linear differential-algebraic equations

Asymptotic integration of

linear differential-algebraic equations

Vu Hoang Linh

and Nguyen Ngoc Tuan

1Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, Hanoi, Vietnam

2Department of Mathematics, Hung Yen University of Technology and Education, Hung Yen, Vietnam

Dedicated to the memory of Professor Katalin Balla (1947–2005) Received 30 August 2013, appeared 21 March 2014

Communicated by Tibor Krisztin

Abstract. This paper is concerned with the asymptotic behavior of solutions of lin- ear differential-algebraic equations with asymptotically constant coefficients. Some re- sults of asymptotic integration which are well known for ordinary differential equations (ODEs) are extended to differential-algebraic equations (DAEs).

Keywords:linear differential-algebraic equation, asymptotic integration, regular pencil, index, Weierstraß–Kronecker canonical form.

2010 Mathematics Subject Classification:34A09, 34D05, 34E10.

1 Introduction

Linear differential-algebraic equations (DAEs) are equations of the form

E(t)x⁰(t) = A(t)x(t), t ∈_I, (1.1) where E,A ∈ C(_I,C^n×n)withn ∈ _{N, I} = [t0,∞), and E(t)is assumed to be singular for all t ∈ I. Linear systems of the form (1.1) may occur when one linearizes a general nonlinear system of DAEs

F(t,x(t),x⁰(t)) =0, t ∈_I, (1.2) along a particular solution x^∗(t), whereF: I×_Cⁿ×_Cⁿ −→ _Cⁿ is assumed to be sufficiently smooth.

Differential-algebraic equations are also called singular differential equations which are generalizations of ordinary differential equations (ODEs). They play an important role in math- ematical modeling arising in multibody mechanics, electrical circuits, prescribed path control, chemical engineering, etc., see [4,16,25].

The qualitative theory and numerical analysis of DAEs are more difficult than ODEs be- cause the equations cannot be solved explicitly for the derivative and hidden algebraic con- straints may be involved. The difficulties are usually characterized by different index notions.

BCorresponding author. Email: linhvh@vnu.edu.vn

In the last two decades, the existence and uniqueness theory, the stability analysis, and the numerical treatment for DAEs, particularly for lower-index systems, have already been fairly well established, see [16,18,24].

In many problems, detailed information about the asymptotic behavior of solutions nearby singular points is useful. For example, it becomes desirable when one tries to formulate an approximate initial or boundary condition in the neighbourhood of singular points. The first asymptotic integration results for ODEs were given a long time ago by Levinson and others, see [6,15,19]. Later, further extensions of these classical results were carried out by many au- thors [2,11,13,14,22,27]. Recently, there have been many contributions to the stability and the asymptotic behavior of solutions of DAEs, e.g. see [1,3,5,7,8,9,10,17,20,21,26] and refer- ences therein. However, up to our knowledge, asymptotic integration results are still missing in the DAE literature. Therefore, the purpose of this paper is to extend classical asymptotic integration results from linear ODEs to linear DAEs.

In this paper, we consider linear asymptotically constant coefficient differential-algebraic equations of the form

[E+F(t)]x⁰(t) = [A+B(t) +R(t)]x(t), t≥t₀, (1.3) where E,A ∈ _C^n×n_, F, B, R ∈ C(_I;C^n×n), and constant matrix E is assumed to be singular.

Typically, the termsF, BandRplay the role of perturbations which may arise, for example, in the linearization process or in the course of modeling. The main question is that if perturbations F, BandRare supposed to be sufficiently small in some sense, how certain solutions of (1.3) are related to those of the unperturbed DAEs, which are with constant coefficients and quite well understood. In particular, the behavior of solutions asttends to infinity is of interest.

In order to characterize the asymptotic behavior of solutions of (1.3), we first transform the system into the semi-implicit form, i.e., the system is transformed into a coupled system consisting of an implicit differential equation and an algebraic one. Here, we use the decom- posing procedure for index-1 DAEs, e.g. see [23], and the well-known Kronecker–Weierstraß canonical form [4, 12, 16] for the higher index case. Then, conditions for perturbations F, B andRare given so that asymptotic formulas for solutions of (1.3) are explicitly obtained, which show the asymptotic equivalence between the solutions of (1.3) and those of the corresponding constant-coefficient DAEs. These results generalize the well-known asymptotic integration re- sults for linear ODEs. In addition, we show that perturbations arising in the leading term and for higher-index DAEs must be of appropriate structure. Otherwise, the asymptotic behavior of solutions of perturbed DAEs may be completely different from that of solutions of unper- turbed DAEs. This is the main difference between the asymptotic integration results for ODEs and those for DAEs.

The paper is organized as follows. In the next section, we summarize some basic results from the theory of DAEs. In Section3, we present the main result on the asymptotic integration for index-1 DAEs with perturbations arising only on the right hand side. Then, extensions to the case of the perturbed leading term and to the case of higher index DAEs are investigated in Sections4and5. Some examples are also included for illustration. We close the paper by a conclusion and a suggestion for future works.

