DSpace at VNU: Adjoint pairs of differential-algebraic equations and Hamiltonian systems

Assuming that the equations are tractable with index less than or equal to 2, we give a criterion ensuring the inherent ordinary differential equations of the pair to be adjoint each to

Adjoint pairs of differential-algebraic equations

and Hamiltonian systems✩

Katalin Ballaa,∗, Vu Hoang Linhb

aComputer and Automation Research Institute, Hungarian Academy of Sciences, H-1518 Budapest P.O Box 63, Hungary

bFaculty of Mathematics, Mechanics and Informatics, Vietnam National University, Hanoi, 334 Nguyen Trai Str., Vietnam

Abstract

We consider linear homogeneous differential-algebraic equations A(Dx) + Bx = 0 and their adjoints

−D∗(A∗x)+B∗x= 0 with well-matched leading coefficients in parallel Assuming that the equations are tractable with index less than or equal to 2, we give a criterion ensuring the inherent ordinary differential equations of the pair to be adjoint each to other We describe the basis pairs in the invariant subspaces that yield adjoint pairs of essentially underlying ordinary differential equations For a class of formally self-adjoint equations, we charac-terize the boundary conditions that lead to self-adjoint boundary value problems for the essentially underlying Hamiltonian systems.

 2004 IMACS Published by Elsevier B.V All rights reserved.

Keywords: Differential-algebraic equations; Adjoint pairs of differential-algebraic equations; Self-adjoint boundary value

problems

1 Introduction

Recently, differential-algebraic equations (DAEs) of the form

✩

This work was supported by OTKA (Hung National Sci Foundation) Grants # T043276, T031807.

* Corresponding author.

E-mail addresses: balla@sztaki.hu (K Balla), vhlinh@hn.vnn.vn (V.H Linh).

URL: http://www.sztaki.hu/~balla.

0168-9274/$30.00  2004 IMACS Published by Elsevier B.V All rights reserved.

doi:10.1016/j.apnum.2004.08.015

where A, D, B : C([a, b], L(C m )), f : C([a, b], C m ), were introduced in [4] and their analysis was

launched A motivation to considering this class of implicit equations was that under very mild con-ditions the formal adjoint equation

−D∗

A∗x∗

+ B∗x

g : C([a, b], C m ), shares the basic properties with (1) In the paper, we intend to study two kinds of

ordi-nary differential equations (ODEs) that are derived from (1) and (2) in parallel and called inherent ODEs (INHODEs) and essentially underlying ODEs (EUODEs) Assuming that Eqs (1) and (2) are homoge-neous, we focus on the relationship within the INHODE pairs and EUODE pairs Special attention will

be paid to a particular subclass of (formally) self-adjoint homogeneous equations different from the one analyzed in [1] and similar to one in [7,6,3] For this class, we formulate boundary conditions that may

be considered as ones giving to rise to a (formally) self-adjoint BVP

The results of the paper rely strongly upon the analysis in [4,5] We adopt the notions from [5] that differ slightly from those in [4] Some of the ideas used here were inspired by constructions in [1] dealing with a special class of (formally) self-adjoint equations

Our general results if applied to (3) coincide with those proved in [1] If specialized into the equation examined in [3], the statements here are complementary to those in [3] Some special cases of adjoint DAE pairs with their specially chosen EUODEs appear in [8] and the results were implemented in [9] Our results cover these special cases, too

The outline of the paper is as follows For the convenience of the reader in Section 2 we recall the notions and notations related to next sections For an explanation of the relevance, we refer to [4,5] The basic results concerning the notions will be used here freely Section 3 is addressed to INHODEs Section 4 deals with the construction of the bases and pairs of bases in subspaces connected with (1) and (2) Section 5 derives the EUODEs and the pairs of EUODEs Section 6 defines a class of (formally) self-adjoint DAEs In terms of the boundary conditions posed for these (formally) self-adjoint DAEs, we formulate a condition sufficient to give a self-adjoint BVP for the EUODE Finally, in Section 7 we point out the practical importance of associated ODEs

2 Preliminaries

In the paper, the source functions f, g occurring in the general linear differential-algebraic equations

of the form (1) and (2) will not be involved into the analysis Therefore, without further mentioning we

will assume that f = 0, g = 0, i.e., the equations are homogeneous As to the remaining coefficients

defined onI := [a, b], first we assume that the leading coefficients A and D are “well-matched”, i.e.,

they fulfill as follows

and there exist functions a1, , a m −r , d1, , d r ∈ C1( I, C m ) such that for all t ∈ [a, b],

ker A(t)= spana1(t), , a m −r (t)

, im D(t)= spand1(t), , d r (t)

We define the projector function R ∈ C1( I, L(C m )) by R2= R, ker R(t) = ker A(t), im R(t) = im D(t).

