, there exists a sufficiently small ε2 = ε2 (δ) such that
for ε ε2 , the inequality
sup

|s| εT

G F (s)

sup G F (s) − δ

s∈iR

holds. Hence, we obtain
sup G ε (t)

|t| T

sup G F (s) − 2δ.

s∈iR

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(a) First, we prove the last inequality in (4.6).
Due to the uniform convergence verified above, there exists a sufficiently large number T = T (δ)
(T is independent of ε) such that


80

N. H. DU AND V. H. LINH

On the other hand, since supt∈iR G S (t) is finite, there exists a number t0 = t0 (δ) ∈ iR such
that
G S (t0 )
sup G S (t) − δ.
t∈iR

Moreover, because of the continuity of G ε (t0 ) as a function of ε, there exists a sufficiently
small ε3 = ε3 (δ) such that for ε ε3 , we obtain
sup G ε (t)

t∈iR

G S (t0 ) − δ

sup G S (t) − 2δ.

t∈iR

min{ε2 , ε3 }, the inequality
sup G ε (t)

t∈iR

max sup G S (t) , sup G F (s)
t∈iR

s∈iR

− 2δ

holds.
Then, for ε

min{ε1 , ε2 , ε3 }, the inequalities in (4.6) hold. The proof of Theorem 1 is complete.

R EMARK 6 The results in this section let us conclude that under the conditions supposed here on F22
and for sufficiently small ε, the blocks F11 , F12 , F21 have almost no effect on the robust stability of the
singular perturbation problem.
R EMARK 7 Assumption A2 is sufficient to provide an index-change perturbation problem, only. Index
change may occur with a singular F22 as well. A right question is that what we can do in such a case?
In the particular case F22 = 0, by transforming the leading term into the block-diagonal form as in (5.1)
below, one can easily find sufficient conditions on the other blocks Fi j to ensure that the pencil {E +

εF, A} be index-0 and remain stable for all sufficiently small ε. However, if F22 is a non-zero and
singular matrix, the situation becomes more complicated.
4.2 The rank-change case
A special problem of the rank-change case was discussed in Du & Linh (2005), where F11 , F12 , F21 are
zero matrices. The index of the parameterized system is invariant (and equal to 1) if index {F22 , A22 } =
1. Then, under the stability condition A3, statements similar to Proposition 5 and Theorem 1 in the
previous subsection were obtained in Du & Linh (2005).
The restriction on the direction matrix F may be somewhat relaxed. In this subsection, instead of
A2, we suppose the following assumption.
A SSUMPTION A2# Matrix F is of block-triangular form, i.e. F21 (or F12 ) is a zero matrix and index
{F22 , A22 } = 1.
As well as in the index-change case, the stability condition A3 is supposed to be true. First, the
problem is transformed into an equivalent one with a block-diagonal leading term. Under the assumption
that F21 = 0 (the case of F12 = 0 is treated similarly), we have
E + εF =
with
Qε =

In 1
0

E 11 + ε F11
0

0
Qε ,
ε F22

ε(E 11 + εF11 )−1 F12
.

In 2

(4.8)

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Therefore, for ε

G ε (t0 )


ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS

81

Then, after modifying the data set as in the previous subsection, we obtain an equivalent problem with
the leading term
E 11 + εF11
0
.
Eε =
0
ε F22
Let Aε , B ε , C ε denote the transformed matrices, respectively. We have
−1

Aε = AQ ε ,

B ε = B,

−1

C ε = C Qε .