2 Preliminaries

Consider linear constant-coefficient DAEs

Ex⁰(t) =Ax(t)_, t∈_{I, (2.1)}

whereE,Aare as in (1.3).

The matrix pencil{E,A}is said to beregularif there existsλ∈_Csuch that the determinant det(λE−A)is nonzero. Otherwise, if det(λE−A) = 0, for allλ ∈C, then we say that{E,A} is irregular or non-regular. If{E,A}is regular, thenλ ∈ Cis a (generalized finite) eigenvalue of{E,A}and a nonzero vectorζis the associated eigenvector ifλEζ = Aζ. It is known that the system (2.1) is solvable if and only if the matrix pencil{E,A}is regular [4,12,16]. The following theorem is known as the Kronecker–Weierstraß canonical form, which plays an important role in the analysis of linear constant-coefficient DAEs.

Theorem 2.1. Suppose that{E,A}is a regular pencil. Then, there exist nonsingular matrices G and H such that

GEH =

I_n1 0

0 N

, GAH=

J_n1 0 0 I_n2

, (2.2)

where n₁+n2 = n, Jn₁ is a n₁×n₁ matrix and N is a matrix of nilpotency index k, i.e., N^k = 0, but N^k−1 6=0. If N is a zero matrix, then we define k =1.

Without loss of generality, we may assume thatNandJ_n1 are given in the Jordan canonical form. Theindexof the pencil{E,A}is defined by the nilpotency index of the matrixNin (2.2).

For index-1 DAEs, the following reduction of (2.1) can be realized in practice, e.g., see [23].

Let the matrixEin (2.1) satisfy rank(E) =n₁, where 1≤n₁ <nand let the matricesU ∈_C^n×n2 and V ∈ _C^n×n2 be such that their columns form (minimal) bases for the left and right null- spaces ofE, respectively, i.e.,

U^TE=_{0, EV} =_{0. (2.3)}

Then, we define the matrices

U= U^⊥ U

, V= V^⊥ V

, (2.4)

whereU^⊥andV^⊥are the bases of the orthogonal subspaces associated withUandV. Letting x=_V u^T v^T T

,

whereu(t)∈_Cⁿ¹ andv(t)∈_Cⁿ², and multiplying (2.1) byU^T, we obtain E₁₁u⁰ = A₁₁u+A₁₂v,

0= A₂₁u+A₂₂v, (2.5)

where

E₁₁ =U^⊥TEV^⊥, (2.6)

and

A₁₁ =U^⊥TAV^⊥, A₁₂=U^⊥TAV, A₂₁=U^TAV^⊥, A₂₂=U^TAV. (2.7) The matrix E₁₁ is invertible since rank(E₁₁) = rank(_U^TEV) = rank(E) = n₁. In practice, the transformation matrices U and V can be computed from the singular value decomposition (SVD) ofE. Namely, their columns are left and right singular vectors of E, respectively. Thus, the transformation matricesUandVare orthogonal.

It is easy to see that{E,A}is regular of index-1 if and only if the matrixA22is nonsingular.

In this case, then from the second equation of the system (2.5), we imply that

v=−A₂₂⁻¹A₂₁u. (2.8)

Substituting the equation (2.8) into the first equation of the system (2.5) and then multiplying byE₁₁⁻¹, we obtain an ODE

u⁰ = E₁₁⁻¹ A₁₁−A₁₂A⁻₂₂¹A₂₁

u, (2.9)

which is called theessential underlyingODE. The asymptotic integration of ODEs under small perturbations is a well-established topic of the qualitative theory. In the next section, by using the transformed system (2.5), first we extend the classical ODE results of asymptotic integra- tion, e.g. see [6], to index-1 DAEs of the form (2.1) with perturbations arising on the right hand side.

3 Asymptotic solutions for index-1 DAEs

In this section, first we consider the perturbed DAEs of the form

Ex⁰(t) = [A+R(t)]x(t), t∈_I, (3.1) whereE,A ∈ _C^n×n, the pencil{E,A}is of index-1, andR ∈ C(_R+;C^n×n). We will show that ifRis sufficiently small in some sense, then the asymptotic behavior of the solutions of (3.1) is determined by the solutions of the unperturbed system (2.1).

Let the matricesU, V, U, andVbe defined by (2.3) and (2.4) in Section2. Multiplying (3.1) byU^T and substituting

x=_V u^T v^T T

, we obtain

E₁₁ 0

0 0

u⁰ v⁰

=

A₁₁ A₁₂ A₂₁ A₂₂

+

R₁₁(t) R₁₂(t) R₂₁(t) R₂₂(t)

u v

, (3.2)

whereE₁₁,A_ij,i,j=1, 2, are defined as in (2.6) and (2.7), and

R₁₁(t) =U^⊥TR(t)V^⊥, R₁₂(t) =U^⊥TR(t)V, R₂₁(t) =U^TR(t)V^⊥, R₂₂(t) =U^TR(t)V. (3.3) Since matrixE₁₁is invertible, then we obtain

u⁰ =E⁻₁₁¹(A₁₁+R₁₁(t))u+E⁻₁₁¹(A₁₂+R₁₂(t))v,

0= (A₂₁+R₂₁(t))u+ (A₂₂+R₂₂(t))v, (3.4) which is a DAE system in semi-explicit form. In order to investigate the asymptotic behavior of solutions of equation (3.1), we make some assumptions.