The leading coefficients of Eq (2) are also well-matched, the corresponding projector function is R∗∈

C1( I, L(C m )).

With Eq (1), the following chain is associated:

G0:= AD, B0:= B;

For i = 0, 1, Q i , P i , W i ∈ C(I, L(C m

)):

Q2i = Q i , W i2= W i ,

N i := ker G i = im Q i , P i = I − Q i ,

ker W i = im G i ,

G i+1:= G i + B i Q i , B i+1= B i P i ,

S i := {z : I → C m , B i z ∈ im G i}

Similarly, Eq (2) gives rise to the chain

G∗0:= −D∗A∗, B

∗0:= B∗;

For i = 0, 1, Q ∗i , P ∗i , W ∗i ∈ C(I, L(C m

)):

Q2∗i = Q ∗i , W ∗i2 = W ∗i ,

N ∗i := ker G ∗i = im Q ∗i , P ∗i = I − Q ∗i ,

ker W ∗i = im G ∗i ,

G ∗i+1 := G ∗i + B ∗i Q ∗i , B ∗i+1 = B ∗i P ∗i ,

S := {z : I → C m , B ∗i z ∈ im G ∗i}

To characterize DAEs we need a further assumption on smoothness:

Condition C2:

dim D(t)S1(t) = const = : ρ and dim D(t)N1(t) = const = : ν, (8)

and there exist functions s D

1, , s D

ρ , n D

1, , n D

ν ∈ C1( I, C m ) such that for all t ∈ I,

D(t)S1(t)= spans1D (t), , s ρ D (t)

, D(t)N1(t)= spann D1(t), , n D ν (t)

.

Let conditions C1 and C2 be valid Eq (1) is said to be

an index-0 tractable DAE if

an index-1 tractable DAE if

an index-2 tractable DAE if

Denote the reflexive generalized inverses (RGIs) of D and A∗by D−and A∗−if defined by

D−D = P0, A∗−A∗= P∗0,

DD−= R, A∗A∗−= R∗.

While chain (6) depends on projectors Q0, Q1, the index does not If (1) has an index µ, µ ∈ {0, 1, 2},

then (2) is also tractable with index µ and vice versa.

In index-1 case, projector function Q0 (P0) is called canonical and it is denoted by Q 0c (P 0c ), if

ker Q0= S0 (im P 0c = S0) In index-2 case, the subspace N1 depends on the special choice of Q0(P0)

while S1does not Projector Q1is marked byif ker Q1= S1holds All terms in chain (6) derived by the use of Q1are marked by, too In index-2 case, canonical projector function Q 0c onto N0is defined as

Q 0c = Q0P1G−1

2 B + Q0Q1D−

D  Q1D−

D,

where Q0 is an arbitrary projector function onto N0; Q 0c does not depend on the special choice of Q0

All terms in chain (6) derived by the use of Q 0care marked with subscriptc, too Q 1c= Q1holds Note that if P1= I is set, then we get the formula valid for index-1 case Therefore, there is no need for special

notation segregating the index-1 and index 2 canonical projectors Q 0c

In index-2 case, decomposition D(t)S1(t) ⊕ D(t)N1(t) ⊕ ker A(t) = C m induces projector

func-tions D  P1D−, D  Q1D−, I − R ∈ C1( I, L(C m )) onto the subspaces in the decomposition along the

other couple Similarly, projectors A∗P∗1A∗−, A∗Q∗1A∗− and I − R∗ correspond to decomposition

A∗(t)S∗1(t) ⊕ A∗(t)N

∗1(t) ⊕ ker D∗(t)= Cm and identity (D  P1D−) = A∗P∗1A∗−holds

For an arbitrary V ∈ C(I, L(C m )), C1

V denotes function space{v ∈ C(I, C m ): V v ∈ C1( I, C m )} With

Eq (1) we associate operator

L : C1

D → C1

D  Q1 G −1

2

D ,

and similarly, Eq (2) is related to operator

L∗:C1

A∗→ C1

A∗Q ∗1G −1

∗2, L∗x∗:= −D∗(A∗x

∗ + B∗x

∗, x∗∈ C1

A∗.

