Asymptotic Integration of Linear Differential-algebraic Equations

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


Introduction
Linear differential-algebraic equations (DAEs) are equations of the form E(t)x (t) = A(t)x(t), t ∈ I, (1.1) where E, A ∈ C(I, C n×n ) with n ∈ N, I = [t 0 , ∞), 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), x (t)) = 0, t ∈ I, along a particular solution x * (t), where F : I × C n × C n −→ C n 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 mathematical 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 because the equations cannot be solved explicitly for the derivative and hidden algebraic constraints may be involved.The difficulties are usually characterized by different index notions.

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V. H. Linh and N. N. Tuan 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 authors [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 references 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 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 terms F, B and R play 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, B and R are 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 as t tends 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 decomposing 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 and R are 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 results 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 unperturbed 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 Section 3, 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 Sections 4 and 5. Some examples are also included for illustration.We close the paper by a conclusion and a suggestion for future works.

Preliminaries
Consider linear constant-coefficient DAEs Asymptotic integration of linear DAEs where E, A are as in (1.3).The matrix pencil {E, A} is said to be regular if there exists λ ∈ C such that the determinant det(λE − A) is nonzero.Otherwise, if det(λE − A) = 0, for all λ ∈ C, then we say that {E, A} is irregular or non-regular.If {E, A} is regular, then λ ∈ C is a (generalized finite) eigenvalue of {E, A} and a nonzero vector ζ is the associated eigenvector if λEζ = 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 where n 1 + n 2 = n, J n 1 is a n 1 × n 1 matrix and N is a matrix of nilpotency index k, i.e., N k = 0, but N k−1 = 0.If N is a zero matrix, then we define k = 1.
Without loss of generality, we may assume that N and J n 1 are given in the Jordan canonical form.The index of the pencil {E, A} is defined by the nilpotency index of the matrix N in (2.2).
For index-1 DAEs, the following reduction of (2.1) can be realized in practice, e.g., see [23].Let the matrix E in (2.1) satisfy rank(E) = n 1 , where 1 ≤ n 1 < n and let the matrices U ∈ C n×n 2 and V ∈ C n×n 2 be such that their columns form (minimal) bases for the left and right null- spaces of E, respectively, i.e., U T E = 0, EV = 0. ( Then, we define the matrices where U ⊥ and V ⊥ are the bases of the orthogonal subspaces associated with U and V. Letting where u(t) ∈ C n 1 and v(t) ∈ C n 2 , and multiplying (2.1) by U T , we obtain where and The matrix E 11 is invertible since rank(E 11 ) = rank(U T EV) = rank(E) = n 1 .In practice, the transformation matrices U and V can be computed from the singular value decomposition (SVD) of E. Namely, their columns are left and right singular vectors of E, respectively.Thus, the transformation matrices U and V are orthogonal.
It is easy to see that {E, A} is regular of index-1 if and only if the matrix A 22 is nonsingular.In this case, then from the second equation of the system (2.5), we imply that Substituting the equation (2.8) into the first equation of the system (2.5) and then multiplying by which is called the essential underlying ODE.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 integration, e.g.see [6], to index-1 DAEs of the form (2.1) with perturbations arising on the right hand side.