Assumption 3.1. Suppose that sup_t≥t

0kR₂₂(t)k<kA⁻₂₂¹k⁻¹holds.

Then, it is easy to see that (A₂₂+R₂₂(t)) is invertible for all t ≥ t₀ and the inverse is uniformly bounded. From now on, we omit the argumentt of the coefficients for simplicity, where no confusion arises.

It follows from the second equation of (3.4) that

v=−(A₂₂+R₂₂)⁻¹(A₂₁+R₂₁)u. (3.5) By reformulating

(A₂₂+R₂₂)⁻¹= A⁻₂₂¹−(A₂₂+R₂₂)⁻¹R₂₂A⁻₂₂¹,

the equation (3.5) can be rewritten as

v= −A⁻₂₂¹A₂₁+Re₂₁(t)u, (3.6) whereRe₂₁(t) =A₂₂⁻¹R₂₁−(A₂₂+R₂₂)⁻¹R₂₂A⁻₂₂¹(A₂₁+R₂₁).

Substituting (3.6) into the first equation of system (3.4), we obtain the following ODE for the differential componentu

u⁰ =E₁₁⁻¹h

A₁₁−A₁₂A⁻₂₂¹A₂₁+R₁₁−A₁₂A₂₂⁻¹R₂₁−R₁₂A⁻₂₂¹(A₂₁+R₂₁) +(A₁₂+R₁₂)(A₂₂+R₂₂)⁻¹R₂₂A⁻₂₂¹(A₂₁+R₂₁)ⁱu.

Let us denote

Ae₁₁= E⁻₁₁¹(A₁₁−A₁₂A₂₂⁻¹A₂₁), and

Re₁₁=E₁₁⁻¹h

R₁₁−A₁₂A₂₂⁻¹R₂₁−R₁₂A⁻₂₂¹−(A₁₂+R₁₂)(A₂₂+R₂₂)⁻¹R₂₂A⁻₂₂¹

(A₂₁+R₂₁)ⁱ. Then we obtain

u⁰ = [Ae₁₁+Re₁₁(t)]u. (3.7) Assumption 3.2. LetR_2j(t)→0,j=1, 2, ast→_∞.

Assumption 3.3. Let the matrix functionRbe absolutely integrable on[0;∞), i.e., Z _∞

t0

kR(t)kdt<_∞. (3.8)

Theorem 3.4. Let Assumptions 3.1, 3.2, and 3.3 hold and the matrix Ae₁₁ be similar to a diagonal matrix J.Suppose thatξ_j is an eigenvector associated with an eigenvalueµ_j of the pencil{E,A},i.e., µ_jEξ_j= Aξ_j.Then, the system (3.1) has a solution ϕ_j(t)such that

tlim→_∞ϕ_j(t)e^−µjt =ξ_j. Proof. Let

ξ_j =_V^h ξ¹_j^Tξ²_j^T iT

,

whereξ¹_j ∈_Cⁿ¹ _andξ²_j ∈_Cⁿ². From the equalityµ_jEξ_j = Aξ_j, we imply that

µ_jU^TEV

"

ξ¹_j ξ²_j

#

=_U^T_AV

"

ξ¹_j ξ²_j

#

or equivalently,

µ_jE₁₁ξ¹_j = A₁₁ξ¹_j +A₁₂ξ²_j,

0= A₂₁ξ¹_j +A₂₂ξ²_j. (3.9) Since the matrix A₂₂is invertible, and from the second equation of the system (3.9), we obtain

ξ²_j =−A⁻₂₂¹A₂₁ξ¹_j. (3.10)

Substituting (3.10) into the first equation of the system (3.9), we have µ_jξ¹_j = E⁻₁₁¹ A₁₁−A₁₂A⁻₂₂¹A₂₁

ξ¹_j. (3.11)

Hence,µ_j is an eigenvalue andξ¹_j is an associated eigenvector of the matrix Ae₁₁ = E₁₁⁻¹(A₁₁− A₁₂A₂₂⁻¹A₂₁).

Now, we consider the essential underlying system (3.7). It is easy to show that, taking into account the formula ofRe₁₁, Assumptions3.1and3.3imply thatRe₁₁is absolutely integrable. It follows from [6, p. 104, Prob. 29] that the system (3.7) has a solutionu_j(t)such that

tlim→_∞u_j(t)e^−µjt =ξ¹_j.