A function x ∈ C1

D is called the solution if it satisfies (1) pointwise For each t ∈ I, solutions to (1) form

a linear subspaceSµ (t)⊆ Cm,

Sµ (t) := im Π can µ (t),

Πcan 0= I, Πcan 1= P 0c , Πcan 2= P 0c P1c = P 0c P1= UP0P1,

here

U = I − Q0P1G−1

2 B − Q0P1D−

D  Q1D−

D,

U is invertible, Q0is an arbitrary projector function onto N0 Solutions x∗∈ C1

A∗and canonical projector functions for (2) are introduced in an analogous way with changes corresponding to the chain for (2)

Thorough the paper we use the notational convention: If H ∈ L(C n ) is invertible, then H−∗:= (H−1) ,

if H is not invertible then H−is a (fixed) RGI, H−∗:= (H−) ; in the latter case, H−∗= H∗− does not

hold, in general If H : I → L(C n ), then H−(t) := [H (t)]−, H∗(t) := [H (t)]∗.

3 Inherent ODEs for adjoint pairs

We remind that the equations under consideration are homogeneous

3.1 Index-0 equations

Now, Condition C1 combined with (9) ensures that both A and D are invertible Let u := Dx, u∗:=

A∗x∗ Equations

−u

∗+ D∗−1B∗A∗−1u

are called inherent ODEs (INHODEs) associated with (1) and (2), respectively On the one hand, they are equivalent to equations from which they are derived On the other hand, (14) and (15) form an adjoint pair

of ODEs Trivially, x ∈ C1

D (x∗∈ C1

A∗) is a solution of (1) ((2)) if and only if Dx (A∗x∗) is a solution of (14) ((15)) Practically, we do not treat equations of index-0 anymore, we included them for completeness, only

3.2 Index-1 equations

In index-1 case the canonical projectors and the arbitrary ones belonging to pair (1) and (2) are con-nected by the following proposition

Lemma 1 Let (1) (and/or (2)) be an index-1 DAE and let Q0and Q∗0be arbitrary projector functions onto N0and N∗0, respectively Then, P 0c = −(D∗A∗G−1

1 )

Proof The representation Q 0c = G∗−1

∗1 Q∗∗0G∗∗1 is a simple consequence of the trivial identities

ker Q0G−11 B = ker W0B and Q∗∗0G1= G∗

∗1Q0 Indeed, G∗−1∗1 Q∗∗0G∗∗1= G∗−1

∗1 Q∗∗0B = Q0G−11 B We

im-mediately get P 0c = I − G∗−1

∗1 Q∗∗0G∗∗1= G∗−1

∗1 P∗0∗G∗∗1= −G∗−1

∗1 AD The second representation can be

checked in a similar way 2

The lemma results in identities Q∗∗0c BP 0c = 0 and Q∗

0c B∗P ∗0c = 0, which, in turn, yield BP 0c=

P ∗0c∗ BP 0c = (P∗

0c B∗P ∗0c ) = (B∗P ∗0c ) = P∗

∗0c B Therefore, (1) can be rewritten as

0= A(Dx)+ BP 0c x + BQ 0c x = A(Dx)+ P∗

∗0c BP 0c x + Q∗

Take the projections onto im Q∗∗0c and ker Q∗∗0c and use RGIs A∗−c , D−c : P ∗0c = A∗−

c A∗, P 0c = D−

c D and

A∗−∗c A = R = DD−

c , R ∈ C1( I, L(C m )):

(Dx)− RDx + A∗−∗

A particular case of the above lemma is representation Q ∗0c = G−∗

1c Q∗0c G∗1c Thus, Q∗∗0c (BQ 0c )=

(G−∗1c Q∗0c G∗1c ) G 1c Q 0c = G 1c Q 0c, and therefore,

Q 0c x = 0,

is an equivalent to (17) Set u = DP 0c x = Dx and get the INHODE for (1) from (18):

u− Ru + A∗−∗

The next “invariance” gives importance to the INHODE:

(I1) If u(t0) ∈ im D(t0) holds for an arbitrary t0, then u ∈ im D.

