Linear spaces for index 2 differential-algebraic equations

equations

Thomas Petry

^y

October 25, 1995

Abstract

In this paper we consider solution spaces of linear index-2-tractable dif- ferential algebraic equations. Relations between the solutions of the adjoint equations and the corresponding solution spaces are derived and, thus, a sim- ple method for computing consistent initial values is provided.

Key words. Dierential-algebraic equations, linear solution spaces, consistent initial values

AMSsubjectclassi cation. 65L10

1 Introduction

We consider the homogeneous index-2-tractable dierential-algebraic equation

A(t)y⁰(t) + B(t)y(t) = 0 (1)

with the coe cients A ² C¹(^I L(^Rm)), B ² C(^I L(^Rm)), and ^{I R} a given interval.

The adjoint equation of (1) reads

A^T(t)⁰(t)^;(B^T(t)^;A^T(t)⁰)(t) = 0: (2) The solution spaces of the initial equation (1) and the adjoint equation (2), re- spectively,can be described by means of canonical projectors, which will be described in the section 3. It is necessary to have a practicable representation of subspaces of these solution spaces for the problems (1) and (2) in order to determine the solution spaces belonging to the corresponding initial resp. boundary value problems. By

This research was supported by German Scientic Research Fund, Grant Ma 1347/3-1

yHumboldt-Universitat zu Berlin, Institut fur Angewandte Mathematik, Unter den Linden 6, D-10099 Berlin, Germany

1

means of this representation it becomes possible to apply transfer methods (see 1], 3], 10]), i.e. to solve boundary value problems by modied adjoint initial value problems. The solution spaces can be represented as solutions of linear systems of equations, hence, we obtain a simple method for calculating consistent initial values (section 6).

First, in section 2, we will prove that equation (2) is also index-2 tractable and then we dene the matrix chains for the adjoint problem (2).

In section 4 we investigate solution representations by means of the solution of the adjoint equation. The theorem given there provides a suitable representation of solution spaces of (1) and, thus, justies the application of transfer methods due to Abramov (see 1], 3], 10]). An analogous result for index 1 DAEs was already proved in 2].

In section 5 we apply the results to inhomogeneous systems.

Finally, in section 6, we derive relations among the representation of the solution spaces, the constraint manifolds, a suitable index reduction and the computation of consistent initial values.

At the end of the paper, in the Appendix, we gathered some statements of the theory of linear algebra.

2 The index of the adjoint equation

Let us rst consider the relations between the initial equation and the adjoint prob- lem with respect to the index.

Let the nullspace N(t) := ker(A(t)) be smooth, P(t) = A^y(t)A(t) be the ortho- projector along N(t) onto N(t)^?. The tractability with index 2 is characterized by the fact that the modied matrix pencil (A B^;AP⁰) has index 2 uniformly on ^I 5].In order to investigate the index, we have to nd the matrix pencil that is relevant for the index. For the initial equation it reads (A B^;AP⁰), which corresponds to the equation

A(Py)⁰+ (B^;AP⁰)y = 0: (3)

Let P?(t) = A^Ty(t)A^T(t) = A(t)A^y(t) be the projector along kerA^T(t). On account ofP?⁰= (P?)^0;P?⁰, a representation of the adjoint equation (2) that is analogous to the equation (3) now reads

A^T(P?)^0;(B^{T ;}A^T0P?) = 0:

Hence, (A^T B^T;A^T0P?) is the matrix pencil to be investigated.

Theorem 1

Let the DAE (1) be tractable with index 2. Then the DAE A^T0; B^{T ;}A^T0 = 0 has index 2, too.

2

Proof. The index-2 condition reads

ind(A B^;AP⁰) = 2 = ind A^T B^{T ;}P⁰A^T= ind A^{T ;}B^T +P⁰A^T where we use the special orthoprojectors P = A^yA, P? = AA^y. Because of A^T = PA^TP?, we have

A^T0P? =P⁰A^TP?+PA^T0P? =P⁰A^T+PA^T0P?:

Now we consider the matrix pencil determining the index of equation (2) A^{T ;}B^T +A^T0P?