Under Assumptions3.1and3.2, it is easy to check thatRe₂₁(t)→ 0 ast → ∞. Therefore, from the equality (3.10), the corresponding algebraic componentv_j(t)determined by

v_j(_t) =− A⁻₂₂¹A21+Re21(_t)_uj(_t) satisfies

tlim→_∞v_j(t)e^−µjt =−lim

t→_∞ A⁻₂₂¹A₂₁+Re₂₁(t)u_j(t)e^−µjt =−A⁻₂₂¹A₂₁ξ¹_j =ξ²_j. Thus, the function

ϕ_j(t) =_V

u_j(t) v_j(t)

is a solution of equation (3.1) and it satisfies

tlim→_∞ϕ_j(t)e^−µjt = _lim

t→_∞V

u_j(t)e^−µjt v_j(t)e^−µjt

=_V

"

ξ¹_j ξ²_j

#

=ξ_j. The proof of Theorem3.4is complete.

Remark 3.5. It is well known thatAe₁₁is similar to a diagonal matrix if and only if the number of linearly independent eigenvectors of index-1 pencil{E,A}is exactlyn₁, the rank of matrixE.

This holds true, for example, if all the eigenvalues of{E,A}are distinct. Further, Assumptions 3.2and3.3may be relaxed somewhat. Namely, it is sufficient to give the analogous conditions forRe₂₁andRe₁₁. However, here we aim to formulate as-simple-as-possible sufficient conditions for the asymptotic integration.

If the matrix pencil{E, A}has multiple eigenvalues and the matrixAe₁₁is similar to a block diagonal matrixJ with Jordan blocks J_k, 1 ≤ k ≤ land the maximal size of the Jordan blocks J_k is r+1, r ≥ 1, then we need the following stronger assumption on R in order to obtain asymptotic formulas for the solutions of (3.1).

Assumption 3.6. Let the matrixR(t)satisfy that Z _∞

t₀ t^rkR(t)kdt<+_{∞. (3.12)}

Theorem 3.7. Assume that the matrix Ae₁₁ is similar to a block diagonal matrix J with Jordan blocks J_k, 1≤k≤l and the maximal size of the Jordan blocks J_kis r+1, r≥1. Assume also that Assumptions 3.1,3.2, and3.6hold. Letµ_jbe an eigenvalue of the matrix pencil{E,A}and let the unperturbed DAE system (2.1) have a solution of the form

e^µjtt^mc+O e^µjtt^m−1

, (3.13)

where c is a vector and0≤m≤r. Then, system (3.1) has a solutionϕ_j(t)such that

tlim→_∞

ϕ_j(t)e^−µjtt^−m−c

=0.

Proof. As we show in the proof of Theorem3.4,µ_j is also an eigenvalue of matrixAe₁₁. Denote c=_V^h _c^1T _c^{2T iT},

wherec¹ ∈_Cⁿ¹ _andc²∈ _Cⁿ²_.

We again consider the EUODE system (3.7). From the assumption (3.13) on the solution of the unperturbed DAE, the corresponding unperturbed EUODE system has a solution of the forme^µjtt^mc¹+O(e^µjtt^m−1). Furthermore,c² =−A₂₂⁻¹A₂₁c¹holds. Under Assumptions3.1and 3.6, it can be shown thatRe₁₁satisfiesR_∞

t0 t^rkRe₁₁(t)kdt< +∞. Hence, by the result of [6, p. 106, Problem 35], the system (3.7) has a solutionu_j(t)such that

tlim→_∞

u_j(t)e^−µjtt^−m−c¹

=0.

On the other hand, again using (3.6), the corresponding algebraic componentv_j(t)satisfies

tlim→_∞v_j(t)e^−µjt =−lim

t→_∞ A⁻₂₂¹A₂₁+Re₂₁(t)u_j(t)e^−µjtt^−m = −A⁻₂₂¹A₂₁c¹=c². Thus,

ϕ_j(t) =_V

u_j(t) v_j(t)

is a solution of system (3.1), and

tlim→_∞ϕ_j(t)e^−λjtt^−m = lim

t→_∞V

u_j(t)e^−λjtt^−m v_j(t)e^−λjtt^−m

=_V c¹

c²

= c.

The proof of Theorem3.7is complete.

Remark 3.8. Assumption3.3(or3.6) cannot be replaced by the condition limt→_∞R(t) =0 since it is known that even in the ODE case that the statements of Theorems 3.4and3.7 fail under this relaxed condition. Further, ifEis nonsingular, then the results of Theorems3.4and3.7are reduced to the well-known results for ODEs [6].

In many applications, perturbations arising in the systems can be decomposed into two parts: one tends to zero as t → _∞ and the other is absolutely integrable. Now, consider the DAEs of the form

Ex⁰(t) = [A+B(t) +R(t)]x(t) (3.14) where E,A ∈ _C^n×n, and B, R ∈ C(_R+;C^n×n) which are assumed to be sufficiently small in some sense.