Indeed, A∗−c (I −R∗ = 0 and (19) involve that function w := (I −R)u satisfies the ODE w= (I −R)w,

therefore, w(t0) = 0 for an arbitrary t0yields w= 0

The similar procedure applied to (2) with the same RGIs yields the decomposition

Q ∗0c x∗= 0,

−
A∗x∗

+ R∗ A∗x

∗+ D−∗

while setting u∗= A∗P ∗0c x∗= A∗x∗results in the inherent ODE for (2)

−u

∗+ R∗u

∗+ D−∗

Now, the invariance of the inherent ODE reads as

(I1∗) If u∗(t0) ∈ im A∗(t

0) holds for an arbitrary t0, then u∗∈ im A∗.

Let us return to arbitrary projector functions Q0and Q∗0 Due to the properties of RGIs, the identities

A∗−c = P ∗0c A∗−c R∗= P ∗0c A∗−c R∗R∗= P ∗0c A∗−c

A∗A∗−

DD−∗

= P ∗0c
A∗−c A∗

A∗−

DD−∗

= P ∗0c P ∗0c A∗−

DD−∗

= P ∗0c A∗−

DD−∗

,

D−c = P 0c D−c R = P 0c D−c DD−= P2

0c D−= P 0c D−,

are valid Therefore,

A∗−∗c BD c−= DD−A∗−∗P∗

∗0c BP 0c D−= DD−A∗−∗P∗

∗0c BD−= DD−A∗−∗G

1P0G−11 BD−

= DD−A∗−∗ADG−1

1 BD−= DD−RDG−1

1 BD−= DG−1

1 BD−,

and the inherent equation turns to be identical to that in [4] Similarly,

D−∗c B∗A∗−c = A∗G−1

∗1B∗A∗−.

The new forms (19) and (21) of INHODEs show transparently that these ODEs depend only on the geometric characteristics of the problem: Eqs (19) and (21) contain terms dependent on the original

data, only: projector functions R, P 0c , P ∗0care defined uniquely by them and RGIs are induced uniquely

by the latter

Furthermore, the following statement becomes transparent

Theorem 2 In the index-1 case, the adjoint of the inherent ODE (19) of the DAE (1) coincides with the

inherent ODE (21) of the adjoint DAE (2) if and only if neither im D(t ) nor ker A(t ) depend on t

Proof Let im D and im A∗ be constant subspaces Then, so are their orthogonal complements, ker D∗ and ker A Then, R= 0 and vice versa 2

3.3 Index-2 equations

In index-2 case, we use again canonical projectors Q 0c ,  Q1, Q ∗0c ,  Q∗1, Πcan 2, Π∗can 2and the decom-positions relying upon them In the analysis we also utilize the identity Q = Q , i.e., Q = Q P

and its counterpart marked by ∗ We take the projections of the equations onto im G 2c Πcan 2G−12c =

im AD  P1G−12c and im G 2c (I − Πcan 2) G−12c = im G 2c (Q 0c + P 0c Q1) G−12c In the solution x part Πcan 2x=

P 0c P1x is segregated Terms D  P1x = DP 0c P1x = D  P1D−

c Dx and D  Q1x = D  Q1D c−Dx are

differen-tiable for x ∈ C1

D and DQ 0c P1x = 0, so we may split their sum Dx Note that due to condition C1,

ker AD = ker D holds Relation  G−12c A= P1D−c also will be useful After partial differentiation, the first projection simplifies to

D  P1x

−
D  P1D−c

D  P1x−
D  P1D c−

D  Q1x + D  P1G−1

2c BD−c D  P1D c−D  P1x = 0. (22)

We undertake the second projection to scaling by G−12c that yields

0= Q 0c P1D−

c (Dx)+
Q 0c + P 0c Q1G−1

2c Bx

= −Q 0c Q1D−

c (Dx)+ Q 0c
P1+ Q1G−12c Bx + P 0c Q1x

= −Q 0c Q1D c−(Dx)− Q 0c Q1D c−

D  Q1D c−

Dx

+
Q 0c Q1D c−

D  Q1D c−

D + Q 0c P1G−12c B

x+ Q1x

= −Q 0c Q1D−

c

D  Q1x

A consequence of (23) is that the projections of the right side also vanish We obtain that in fact, (23) is equivalent to a system of algebraic equations:

The hidden constraint now sits inside the system, it becomes transparent if one returns to an arbitrary Q0

On the other hand, if (24) is taken into account, (22) yields the INHODE for u := D  P1x

u−
D  P1D−c

u + D  P1G−1

This is the same equation we got in [4] Indeed, the multiplier D  P1D c− of u in the last term could be

inserted in [4] Subscriptcwas not present there However, it was shown that the equation is independent

of the choice of Q0

Now, we check that D  P1G−12c BD−c D  P1D−c = D  P1D c−A∗−∗c BD c−D  P1D c−.