= A^{T ;}B^T+P⁰A^T +PA^T0P?

: This matrix pencil has index 2 ( a simple conclusion from Theorem 24). ²

Now we can construct the related matrix chains and spaces for the DAEs (1) and (2) (the arguments are omitted):

A A? = A^T

B B? = ^;B^T+A^T0

P = A^yA P? = A^TyA^T

Q = I^;P Q? = I^;P?

N(t) = kerA(t) N?(t) = kerA?(t)

S(t) = ^{f 2Rm}: B(t)²imA(t)^g S?(t) = ^{f 2Rm}:B?(t)²imA?(t)^g B⁰ = B^;AP⁰ B?⁰ = B?^;A?P_?⁰=^;B^T +A^T0P?

A¹ = A + B⁰Q A?¹ = A?+B?⁰Q? =A^{T ;}B^TQ?

N¹(t) = kerA¹(t) N?¹(t) = kerA?¹(t) S¹(t) = ^{f 2Rm}: B⁰(t)P(t)²imA¹(t)^g

S?¹(t) =^f2Rm:B?⁰(t)P?(t) ²imA?¹(t)^g Q¹(t) the projector onto N¹(t) along S¹(t)

Q?¹(t) the projector onto N?¹(t) along S?¹(t)

P¹ = I^;Q¹ P?¹ = I^;Q?¹

G² = A¹+B⁰PQ¹ G?² = A?¹+B?⁰P?Q?¹

ForQ¹ and Q?¹ it holds

Q¹ =Q¹G^;12 B⁰P and Q?¹ =Q?¹G^;1?²B?⁰P? (see e.g. 8]):

The condition that A and A¹ are singular and of constant rank on ^I, and G² is nonsingular on ^I is equivalent to the tractability with index 2 7, Theorem 2.6].

This holds analogously for the adjoint system "?".

The dierentiability of A and the rank consistency imply the dierentiability of the orthoprojectors P, Q, P? and Q?.

3

3 The projectors

can

and

?can

LetQ¹ and Q?¹ be dierentiable. The linear space imcan(t), where can(t) := (I ^;(QQ¹)⁰(t)^;(QP¹G^;12 B⁰)(t))(PP¹)(t)

consists of the values of the solutions of the homogeneous equation (1) 6, Theo- rem 1.2], 9].

Because of PP¹can =PP¹ and canPP¹ = can it holds that

kercan= kerPP¹: (4)

Now we investigate the adjoint equation (2) rewritten as A?(P?)⁰+B?⁰ = 0:

This equation has index 2. We split this equation by means of standard methods in order to derive a similar representation for a projector ?can. The above equation is equivalent to

(A?+B?⁰Q?)

| {z }

A?¹

(P?(P?)⁰+Q?) + B?⁰P? = 0 and, nally, to

(A?¹+B?⁰P?Q?¹)

| {z }

G?²

(P?¹(P?(P?)⁰+Q?) + Q?¹) + B?⁰P?P?¹ = 0:

Multiplying by P?P?¹G^;1?², Q?P?¹G^;1?² and Q?¹G^;1?² yields the following equivalent representation of equation (2)

(P?P?¹)^0;(P?P?¹)⁰P?P?¹ + P?P?¹G^;1?²B?⁰P?P?¹ = 0 Q? + (Q?Q?¹)⁰P?P?¹ + Q?P?¹G^;1?²B?⁰P?P?¹ = 0 Q?¹ = 0:

Hence, = P?P?¹ + Q? = I^;(Q?Q?¹)^0;Q?P?¹G^;1?²B?⁰

P?P?¹. Obviously, ?can := I^;(Q?Q?¹)^0;Q?P?¹G^;1?²B?⁰

P?P?¹ is a projector and de- nes the space of the solutions of the adjoint homogeneous index 2 equation.

4 Justication theorem for transfer methods

In this section we introduce and prove a representation theorem for the solution subspaces of linear DAEs of index 2.