Applying again the transformation withUandVto (3.14) as above, we obtain E₁₁u⁰ = (A₁₁+B₁₁(t) +R₁₁(t))u+ (A₁₂+B₁₂(t) +R₁₂(t))v,

0= (A₂₁+B₂₁(t) +R₂₁(t))u+ (A₂₂+B₂₂(t) +R₂₂(t))v, (3.15) whereE₁₁,A_ij, andR_ij,j=1, 2, are defined as in (2.6), (2.7), and (3.3), and

B₁₁(t) =U^⊥TB(t)V^⊥, B₁₂(t) =U^⊥TB(t)V, B₂₁=U^TBV^⊥, B₂₂ =U^TB(t)V. (3.16) In order to investigate the asymptotic behavior of solutions of equation (3.14), we present the following assumptions.

Assumption 3.9. Suppose that sup_t≥t

0(kB₂₂(t)k+kR₂₂(t)k)<kA⁻₂₂¹k⁻¹holds.

Assumption 3.9 implies that the inverse matrix (A₂₂+B₂₂+R₂₂)⁻¹ exists and it is uni- formly bounded. From the second equation of the system (3.15), we find that

v= −(A₂₂+B₂₂(t) +R₂₂(t))⁻¹(A₂₁+B₂₁(t) +R₂₁(t))u. (3.17) By an elementary reformulation, we have

(A₂₂+B₂₂(t) +R₂₂(t))⁻¹ = (A₂₂+B₂₂(t))⁻¹+C(t), whereC(t) =−(A₂₂+B₂₂(t) +R₂₂(t))⁻¹R₂₂(t)(A₂₂+B₂₂(t))⁻¹.

Substituting (3.17) into the first equation of system (3.15), we obtain an ODE system foru of the form

u⁰ = [Ae₁₁+Be₁₁(t) +Rb₁₁(t)]u, (3.18) whereAe₁₁= E₁₁⁻¹(A₁₁−A₁₂A⁻₂₂¹A₂₁), and

Be₁₁= E⁻₁₁¹h

B₁₁−B₁₂A⁻₂₂¹A₂₁−(A₁₂+B₁₂)A⁻₂₂¹B₂₁−(A₂₂+B₂₂)⁻¹B₂₂A⁻₂₂¹(A₂₁+B₂₁)ⁱ, and

Rb₁₁ =E⁻₁₁¹h

R₁₁−(A₁₂+B₁₂)(A₂₂+B₂₂)⁻¹R₂₁

+(A₁₂+B₁₂)C(t) +R₁₂ (A₂₂+B₂₂)⁻¹+C(t)(A₂₁+B₂₁+R₂₁)ⁱ. On the other hand, (3.17) is equivalent to

v =−(A₂₂+B₂₂)⁻¹+C(t)(A₂₁+B₂₁+R₂₁)u and thus it can be rewritten in the form

v=− A⁻₂₂¹A₂₁+Be₂₁(t)u, (3.19) whereBe21(_t) =_A22⁻¹(_B21+_R21)−((_A⁻₂₂¹+_B22)⁻¹_B22A⁻₂₂¹+_C(_t))(_A21+_B21+_R21)_.

Assumption 3.10. LetB(t)be differentiable on[t₀,∞)such that Z _∞

t0

kB⁰(t)kdt<_∞ (3.20)

andB(t)→0 ast →_∞.

Assumption 3.11. Let the matrixR(t)be absolutely integrable on[t₀,∞), i.e., Z _∞

t0

kR(t)kdt<_∞, (3.21)

and letR_2j(t)→0 ast→_∞hold, j=1, 2.

Lemma 3.12. Let Assumptions3.9,3.10, and3.11hold. Then the following statements are true.

(i) Be₂₁(t)→0as t→_∞;

(ii) Be₁₁(t)is differentiable on[t₀,∞)and satisfiesR_∞

t0 kBe₁₁⁰ (t)kdt<∞; Furthermore,Be₁₁(t)→0as t →_∞;

(iii) Rb₁₁is absolutely integrable on[t0,∞), i.e.,R_∞

t0 kRb₁₁(t)kdt<_∞.

Proof. By taking into account the explicit formulas ofBe₂₁, Be₁₁, Rb₁₁, the verifications are straight- forward.

Theorem 3.13. Let Assumptions3.9,3.10, and3.11hold and let the matrix pencil{E,A}have distinct eigenvaluesµ_j, j=1, 2, . . . ,n₁. Furthermore, letλ_j(t)be the roots of det(A+B(t)−λE) =0. Clearly, by reordering theµ_jif necessary, we have lim

t→_∞λ_j(t) =µ_j.For a given k,let D_kj(t) =Re(λ_k(t)−λ_j(t)).