D  P1G−12c BD c−D  P1D−c − D  P1D−c A∗−∗c BD−c D  P1D−c

= D  P1D c−

D − A∗−∗

c G2cG−1

2c BD−c D  P1D c−

= −D  P1D c−A∗−∗c 

BQ 0c
P1+ Q1G−12c BD c−D  P1D c−+ BP 0c Q1G−12c BD−c D  P1D−c

= −D  P1D c−A∗−∗c 

BQ 0c P1G−1

2c BD c−D  P1D c−+ BQ 0c Q1P 0c P1D−

c + BP 0c Q1P 0c P1D−

c



= −D  P1D c−A∗−∗c B

Q 0c − Q 0c Q1D−

c

D  Q1D c−

D

P 0c P1D−

c

= −D  P1D c−A∗−∗c BQ 0c Q1D−c

D  Q1D−c

DP 0c P1D c−

= +D  P1D c−A∗−∗c AD  Q1D−c

D  Q1D−c

DP 0c P1D c−

= +D  P1Q1D−

c

D  Q1D c−

DP 0c P1D−

c = 0.

Thus, the transparent form of (26) is

u−
D  P1D−

u+
D  P1D−

A∗−∗BD−

D  P1D−

We can treat the adjoint equation (2) in a quite similar way The final result for u∗= A∗P∗1x∗is

−u

∗+
A∗P∗1A∗−

c



u∗+
A∗P∗1A∗−

c



D−∗c B∗A∗−c

A∗P∗1A∗−

c



while the constrains are

Recall that D  P1D c−= D  P1D−and A∗P∗1A∗−

c = A∗P∗1A∗− Eq (27) has the “invariance” property:

(I2) If u(t0) ∈ im D(t0)  P1(t0) is valid for an arbitrary t0, then u ∈ im D  P1,

while for (28) the next claim holds:

(I2∗) If u∗(t0) ∈ im A∗(t

0)  P∗1(t0) is valid for an arbitrary t0, then u∗∈ im A∗P∗1.

The proof is similar to the index-1 case, projector R is replaced by D  P1D−

Relation (D  P1G−1

2c BD−c D  P1D c−) = A∗P∗1A∗−

c D c−∗B∗A∗−c A∗P∗1A∗−

c connecting the last terms in (27) and (28) calls for comparing the adjoint of the INHODE (27) of the DAE (1) with the INHODE (28) of the adjoint DAE (2) The next Theorem gives the trivial answer

Theorem 3 In the index-2 case the adjoint of the inherent ODE (27) of the DAE (1) coincides with the

inherent ODE (28) of the adjoint DAE (2) if and only if projector D  P1D−is constant.

Proof We again recall that (A∗P∗1A∗−) = D  P1D− 2

We formulate this result in terms of the basic subspaces

Theorem 4 In the index-2 case the adjoint of the inherent ODE (27) of the DAE (1) coincides with the

inherent ODE (28) of the adjoint DAE (2) if and only if neither the subspace D(t )S1(t) nor the subspace

A∗(t)S∗1(t) depend on t

Proof Let D(t)S1(t) and A∗(t)S∗1(t) be constant subspaces Then, the orthogonal complement

A∗(t)N∗1(t) ⊕ ker D∗(t) of the first one does not depend on t either It means that neither the

im-age nor the kernel of projector A∗(t)  P∗1(t)A∗−(t) depend on t , i.e., A∗(t)  P∗1(t)A∗−(t) ≡ const Then,

D(t)  P1(t)D−(t) is constant, as well.