4

Theorem 2

^{Mis a} k-dimensional linear subspace of solutions of the index 2 equa-

tion Ay⁰+By = 0

if and only if there exists a m -dimensional matrix-valued function on ^I , = rankPP^1;k, and it holds that

(i) rank(t),

(ii) for all t^{2I we have}

M(t) =^{n 2RmT}(t)A(t) = 0 (I^;can(t)) = 0^o, (iii) and is a solution of the equation (2).

Remark 3

This theorem is an immediate extension of the result 2] to the index 2 with similar smoothness assumptions.

Proof. First, let ^M be a k-dimensional space of solutions of the equation (1) and lety^2Mbe chosen arbitrarily. Then, for any xedt ^2I,^M(t) is a k-dimensional subspace of ^Rm and it holds that ^M(t)imcan(t). Let us x a t. Hence, there is a-dimensional basis^fq¹ ::: q^gof the orthogonal complement of ^M(t) relative to imcan(t). Let q := (q¹ ::: q). Then, for this xed t, y(t) = can(t)y(t)^{2 M}(t) fulls the conditions

8

<

:

(I^;can(t))y(t) = 0 q^Tcany(t) = 0:

On account of the representation of can = canP = canA^yA the latter equation can be formulated as follows

^T0A(t)y(t) = 0 (5)

where ^T0 = q^Tcan(t)A^y(t). Clearly, equation (1) implies Q?By 0. Because of 0 =Q_T?¹A_T?¹=Q_T?¹(A^;Q?B) it holds

QT?¹Ay = QT?¹Q?By = 0:

Now equation (5) can be written in the following way ^T0A(t)y(t) = ^T0P_T?¹(t)A(t)y(t) =

^T0P_T?¹(t)P?(t)A(t)y(t) = ^T0_T?can(t)A(t)y(t) = 0:

Let(t⁰) := ?can⁰. Each column vector of(t⁰) is a consistent initial value for the adjoint problem(2). Consequently, the columns of remain linearlyindependent

5

on the whole interval ^I and for the solutions y ²CA¹ := ^fy ²C^j(Py)² C^1g of (1) and of (2) we have

^T(t)A(t)y(t)⁰= ((t)^TA(t))⁰P(t)y(t) + (t)^TA(t)(P(t)y(t))⁰

=^T(t)B(t)P(t)y(t)^;T(t)(B(t)^;A(t)P⁰(t))y(t)

=^;T(t)B(t)Q(t)y(t)+ ^T(t)A(t)P⁰(t)y(t)

=^;(^T(t)A(t))⁰Q(t)y(t)+ ^T(t)A(t)P⁰(t)y(t)

=^T(t)A(t)Q⁰(t)y(t) + ^T(t)A(t)P⁰(t)y(t) = 0:

Thus,^TAy0 holds true.

For the other direction of the proof we assume a linear space ^Mand a function with the properties (ⁱ){(iii) to be given.

Then, = ?can, and A^T is a solution of the dierential equation (A^T)^0;B^T?canA^TyA^T = 0:

Thus, rank^TA . Consequently, ^M(t) is of dimension k for all t ^{2 I}. For an arbitrarily but xed time t we determine a basis ^fy⁰(t) ::: y^k(t)^g of ^M(t).

Each vector of this basis is a consistent initial value for the DAE (1). The linear set spanned by the solutions to these initial values remains k-dimensional on the whole solution interval. Hence,^M(t) = span^fy⁰(t) ::: y^k(t)^g for any t, i.e. ^M= span^fy⁰ ::: y^kg. ²

Corollary 4

Provided that (I ^;can)y = 0 is valid, the conditions PP¹y = 0, PT?¹P?APP¹y = 0 and T?canAPP¹y = 0 are equivalent.

Remark 5

If Q?B ²C¹ and ifQ?¹(t) is any arbitrary di erentiable projector func- tion onto N¹(t), then, for a more practicable realization, the space ^M(t) ^{can easily} be described according to Theorem 13 and Remark 14 as a solution of the system of equations

^T(t)A(t)y(t) = 0 Q?(t)B(t)y(t) = 0 QT?¹(t)P?(t)B(t)^;(Q?(t)B(t))⁰]y(t) = 0:

5 Application to inhomogeneous systems

As the inhomogeneous systems are more general than the homogeneous ones, we consider the inhomogeneous index 2 DAE

Ay⁰+By = f: (6)

6

This DAE is homogenized by a simple trick 3], and then we can apply Theorem 2 to this special case.