Suppose that each j falls into one of two classes I₁and I₂,where j∈ I₁ifRt

0D_kj(τ)dτ→+_∞as t→_∞ and

Z _t2

t1

D_kj(τ)dτ>−K (t₂ ≥t₁≥0), (3.22) j∈ I₂if

Z _t2

t1

D_kj(τ)dτ<K (t₂≥t₁≥0), (3.23) where k is fixed and where K is a constant. Let ξ_k be the eigenvector associated withµ_k of the pencil {E,A},so thatµ_kEξ_k = _Aξk.Then equation (3.14) has a solutionsϕ_kand there exists t1(_t0 ≤t1≤_∞) such that

tlim→_∞ϕ_k(t)e⁻

R_t

t1λ_k(τ)dτ

=ξ_k. Proof. Let again

ξ_k =_V^h _ξ^1T

k ξ²_k^T iT

,

where ξ¹_k ∈ _Cⁿ¹ and ξ_k² ∈ _Cⁿ², as in the proof of Theorem 3.4. Then, from the equality µ_kEξ_k = Aξ_k, we again have the formulas (3.10) and (3.11), which means thatξ¹_k is an eigen- vector associated with the eigenvalueµ_kof the matrix Ae₁₁ =E₁₁⁻¹(A₁₁−A₁₂A⁻₂₂¹A₂₁)_{. Now, let} p_k(t)be an eigenvector associated with the eigenvalue λ_k(t)of the pencil{E,A+B(t)}, i.e., λ_kEp_k(t) = (A+B(t))p_k(t)for allt≥t₀with a sufficiently larget₀. Let

p_k(t) =_V^h _p¹_k^T(t) p²_k^T(t) iT

, where p¹_k(t)∈_Cⁿ¹ andp²_k(t)∈_Cⁿ². From the definition, we have

λ_k(t)_U^T_EVpk(t) =_U^T(A+B(t))_Vp_k(t),

or equivalently

λ_kE₁₁p¹_k = (A₁₁+B₁₁)p¹_k+ (A₁₂+B₁₂)p²_k,

0= (A₂₁+B₂₁)p¹_k+ (A₂₂+B₂₂)p²_k. (3.24) From Assumption3.9, it follows that kB₂₂(t)k < (kA⁻₂₂¹k)⁻¹ for all t ≥ t₀. Hence, there exists the inverse matrix(A₂₂+B₂₂)⁻¹and we have

(A₂₂+B₂₂)⁻¹ = A⁻₂₂¹−(A₂₂+B₂₂)⁻¹B₂₂A⁻₂₂¹. From the second equation of the system (3.24), we obtain

p²_k =−(A₂₂+B₂₂)⁻¹(A₂₁+B₂₁)p¹_k, or equivalently

p²_k = −A⁻₂₂¹A₂₁+A⁻₂₂¹B₂₁−(A₂₂+B₂₂)⁻¹B₂₂A⁻₂₂¹(A₂₁+B₂₁)p¹_k. (3.25) Substituting (3.25) into the first equation of the system (3.24), we obtain

λ_kp¹_k = E⁻₁₁¹h

A₁₁−A₁₂A⁻₂₂¹A₂₁+B₁₁−B₁₂A⁻₂₂¹A₂₁

−(A₁₂+B₁₂)(A⁻₂₂¹B₂₁−(A₂₂+B₂₂)⁻¹B₂₂A⁻₂₂¹(A₂₁+B₂₁))ⁱp¹_k, or equivalently

λ_kp¹_k =Ae₁₁+Be₁₁(t)p¹_k. (3.26) This means thatλ_k(t)is also an eigenvalue of the matrixAe₁₁+Be₁₁(t)andp¹_k is an eigenvector associated withλ_k(t). By the assumption on the eigenvalues of {E, A}, the matrix Ae₁₁ has distinct eigenvalues. On the other hand, by Lemma3.12, we have thatBe₁₁(t) → 0 ast → _∞.

Thus, λ_k(t) → µ_k as t → _∞ (by reordering if necessary). Furthermore, the conditions of [6, Chapter 3, Theorem 8.1] are satisfied by the underlying ODE system (3.18). It follows that there existst₁≥t₀such that (3.18) has a solutionu_k(t)satisfying

tlim→_∞u_k(t)e⁻

R_t

t1λ_k(τ)dτ

= ξ¹_k. By the equality (3.19), there existsv_k(t)defined by

v_k(t) =− A⁻₂₂¹A₂₁+Be₂₁(t)u_k(t), which fulfills

t→+lim_∞v_k(t)e⁻

R_t

t1λ_k(τ)dτ

=− lim

t→+_∞ A⁻₂₂¹A₂₁+Be₂₁(t)u_k(t)e⁻

R_t

t1λ_k(τ)dτ

=−A⁻₂₂¹A₂₁ξ_k¹=ξ²_k. Let us define

ϕ_k(t) =_V

u_k(t) v_k(t)

.

By its construction,ϕ_k(t)is obviously a solution of system (3.14) and it satisfies

t→+lim_∞ϕ_k(t)e⁻

Rt t1λk(τ)dτ

= lim

t→+_∞V





u_k(t)e⁻

R_t

t1λ_k(τ)dτ

v_k(t)e⁻

R_t

t1λ_k(τ)dτ





=_V ξ_k¹

ξ_k²

=ξ_k. This completes the proof of Theorem3.13.