Asymptotic solutions for index-1 DAEs
In this section, first we consider the perturbed DAEs of the form where E, A ∈ C n×n , the pencil {E, A} is of index-1, and R ∈ C(R + ; C n×n ).We will show that if R is 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 matrices U, V, U, and V be defined by (2.3) and (2.4) in Section 2. Multiplying (3.1) by U T and substituting we obtain where E 11 , A ij , i, j = 1, 2, are defined as in (2.6) and (2.7), and Since matrix E 11 is invertible, then we obtain 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.
Then, it is easy to see that (A 22 + R 22 (t)) is invertible for all t ≥ t 0 and the inverse is uniformly bounded.From now on, we omit the argument t of the coefficients for simplicity, where no confusion arises.
It follows from the second equation of (3.4) that the equation (3.5) can be rewritten as where ). Substituting (3.6) into the first equation of system (3.4),we obtain the following ODE for the differential component Let us denote , and Theorem 3.4.Let Assumptions 3.1, 3.2, and 3.3 hold and the matrix A 11 be similar to a diagonal matrix J. Suppose that ξ j is an eigenvector associated with an eigenvalue µ j of the pencil {E, A}, i.e., µ j Eξ j = Aξ j .Then, the system (3.1) has a solution ϕ j (t) such that lim t→∞ ϕ j (t)e −µ j t = ξ j . Proof.Let where From the equality µ j Eξ j = Aξ j , we imply that or equivalently, (3.9) Since the matrix A 22 is invertible, and from the second equation of the system (3.9),we obtain V. H. Linh and N. N. Tuan Substituting (3.10) into the first equation of the system (3.9),we have Hence, µ j is an eigenvalue and ξ 1 j is an associated eigenvector of the matrix . Now, we consider the essential underlying system (3.7).It is easy to show that, taking into account the formula of R 11 , Assumptions 3.1 and 3.3 imply that R 11 is absolutely integrable.It follows from [6, p. 104, Prob.29] that the system (3.7) has a solution u j (t) such that lim t→∞ u j (t)e −µ j t = ξ 1 j .
Under Assumptions 3.1 and 3.2, it is easy to check that R 21 (t) → 0 as t → ∞.Therefore, from the equality (3.10), the corresponding algebraic component v j (t) determined by Thus, the function is a solution of equation (3.1) and it satisfies lim t→∞ ϕ j (t)e −µ j t = lim t→∞ V u j (t)e −µ j t v j (t)e −µ j t = V The proof of Theorem 3.4 is complete.
Remark 3.5.It is well known that A 11 is similar to a diagonal matrix if and only if the number of linearly independent eigenvectors of index-1 pencil {E, A} is exactly n 1 , the rank of matrix E. This holds true, for example, if all the eigenvalues of {E, A} are distinct.Further, Assumptions 3.2 and 3.3 may be relaxed somewhat.Namely, it is sufficient to give the analogous conditions for R 21 and R 11 .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 matrix A 11 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 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 matrix R(t) satisfy that ∞ t 0 t r R(t) dt < +∞.
(3.12) Theorem 3.7.Assume that the matrix A 11 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 k is r + 1, r ≥ 1. Assume also that Assumptions 3.1, 3.2, and 3.6 hold.Let µ j be an eigenvalue of the matrix pencil {E, A} and let the unperturbed DAE system (2.1) have a solution of the form e µ j t t m c + O e µ j t t m−1 , ( where c is a vector and 0 ≤ m ≤ r.Then, system (3.1) has a solution ϕ j (t) such that lim t→∞ ϕ j (t)e −µ j t t −m − c = 0.
Proof.As we show in the proof of Theorem 3.4, µ j is also an eigenvalue of matrix A 11 .Denote where 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 form e µ j t t m c 1 + O(e µ j t t m−1 ).Furthermore, c 2 = −A −1 22 A 21 c 1 holds.Under Assumptions 3.1 and 3.6, it can be shown that R 11 satisfies ∞ t 0 t r R 11 (t) dt < +∞.Hence, by the result of [6, p. 106, Problem 35], the system (3.7) has a solution u j (t) such that lim t→∞ u j (t)e −µ j t t −m − c 1 = 0.
On the other hand, again using (3.6), the corresponding algebraic component v j (t) satisfies Thus, is a solution of system (3.1), and The proof of Theorem 3.7 is complete.
Remark 3.8.Assumption 3.3 (or 3.6) cannot be replaced by the condition lim t→∞ R(t) = 0 since it is known that even in the ODE case that the statements of Theorems 3.4 and 3.7 fail under this relaxed condition.Further, if E is nonsingular, then the results of Theorems 3.4 and 3.7 are 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 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 with U and V to (3.14) as above, we obtain  and let R 2j (t) → 0 as t → ∞ hold, j = 1, 2.
(i) B 21 (t) → 0 as t → ∞; Proof.By taking into account the explicit formulas of B 21 , B 11 , R 11 , the verifications are straightforward.Theorem 3.13.Let Assumptions 3.9, 3.10, and 3.11 hold and let the matrix pencil {E, A} have distinct eigenvalues µ j , j = 1, 2, . . ., n 1 .Furthermore, let λ j (t) be the roots of det(A + B(t) − λE) = 0. Clearly, by reordering the µ j if necessary, we have lim t→∞ λ j (t) = µ j .For a given k, let Suppose that each j falls into one of two classes I 1 and I 2 , where j where k is fixed and where K is a constant.Let ξ k be the eigenvector associated with µ k of the pencil {E, A}, so that µ k Eξ k = Aξ k .Then equation (3.14) has a solutions ϕ k and there exists t Proof.Let again where as in the proof of Theorem 3.4.Then, from the equality µ k Eξ k = Aξ k , we again have the formulas (3.10) and (3.11), which means that ξ 1  k is an eigenvector associated with the eigenvalue µ k of the matrix . 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 k Ep k (t) = (A + B(t))p k (t) for all t ≥ t 0 with a sufficiently large t 0 .Let where From the definition, we have V. H. Linh and N. N. Tuan or equivalently From Assumption 3.9, it follows that B 22 (t) < ( A −1 22 ) −1 for all t ≥ t 0 .Hence, there exists the inverse matrix (A 22 + B 22 ) −1 and we have From the second equation of the system (3.24),we obtain Substituting (3.25) into the first equation of the system (3.24),we obtain or equivalently This means that λ k (t) is also an eigenvalue of the matrix A 11 + B 11 (t) and p 1 k is an eigenvector associated with λ k (t).By the assumption on the eigenvalues of {E, A}, the matrix A 11 has distinct eigenvalues.On the other hand, by Lemma 3.12, we have that B 11 (t) → 0 as t → ∞.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 exists t 1 ≥ t 0 such that (3.18) has a solution u k (t) By the equality (3.19), there exists v k (t) defined by Let us define By its construction, ϕ k (t) is obviously a solution of system (3.14) and it satisfies lim This completes the proof of Theorem 3.13.