^A^y⁰+ ^B^y = 0 ^A = A 0 0 1

! ^B = B ^;f 0 0

!

: (7)

The vector ^y is by one dimension larger than y, that means (m+1). The DAEs (6) and (7) are equivalent in the sense that (y^T 1)^T is a solution of (7) in case y is a solution of (6) and reversely.

In order to apply Theorem 2, the projectors and matrices have to be computed.

Obviously,

^P = P 0 0 1

! ^A¹ = A¹ 0 0 1

! ^Q¹ = Q^{1 ;}Q¹G^;12 f

0 0

! ^G² = G^{2 ;}B⁰PQ¹G^;12 f

0 1

!

holds, which implies

^can = can QP¹G^;12 f + (QQ¹G^;12 f)⁰+ (I^;(QQ¹)⁰)PQ¹G^;12 f

0 1

!

:

Now, let^Mc be a (k+1)-dimensional solution space of the homogenized DAE (7). By Theorem 2 there exists a function ^ with the properties mentioned in the theorem, = . Let ^^ ^T = (^Tj;h) be a solution of the equation that is adjoint to the homogenized DAE (7). Then (^Tj;h)^T satises the dierential equations

A^T0;(B^{T ;}A^T0) = 0 h^0;Tf = 0:

Now ^y is a solution of (7) if

(I^;can)y = QP¹G^;12 f + (QQ¹G^;12 f)⁰+ (I ^;(QQ¹)⁰)PQ¹G^;12 f:

These conditions are now retransformed to the non-homogenized system (6), (I^;can)y = QP¹G^;12 fym⁺¹

^TAy = hym⁺¹:

Taking into account that ym⁺¹ 1 has to be valid, i.e. that the space ^Mcym⁺¹¹ is only k-dimensional, we obtain

Theorem 6

^M is a k-dimensional ane set of solutions of the equation (6) if and only if there exist am-dimensional matrix-valued function , = rankPP^1;k, and a vector-valued function h on ^I such that

(i) rank(t)^,

7

(ii) for all t^{2I we have}

M(t)=^{n 2RmT}(t)A(t) = h(t)

(I^;can(t)) = QP¹G^;12 f + (QQ¹G^;12 f)⁰ +(I ^;(QQ¹)⁰)PQ¹G^;12 f(t)^o (iii) and ^, h full the system of equations

A^T0;(B^{T ;}A^T0) = 0 h^0;Tf = 0:

This has already been proved.

Remark 7

If Q?B Q?f ²C¹ and ifQ?¹(t) is any di erentiable projector function onto N¹(t), then, for a practicable realization, the set ^M(t) can easily be described by Theorem 13 and Remark 14 of the next section as the solution of the system of equations

^T(t)A(t)y(t) = h(t) Q?(t)B(t)y(t) = Q?(t)f(t)

Q_T?¹(t)P?(t)B(t)^;(Q?(t)B(t))⁰]y(t) = Q_T?¹(t)P?(t)f(t)^;(Q?(t)f(t))⁰]:

6 Representation of im

can

by a reduction step

The fact that A_T?¹ is the leading coe cient matrix after a reduction of the equa- tion (1) permits to prove the following reduction theorem

Theorem 8

Let Q?B and Q_T?¹P?B ^;(Q?B)⁰] be di erentiable. Then the DAE Ay⁰+By = 0 is equivalent to the implicit regular ordinary di erential equation

(A^;Q?B^;Q_T?¹P?B^;(Q?B)⁰])y⁰

+P_T?¹(P?B^;(Q?B)⁰)^; Q_T?¹P?B ^;(Q?B)⁰]⁰y = 0 (8) with the additional conditions

Q_T?¹(~t)^hP?(~t)B(~t)^;(Q?(~t)B(~t))⁰ⁱy(~t) = 0 ~t^2I (9a) Q?(^t)B(^t)y(^t) = 0 ^t^2I: (9b) Proof. By Theorem 1 the DAE (2)

A^T0; B^T;A^T0 = 0 8

has index 2, too.