P ROPOSITION 6 Assume that A1, A2# and A3 hold true. For all sufficiently small ε, the pencil {E +
εF, A} remains of index-1 and stable, i.e. there exists ε such that
index {E + εF, A} = 1 and

σ {E + εF, A} ⊂ C−

hold for all ε ∈ [0, ε].
Proof. It is clear that the transformed pencil has the same index as the original one. Since index {F22 ,
A22 } = 1, we have index {F22 , Aε,22 } = 1 for all sufficiently small ε, see Remark 2. Here, Aε,22
denotes the corresponding submatrix of Aε . Then, one can easily verify that the transformed matrix
pencil {E ε , Aε } is of index-1, too. To verify the stability, one may repeat the arguments used in the
proof of Proposition 5. We note that, using (2.3), the number of generalized finite eigenvalues of pencil
{E ε , Aε } is exactly rank E 11 + rank F22 for all sufficiently small ε.
Now we state an analogue of Theorem 1.
T HEOREM 2 Let Assumptions A1, A2# and A3 hold. As ε tends to zero, the stability radius of the
singular perturbation problem (1.1), (3.1) converges to the minimum of the stability radii of the reduced
slow system and of the fast boundary-layer system, i.e.
lim r (E + εF, A; B, C) = min{r (E, A; B, C), r (F22 , A22 ; B2 , C2 )}.

ε→+0

Proof. The scheme of the proof of Theorem 1 can be repeated without difficulties. The only difference
is that now the transformed matrices Aε , C ε contain O(ε) perturbations, too. The uniform boundedness
of the auxiliary functions Aε (t), Bε (t), Cε (t) and Dε (t) defined as in (4.5) remains valid. Furthermore,
we should also take into account, by invoking Proposition 3, the relation
lim sup C ε,2 (s F22 − Aε,22 )−1 B2 = sup C2 (s F22 − A22 )−1 B2 .

ε→0 s∈iR

s∈iR

Similar to Remark 4, we underline again that (only) the block F22 plays a dominant role in the robust
stability of the singular perturbation problem as the parameter tends to zero.
5. The regular perturbation problem
5.1

The case with a non-singular E

Now, we are interested in the behaviour of the stability radius when the parameter perturbation does not
change the algebraic structure of the original pencil. First, we discuss the almost trivial subcase when E

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Noting that Q ε = I + O(ε), ε → 0, hence Aε , C ε are simply obtained from the original ones by using
O(ε) perturbations, only. Furthermore, these two matrices are continuous functions in variable ε. First,
we establish the following statement.


82

N. H. DU AND V. H. LINH

is non-singular. From Lemma 1, we know that the parameterized leading term remains non-singular for
sufficiently small ε. We obtain immediately the following proposition.
P ROPOSITION 7 Assume that E is non-singular and {E, A} is stable. Then the pencil {(E + εF), A} is
of index-0 and stable for all sufficiently small ε, i.e. there exist a positive number ε such that E + εF is

non-singular and σ {E + εF, A} ⊂ C− for all ε ∈ [0, ε].

T HEOREM 3 Let the assumptions in Proposition 7 hold. Then the stability radius r (E + εF, A; B, C)
tends to that of the reduced system as ε tend to zero, i.e.
lim r (E + εF, A; B, C) = r (E, A; B, C).

ε→+0

5.2

The case with a singular E

In this subsection, the case with a singular E is considered. Since (E 11 + εF11 ) is non-singular for
sufficiently small ε, using a decomposition of the form (2.6), we get
E + εF = P ε

E 11 + εF11
0

0
ε[F22 − ε F21 (E 11 + ε F11 )−1 F12 ]

where
Pε =

In 1
εF21 (E 11 + εF11 )−1

0
,

In 2

Qε =

In 1
0

Qε,

(5.1)

ε(E 11 + εF11 )−1 F12
.
In 2

The rank-invariant assumption implies that for sufficiently small ε, the block ε[F22 − ε F21 (E 11 +
εF11 )−1 F12 ] must be identically zero. It holds true, for instance, when F21 , F22 are zero matrices (see
also the characterization of admissible perturbations in Byers & Nichols, 1993). By expanding the expression [F22 − ε F21 (E 11 + εF11 )−1 F12 ] into a power series of ε, we obtain a necessary and sufficient
condition for a regular perturbation.
P ROPOSITION 8 Assume that the pencil {E, A} is of index-1. Then the pencil {E + εF, A} has the same
index and the same number of finite eigenvalues as the reduced pencil {E, A} for all sufficiently small ε
if and only if
−1
−1 i
F22 = 0, F21 E 11
(F11 E 11
) F12 = 0, i = 0, 1, 2 . . . .
(5.2)
Using the transformation technique as in Section 4, the original problem can be transformed into a new
one with leading term

E 11 + ε F11 0
.
Eε =
0
0
Furthermore, the new matrices Aε , B ε , C ε are obtained from the original ones by using O(ε) perturbations, only.
The following theorem characterizes the robust stability of this regular perturbation problem.