In the opposite direction: Let projector function D  P1D− and together with it, projector function

A∗P∗1A∗−be constant Then, their images, DS

1and A∗S∗1are constant 2

Remark 5 In the proof we referred to identities A∗S∗1 = (DN1⊕ ker A)⊥ and DS

1 = (A∗N∗1⊕

ker D∗ ⊥ They allow to formulate the above results in terms of one equation, either (1) or (2):

In the index-2 case, the adjoint of the inherent ODE (27) of the DAE (1) coincides with the inherent

ODE (28) of the adjoint DAE (2) if and only if the subspace pairs D(t )S1(t), D(t)N1(t) ⊕ ker A(t) or

A∗(t)S1(t), A∗(t)N1(t) ⊕ ker D∗(t) are constant.

Remark 6 There is another way to show the connection between the last terms in INHODEs of the

ad-joint pairs of DAEs We provide it since the method used here is applicable in later material, as well The

method is based on an appropriate decomposition of the coefficient matrices and the explicit construction

of matrix chains for the adjoint pair (1), (2) In fact, the technique used here as well as in [2] relies upon

the pointwise reduction to a matrix pair equivalent to the original pair (AD, B).

Let us fix t = t0and omit the argument We begin with a decomposition resulting block matrices of special form

G0= AD = U





0 0 0

0 0 0



 V, B0= B = U





0 0 B33



 V,

where U and V are nonsingular The blocksizes in the decompositions of G0and B0correspond each to

other Let the diagonal blocks be of dimension n1, n2, n3, respectively One always may achieve either

n3= 0 (no third block-row and no block-column appear) or B33 be nonsingular The special case n2= 0

indicates that the matrix pair has index 1 (The decomposition trivially exists: first one should transform

G0to the indicated form, say, by a forward-backward Gauss elimination with row and column changes

if necessary and then by a next transformation that makes the appropriate blocks in B0 to vanish and

does not affect the pattern of G0) For brevity, in the next considerations, we assume the general case,

n2= 0, n3= 0 (and B33 is nonsingular) The special cases may be treated in a similar way A matrix chain corresponding to the scheme in (6) can be constructed explicitly: Set

Q0= V−1





0 0 0

0 I 0

0 0 I



 V, then P0= V−1





0 0 0

0 0 0



 V.

We may choose

W0= U





0 0 0

0 I 0

0 0 I



 U−1.

An elementary computation yields

G1= G0+ B0Q0= U





0 0 B33



 V, B1= B0P0= U





0 0 0



 V,

N1= ker G1=





z=





z1

z2

z3



 ∈ Cm , ˆz1+ B12ˆz2= ˆz3= 0, where ˆz = V z





.

S1=





z=





z1

z2

z3



 ∈ Cm , B21ˆz1= 0, where ˆz = V z





.

By definition in Section 2, if the index exists and it equals 2 then N1∩ S1= {0} One can verify directly

that this is the case if and only if det B21B12= 0 Let this assumption hold For brevity, let us introduce

the notations

If we choose

Q1= V−1





−M 0 0



 V,

then Q1is the projector onto N1along S1 We compute

P1= V−1





I − C 0 0



 V, G2= G1+ B1Q1= U







 V.

Elementary calculations give us

G−12 B = V−1





(I − C)B11(I − C) + C 0 0

M [B11(I − C) − I] I 0



 V.

Let us denote the expression DP1G−12 BD−by H Since D and R were not decomposed, we leave D and

D− in the formula for H However, the structure of H still becomes transparent if we take into account that now P1= P1and, therefore, DP1= DP0P1= DP0P1P0= DP1P0, i.e.,

H = DP1P0G−12 BD−= DV−1





(I − C)B11(I − C) 0 0



 V D−.

Now, let us take the adjoint to H If we make use of the definitions A∗A∗−= D−∗D∗= R∗, D∗R∗=

D∗, R∗A∗= A∗and the decomposition of D∗A∗= (AD)∗, we get

H∗= D−∗V∗





(I − C∗)B∗

11(I − C∗ 0 0



 V−∗D∗A∗A∗−

= D−∗V∗





(I − C∗)B∗

11(I − C∗ 0 0



Let H∗= A∗P∗1G−1

∗2B∗A∗− Similarly to (30), we obtain

H∗= A∗U−∗





(I − C∗)B∗

11(I − C∗ 0 0



 U∗A∗−

= D−∗D∗A∗U−∗





(I − C∗)B∗

11(I − C∗ 0 0



It remains to recall that H and H∗ were shown to be independent of the choice of P0and P∗0[4] and

H = H DP1D−and H∗= H∗A∗P∗1A∗− By this the proof is completed