Multiplying the DAE Ay⁰+By = 0 by Q? from the left yields Q?By = Q?B⁰y = 0

hence,

(A^;Q?B)

| {z }

A_T?¹

y⁰+ (P?B ^;(Q?B)⁰)y = 0 (10) Q?(^t)B(^t)y(^t) = 0

for each ^t^2I. The other direction of the equivalence to equation (1) becomes clear when multiplying the DAE (10) by Q? from the left

(Q?By)⁰=Q⁰?Q?By:

Due to the condition (9b) at time ^t it holds that Q?By0 and we obtain the given DAE 5, A. B, Corollary 7].

The leading matrix in equation (10) is AT?¹. Multiplication by QT?¹ provides equation (9a) and, nally, the equivalent system

(A^;Q?B^;Q_T?¹P?B^;(Q?B)⁰])y⁰

+PT?¹(P?B^;(Q?B)⁰)^; QT?¹P?B^;(Q?B)⁰]⁰y = 0 QT?¹(~t)^hP?B(~t)^;(Q?(~t)B(~t))⁰ⁱy(~t) = 0 Q?(^t)B(^t)y(^t) = 0

~t^2I. Equivalence can be proved analogously to the above.

It remains to show that ^G?² = A^{T ;}B^TQ? ^; B^TP?^;(B^TQ?)⁰Q?¹ is nonsin- gular.

^G?² =A^{T ;}B^TQ?^; B^TP?^;(B^TQ?)⁰Q?¹

=G?²+ (B^TQ?)⁰Q?^1;A^T0P?Q?¹

=G?^2;A⁰?¹Q?¹+A^TP?⁰Q?¹

=G?²+A?¹Q⁰?¹Q?¹+A^TP?⁰Q?¹

=G?²(I + P?¹(Q⁰_?¹+P?P_?⁰)Q?¹):

The two matrices of the product above are nonsingular, hence ^G?² is so, too.

The dierential equation (8) is an implicitly given regular dierential equation. ²

Corollary 9

Under analogous smoothness assumptions the DAE A^T0; B^{T ;}A^T0 = 0 is equivalent to the implicitly given ODE

A^T +QB^0T +Q^{T1 h}P B^{T ;}A^T0;(QB^0T)⁰ⁱ⁰

;

P^1T P B^{T ;}A^T0;(QB^0T)^0; Q^{T1 h}P B^{T ;}A^T0;(QB^0T)⁰ⁱ⁰ = 0 (11) 9

with the conditions

Q^T1(~t)P(~t) B^T(~t)^;A^T0(~t)^; Q(~t)B^0T(~t)⁰(~t) = 0 ~t^2I Q(^t) B^T(^t)^;A^T0(^t)(^t) = 0 ^t^2I:

Proof. It holds that (B ^;A⁰)Q = B⁰Q. The other steps of the proof are the same as in Theorem 8. ²

Remark 10

The equations (9a) and (9b) can also be regarded as additional equa- tions instead of the initial conditions. They determine the manifold where the so- lutions of (8) belong to. For determining the manifold the smoothness assumption can be weakened, i.e. Q?B ²C¹ is sucient.

Lemma 11

The conditions

PP¹y⁰ =PP¹y⁰ (13a) QT?¹P?B^;(Q?B)⁰]y⁰ = 0 (13b)

Q?By⁰ = 0 (13c)

make up a complete set of initial values for the di erential equation (8). This in- cludes the fact that these equations uniquely determine y⁰ and do not contradict each other. Furthermore, they provide a consistent initial value for equation (1), i.e. y⁰ = cany⁰. (All functions and matrices are assumed at a certain time t.)

Remark 12

An analogous statement applies to the adjoint problem (2) and the conditions

P?P?¹⁰ =P?P?¹⁰ Q^T1 P B^{T ;}A^T; QB^0T00 = 0

Q B^{T ;}A^T00 = 0:

Proof. From Q?By⁰ = 0 it follows that By^{0 2} imP? = imAA^y, and, consequently, Q¹y⁰ = 0. We consider the second equation (13b) in the formulation of the lemma.