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We note that this problem can be considered as a particular subcase of that discussed in the previous
section, namely, where n 2 = 0 is taken. The proof of Theorem 1 can be repeated with n 2 = 0, i.e. the
boundary-layer fast system and obviously the function G F (s) are no longer defined. Alternatively, the
parameterized implicit system can be transformed into an equivalent explicit one. The new coefficient
matrix Aε and the new perturbation-structure matrices B ε , C ε can be obtained from those of the reduced explicit system with O(ε) perturbations. Then, one may refer to Proposition 2.2 in Hinrichsen &
Pritchard (1990) as well and easily get the following theorem.


ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS

83

T HEOREM 4 Let Assumption A1 hold again and the perturbation direction F satisfy (5.2). Then the
parameterized pencil {E + εF, A} is stable for all sufficiently small ε. Furthermore, we have
lim r (E + εF, A; B, C) = r (E, A; B, C).

ε→+0

6. Conclusion
In this paper, the robust stability of implicit linear systems containing a small parameter in the leading term is analysed. The problems are classified accordingly to structure changes of the pencil under

the effect of parameter perturbations. We have attempted to examine the asymptotic stability and the
asymptotic behaviour of the stability radius as the small parameter tends to zero. In the cases when
some structure property changes, sufficient conditions on the parameter perturbation are given to ensure
the asymptotic stability. It is shown that the stability radius may be discontinuous as a function of the
parameter. Furthermore, as the parameter tends to zero, the stability radius of the parameterized system
tends to the minimum of those of the reduced slow system and of the boundary-layer fast system. In
case of regular perturbation problems, i.e. when the appearance of a sufficiently small parameter does
not affect the pencil structure, it is proven that the asymptotic stability is preserved for all sufficiently
small parameter. In addition, the stability radius of the parameterized system is asymptotically equal
to that of the reduced system, i.e. the stability radius depends continuously on the parameter. Comparing two kinds of perturbations we arrive at a conclusion that a small parameter perturbation affecting
the ‘differential part’ of an index-1 system has almost no significance in the robustness analysis as the
parameter tends to zero. In contrary, a parameter perturbation affecting on the ‘algebraic part’ plays an
important role in the robust stability. Furthermore, the system affected by a small parameter perturbation
has—in the best case—asymptotically the same stability radius as the reduced system.
The robustness analysis presented in this paper is devoted to linear time-invariant systems of index
less than 1, only. Extending the results to time-invariant systems of higher index or time-varying systems
would be of interest. A similar analysis for the (structured) real stability radius of implicit systems of
the form (1.1) would be another interesting issue, too. It is known that, in contrary to the complex one,
the real structured stability radius has a more complicated formula (see Qiu et al., 1995) and may be a
discontinuous function of data even in the case of explicit systems, see Hinrichsen & Pritchard (1990).
At this moment, these problems are still open.
Acknowledgements
This paper was written during the visit of V. H. Linh at the Computer and Automation Research Institute,
the Hungarian Academy of Sciences, Budapest. He is grateful to all persons arranging the visit. The
authors thank Professor K. Balla for useful comments on the paper.
R EFERENCES
B RACKE , M. (2000) On stability radii of parametrized linear differential–algebraic systems. Ph.D. Thesis, University of Kaiserslautern, Germany.

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Proof. The first statement is obtained as a direct consequence of continuity of the spectrum for the (reduced) index-1 system. It is worth noting that the parameter perturbations occurring in the transformed
system belong to the class of admissible perturbations, see Byers & Nichols (1993). For the second
statement, the arguments in the proof of Proposition 3 can be repeated with a slight modification, only.
Here, the only difference is that E ε,11 = E 11 + εF11 , i.e. E ε,11 contains a O(ε) term, too.


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