The case of perturbed leading coefficient
In this section, we extend the results obtained in Section 3 to DAEs with perturbed leading coefficient where E, A ∈ C n×n and F, R ∈ C(I; C n×n ).
We again suppose that the matrix E is 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 perturbation F arising in the leading term is said to be allowable if ker(E + F(t)) = ker E for all t ∈ I. Otherwise we say F is not allowable.
The following example shows that if ker(E + F(t)) = ker E, then the asymptotic behavior of solutions of the perturbed DAE (4.1) and the asymptotic behavior of solution of the unperturbed one may be quite different, even if the perturbation F is small, e.g., it is convergent to 0 as t → ∞ and absolutely integrable.
It is easy to obtain the solution x 1 (t) = e t−1 x 1 (1) and x 2 (t) = 0.After that, we consider the following perturbed DAE The first component x 1 is unchanged.However, the second component x 2 (t) = e t 3 −1 x 2 (1), which tends to ∞ as t → ∞.That is, a small perturbation in the leading coefficient can completely 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 that F is allowable.Let us apply to (4.1) the transformation with the same U and V as in Section 2.Then, we obtain the transformed DAE Note that under the assumption on F, we have U ⊥ T FV = 0 and U T FV = 0. We make the following set of assumptions.

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 index k ≥ 2 and the Weierstraß-Kronecker canonical form (2.2) holds with a pair of nonsingular matrices G and H. Multiplying both sides of (2.1) by G and introducing a variable transformation x = Hy, y = (u T v T ) T , where u(t) ∈ C n 1 and v(t) ∈ C n 2 , we obtain where Exploiting the nilpotency of N, it is easy to verify that the unperturbed system associated with (5.1) has solution u(t) = e J n 1 (t−t 0 ) u(t 0 ), v(t) = 0 for t ≥ t 0 .The analysis of perturbed system (5.1) is more complicated than the index-1 case.
In general, if the perturbation R(t) is not well-structured, then the asymptotic behavior of solutions is not preserved even under such small perturbations discussed in Section 3.

Example 5.1. First, consider the following DAE
The solution is x 1 (t) = e 2(t−1) x 1 (1), x 2 (t) = x 3 (t) = 0. Let the system be perturbed as follows The solution of the perturbed system is x 1 Similarly, if we consider another perturbed system The perturbation R is said to be allowable for the higher-index case if the component v of (5.1) is identically zero for t ≥ t 0 , i.e., the solution for v is preserved under perturbation.
Let v j (t) = 0 be the second solution component, we have that is a solution of equation (5.1).Hence, it follows that lim t→∞ ϕ j (t)e −µ j t = lim t→∞ H u j (t)e −µ j t v j (t)e −µ j t = H The proof of Theorem 5.4 is complete.
As an analogue of Theorem 3.7, we obtain the following theorem for the case of nondiagonal J n 1 .
Theorem 5.5.Let the matrix J n 1 be a Jordan matrix consisting of blocks J k , k ≥ 1 and r + 1 is the maximal number of rows in any block J k , k ≥ 1.Here r ≥ 1 is considered.Let R satisfy the same assumption as in Theorem 5.4.Let µ j be an eigenvalue of {E, A} and the equation (2.1) has a solution of the form e µ j t t m c + O(e µ j t t m−1 ), where c is a vector and 0 ≤ m ≤ r.Then the perturbed system (5.1) has a solution ϕ j (t) such that lim t→∞ ϕ j (t)e −µ j t t −m − c = 0.
Proof.Using similar arguments as those in the proofs of Theorem 3.7 and Theorem 5.4, the proof is straightforward.
Remark 5.6.An analogue of Theorem 3.13 can be obtained in the higher-index case as well.The asymptotic integration results can also be extended to DAEs with perturbation in the leading term.However, more restrictive structure of perturbation should hold.

Conclusion
In this paper we have extended the classical asymptotic integration results from ODEs to linear DAEs.Checkable conditions are given so that the asymptotic formulas for solution of asymptotically constant coefficient DAE systems are obtained explicitly.It has been shown that the solutions of the perturbed DAEs behave asymptotically like the corresponding solutions of the unperturbed DAEs as time t tends to ∞.As future works, asymptotic results for linear time-varying DAEs, for DAEs with nonlinear perturbations, and for delay DAEs would be of interest.
) where E 11 , A 11 , A 12 , A 21 , A 22 are defined in (2.6) and (2.7), R 11 , R 12 , R 21 , R 22 are defined in(3.3).Further, we have it is easy to see that x 1 (t) remains the same, but both Definition 5.2.Consider the perturbed DAE (5.1) and assume that N is nilpotent of index k ≥ 2.