QT?¹P?B^;(Q?B)⁰]y⁰ = 0

, P?B^;(Q?B)⁰]y^{0 2} imPT?¹ = imAT?¹G^;1?^2T = imAT?¹

(13c⁾

, P?By^0;(Q?B)⁰y⁰ = Az^;Q?Bz for a z ^2Rm

,

( P?By⁰ = Az

Q?(Q?B)⁰y⁰ = Q?Bz

(13c⁾

,

( By⁰ = Az

(Q?B)⁰y⁰ = Q?Bz 10

By the relationG^;12 BQ = Q+P¹PP⁰Q, which can be easily derived, we now conclude the relationQy⁰+P¹PP⁰Qy⁰+G^;12 BPy⁰=P¹Pz from By⁰ =Az. Multiplying this equation by QP¹ from the left yields, after some technical transformations,

Qy^0;(QQ¹)⁰Qy⁰+QP¹G^;12 BPy⁰ =^;QQ¹z:

Let us have a look at the right-hand side of this equation

;QQ¹z =^;QQ¹G^;12 B⁰z =^;QQ¹G^;12 Q?Bz =^;QQ¹G^;12 (Q?B)⁰y⁰

=^;(QQ¹G^;12 Q?B)⁰y⁰ =^;(QQ¹)⁰y⁰: Inserting into the equation providesQy⁰ =^;QP¹G^;12 BPy^0;(QQ¹)⁰Py⁰, hence altogether

y⁰ = cany⁰ = canPP¹y⁰ = cany⁰:

Analogously it is shown that also the equations for the adjoint problem have full rank. Now it remains to show that the equations (13a)-(13c) do not contradict each other. It is su cient to show that rankPP¹ + rankQ?¹+ rankQ? = m. This sum has at least the value m. Supposed it is larger, then, because of rankP?P?¹+ rankQ?¹+ rankQ? =m, it holds that rankPP¹ > rankP?P?¹. Since now rankQ = m^;rankA = rankQ?, it follows that rankQ¹ < rankQ?¹. Obviously, rankP?P?¹+ rankQ¹+rankQ = rankP?P?¹+rankQ¹+rankQ? m, however. This contradicts rankP?P?¹ + rankQ?¹+ rankQ?=m and rankQ¹ < rankQ?¹. ²

Let

^S¹(t) :=^n2Rmj2kerQ?(t)B(t)^\ker ^Q_T?¹(t)(P?(t)B(t)^;(Q?(t)B(t))^0o:

Theorem 13

^S¹ = imcan.

Proof. ^S¹ imcan follows immediately from the proof of Lemma 11.

On the other hand, Lemma 11 and the relation (4) imply ^S¹+ kercan = ^Rm, hence dim ^S¹ m^;dimkercan. We have imcankercan=R^m, dimimcan+ dimkercan =m, and thus dimimcan =m^;dimkercan. The dimensions of ^S¹ and imcan coincide. ²

Consequently, the space of all solutions of the homogeneous index 2 DAE (1) can be represented as the image space of the projector can resp. the kernel of the projector (I^;can), or as the solution set of a system of equations and as the kernel of a matrix, respectively.

Remark 14

Let ^Q?¹(t) be any other di erentiable projector function Q?¹(t) onto N?¹(t) = kerA?¹(t). Because of ^Q?¹(t)Q?¹(t) = Q?¹(t) and Q?¹(t) ^Q?¹(t) = ^Q?¹(t) the statements of Lemma 11 and Theorem 13 hold with ^Q?¹(t)^{, too.}

Remark 15

The above process corresponds to the index reduction considered in 4, Theorem 13].

11

A Linear Algebra

Denition 16

The matrix A has index k if k is the smallest natural number for which imC^k = imC^k+1 holds true.

Lemma 17

For C ²L(^Rm) it holds that imC^k = imC^{k+1 ,}kerC^k = kerC^k+1:

Denition 18

The matrix pencil (A B) is called regular if p(c) := det(cA + B)= 0.

Denition 19

The index of a matrix pencil(A B) is the index of the matrix (cA+

B)^;1A.