4 The case of perturbed leading coefficient

In this section, we extend the results obtained in Section 3 to DAEs with perturbed leading coefficient

[E+F(t)]x⁰(t) = [A+R(t)]x(t), (4.1) whereE,A∈_C^n×n_andF, R∈ C(_I;C^n×n)_.

We again suppose that the matrixEis singular, but the pencil{E,A}is regular of index one.

First, we introduce the concept of allowable perturbations, see [3].

Definition 4.1. The perturbationFarising in the leading term is said to be allowable if ker(E+ F(t)) =kerEfor allt∈I. Otherwise we sayFis not allowable.

The following example shows that if ker(E+F(t)) 6= kerE, then the asymptotic behavior of solutions of the perturbed DAE (4.1) and the asymptotic behavior of solution of the unper- turbed one may be quite different, even if the perturbationFis small, e.g., it is convergent to 0 ast→_∞and absolutely integrable.

Example 4.2. Consider the index-1 DAE 1 0

0 0 x₁⁰ x₂⁰

=

1 0 0 1

x₁ x₂

, t ≥_1.

It is easy to obtain the solution x₁(t) = e^t−1x₁(1)andx₂(t) = 0. After that, we consider the following perturbed DAE





1 0

0 1

3t²



 x⁰₁

x⁰₂

=

1 0 0 1

x₁ x2

.

The first component x₁ is unchanged. However, the second component x2(t) = e^t3−1x2(1), which tends to∞ ast → ∞. That is, a small perturbation in the leading coefficient can com- pletely change the behavior of the solutions. For a related result, see the stability analysis of DAEs containing a small parameter in [9].

In the remainder part of this section we assume thatFis allowable. Let us apply to (4.1) the transformation with the sameUandVas in Section2. Then, we obtain the transformed DAE

(E₁₁+F₁₁(t))u⁰ = (A₁₁+R₁₁(t))u+ (A₁₂+R₁₂(t))v,

F₂₁(t)u⁰ = (A₂₁+R₂₁(t))u+ (A22+R22(t))v, (4.2) where E₁₁, A₁₁, A₁₂, A₂₁, A₂₂are defined in (2.6) and (2.7),R₁₁, R₁₂, R₂₁, R₂₂ are defined in (3.3). Further, we have

F₁₁ =U^⊥TFV^⊥, F₂₁=U^TFV^⊥. (4.3) Note that under the assumption onF, we haveU^⊥TFV =0 andU^TFV =0.

We make the following set of assumptions.

Assumption 4.3. Let sup_t≥t0kF₁₁(t)k<(kE₁₁⁻¹k)⁻¹hold.

Assumption 4.4. LetR_2j(t)→0,j=1, 2, andF₂₁(t)→0, ast→∞. Further, let sup_t≥t0kR_1j(t)k<

∞,j=1, 2.

Assumption 4.5. Let sup_t≥t

0kR₂₂(t)k<kA⁻₂₂¹k⁻¹hold, where

R₂₂(t) =−F₂₁(t)(E₁₁+F₁₁(t))⁻¹(A₁₂+R₁₂(t)) +R₂₂(t)_, provided that(E₁₁+F₁₁(t))⁻¹exists for allt∈_I.

Assumption 4.6. Let bothRandFbe absolutely integrable onI.

We will show that under these assumptions, the perturbed DAE (4.1) can be transformed into the form (3.1). Then, the theorems in Section3can be applied.

Theorem 4.7. Consider the DAE system (4.1) with index-1pencil{E, A}. Let Assumptions4.3–4.6 hold and let Ae₁₁ = E⁻₁₁¹(A₁₁−A₁₂A⁻₂₂¹A₂₁)is similar to a diagonal matrix J. Suppose thatµ_j is an eigenvalue of the pencil{E,A}andξ_jis an eigenvector associated withµ_j, i.e.,µ_jEξ_j = Aξ_j.Then, the perturbed DAE system (4.1) has a solutionϕ_j(t)such that

tlim→_∞ϕ_j(t)e^−µjt =ξ_j.

Proof. Assumption4.3implies that matrices(E₁₁+F₁₁(t))and(I+F₁₁(t)E⁻₁₁¹)are invertible for allt ∈_Iand the inverse matrices are uniformly bounded. We haveE₁₁+F₁₁ = (I+F₁₁E⁻₁₁¹)E₁₁ and(I+F₁₁(t)E₁₁⁻¹)⁻¹ = I−(I+F₁₁(t)E⁻₁₁¹)F₁₁E₁₁⁻¹. In order to bring (4.2) into the form (3.2), we first multiply the first equation of (4.2) by −F₂₁(E₁₁+F₁₁)⁻¹ add the obtained result to the second equation of (4.2). Then, we scale the first equation of (4.2) by multiplying it by (I+F₁₁(t)E₁₁⁻¹)⁻¹. As the result, we obtain a new DAE system as follows