Remark 20

This denition also implies that ind(A B) := ind (cA + B)^;1A

= minkker (cA + B)^;1A^k = ker (cA + B)^;1A^k+1

L17= minkim (cA + B)^;1A^k = im (cA + B)^;1A^k+1:

Lemma 21

Let X Y M ²L(^Rm), and let M be nonsingular. Then imMX = imMY ^,imX = imY ^,imXM^;1 = imY M^;1:

The uniquely determined Moore-Penrose-inverse C^y of a matrixC ²L(^{Rn Rm}) is dened by its properties:

CC^yC = C C^yCC^y=C^y CC^y= CC^yT C^yC = C^yC^T:

We can easily reconstruct that C^Ty = C^yT holds. CC^y is a projector onto imC along (imC)^?, C^yC is a projector onto (kerC)^? along kerC.

Lemma 22

For C ²L(^Rm) we have (imC^T)^? = kerC.

Lemma 23

For regular matrix pencils (A B), A B ²L(^Rm) we have ind(A B) = ind(A^T B^T).

12

Proof. ker

"

cA^T+B^T;1A^Tk

#

= ker

"

cA^T +B^T;1A^Tk+1

#

L17

, im

"

cA^T+B^T;1A^Tk

#

= im

"

cA^T +B^T;1A^Tk+1

#

L21

, im

"

A^T cA^T+B^T;1k

#

= im

"

A^T cA^T +B^T;1k+1

#

, im

"

A^T cA^T+B^T;1k

#

?= im

"

A^T cA^T +B^T;1k+1

#

?

, im

"

(cA + B)^;1A^kT

#

? = im

"

A^T(cA + B)^;1k+1T

#

?

L22

, ker (cA + B)^;1A^k = ker A^T(cA + B)^;1k+1 Thus, ind(A^T B^T) =k ^,ind(A^T B^T) =k. ²

Theorem 24

5, A. A, Theorem 16] If ind(A B) = 2, then (A B + ASA) is regular with index 2 for each matrix S ²L(^Rm).

References

1] A. Abramov, On transfer of boundary conditions for systems of linear ordi- nary di erential equations (a variant of transfer method) (in Russian), USSR Journal of Comput. Math. and Math. Phys., 1(3) (1961), pp. 542{544.

2] K. Balla, On linear subspaces for linear DAEs of index 1, Preprint 94-11, Laboratory of Operation Resarch and Decision Systems, Budapest, 1994. To appear in J. Computers and Mathematics with Applications.

3] K. Balla and R. Marz,Transfer of boundary conditions for DAEs of index 1, Preprint 95-3, Humboldt-Univ., Math. Nat. Fak. II, Budapest, Berlin, 1995.

To appear in SIAM J. Numer. Anal.

4] E. Griepentrog, Index reduction methods for di erential-algebraic equa- tions, in Seminarberichte, Berlin Seminar on Dierential-Algebraic Equations, E. Griepentrog, M. Hanke, and R. Marz, eds., Humboldt-Univ. Berlin, Fach- bereich Mathematik, 1992, pp. 14{29.

5] E. Griepentrog and R. Marz, Di erential-Algebraic Equations and Their Numerical Treatment, Teubner-Texte zur Mathematik 88, Teubner, Leipzig, 1986.

6] M. Hanke, E. I. Macana, and R. Marz,On asymptotics in case of linear index-2-daes, Preprint 94-5, Humboldt-Univ., Fachbereich Mathematik, Berlin, 1994.

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7] R. Marz,Index-2 di erential-algebraic equations, Results in Math., 15 (1989), pp. 149{171.

8] , Numerical methods for di erential-algebraic equations, Acta Numerica, (1992), pp. 141{198.

9] R. Marz,On linear di erential-algebraic equations and linearizations, Preprint 94-14, Humboldt-Univ., Fachbereich Mathematik, Berlin, 1994. To appear in APNUM.

10] T. Petry, On the stability of the Abramov transfer for DAEs of index 1, Preprint 94-8, Humboldt-Univ., Fachbereich Mathematik, Berlin, 1994.

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