E₁₁u⁰ = (A₁₁+R₁₁(t))u+ (A₁₂+R₁₂(t))v,

0= (A₂₁+R₂₁(t))u+ (A₂₂+R₂₂(t))v, (4.4) where

R_1j = (I+F₁₁E₁₁⁻¹)⁻¹(F₁₁E₁₁⁻¹A_1j+R_1j), R_2j =−F₂₁(E₁₁+F₁₁)⁻¹(A_1j+R_1j) +R_2j, forj = 1, 2. Under Assumptions 4.3–4.6, it is not difficult to verify thatR_ij,i,j = 1, 2, satisfy Assumptions3.1–3.3in Section3. Thus, the conditions of Theorem3.4are fulfilled. Applying Theorem3.4to (4.4), the proof is complete.

Similarly, by invoking Theorem3.2, the following theorem is obtained for the case when the matrixAe₁₁is not diagonalizable.

Theorem 4.8. Consider the DAE system (4.1) with index-1pencil{E, A}. Let Assumptions4.3–4.5 hold and letAe₁₁be similar to the block diagonal matrix J with Jordan blocks J_k, 1≤k ≤l. Assume that r+₁is the maximal number of rows in any matrices J_k, 1 ≤ k ≤ l. Here r ≥ ₁is considered. Let R and F satisfy

Z _∞

t₀ t^rkR(t)kdt<_∞, Z _∞

t₀ t^rkF(t)kdt<_∞. (4.5) We suppose thatµ_j is an eigenvalue of the pencil{E,A}and that the corresponding unperturbed DAE has a solution of the form

e^µjtt^mc+O(e^µjtt^m−1),

where c is a nonzero vector and0≤ m≤r. Then, system (4.1) has a solutionϕ_j(t)such that

tlim→_∞[ϕ_j(_t)_e^−µjt_t^−m−c] =_0.

By analogue, the result of Theorem3.13can be extended to DAEs of the form (4.2), as well.

5 The case of higher-index DAEs

In this section, we revisit the DAEs of the form (2.1) and (3.1), but now we assume that the pencil{E,A}has indexk ≥2 and the Weierstraß–Kronecker canonical form (2.2) holds with a pair of nonsingular matricesGandH.

Multiplying both sides of (2.1) by G and introducing a variable transformation x = Hy, y= (u^T v^T)^T, whereu(t)∈_Cⁿ¹ andv(t)∈_Cⁿ², we obtain

u⁰ = [J_n1+R₁₁(t)]u+R₁₂(t)v,

Nv⁰ = R₂₁(t)u+ [I_n2 +R₂₂(t)]v, (5.1) where

GRH=

R₁₁ R₁₂ R₂₁ R₂₂

.

Exploiting the nilpotency ofN, it is easy to verify that the unperturbed system associated with (5.1) has solution u(t) = e^Jn1(t−t0)u(t₀), v(t) = 0 fort ≥ t₀. The analysis of perturbed system (5.1) is more complicated than the index-1 case.

In general, if the perturbationR(t)is not well-structured, then the asymptotic behavior of solutions is not preserved even under such small perturbations discussed in Section3.

Example 5.1. First, consider the following DAE





1 0 0 0 0 1 0 0 0







 x⁰₁ x⁰₂ x⁰₃



=





2 0 0 0 1 0 0 0 1







 x₁ x₂ x3



, t ≥1.

The solution isx₁(t) =e^2(t−1)x₁(₁)_, x₂(t) =x₃(t) =0. Let the system be perturbed as follows





1 0 0 0 0 1 0 0 0







 x⁰₁ x⁰₂ x⁰₃



=





2 0 0

0 1 0

0 −_3t¹₂ 1







 x₁ x2

x₃



.

The solution of the perturbed system is x₁(t) = e^2(t−1)x₁(1), x₂(t) = 3t²e^t3−1x₃(1), x₃(t) = e^t3−1x₃(₁)_{. Clearly,}x₂(t)→_∞andx₃(t)→_∞ast→_{∞except for}x₃(₁) =_0.

Similarly, if we consider another perturbed system





1 0 0 0 0 1 0 0 0







 x⁰₁ x⁰₂ x⁰₃



=







2 0 0

0 1 0

−^sint3

t² 0 1









 x₁ x₂ x3





then, it is easy to see thatx₁(t)remains the same, but both x₃(t) = ^sint

3

t² e^2(t−1)x₁(1) and x₂(t) =

3 cost³e^2t−2sint³

t³ e^2t+^sint

3

t² 2e^2t

x₁(1) are unboundedly oscillating.

Definition 5.2. Consider the perturbed DAE (5.1) and assume thatNis nilpotent of indexk≥2.

The perturbationRis said to be allowable for the higher-index case if the componentvof (5.1) is identically zero fort≥t₀, i.e., the solution forvis preserved under perturbation.