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

Applied Numerical Mathematics 53 (2005) 131–148 www.elsevier.com/locate/apnum Adjoint pairs of differential-algebraic equations and Hamiltonian systems ✩ Katalin Balla a,∗ , Vu Hoang Linh b a Computer and Automation Research Institute, Hungarian Academy of Sciences, H-1518 Budapest P.O Box 63, Hungary b Faculty 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 = and their adjoints −D ∗ (A∗ x) +B ∗ x = 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 characterize 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 Introduction Recently, differential-algebraic equations (DAEs) of the form A(Dx) + Bx = f, ✩ 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 (1) 132 K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 where A, D, B : C([a, b], L(Cm )), f : C([a, b], Cm ), were introduced in [4] and their analysis was launched A motivation to considering this class of implicit equations was that under very mild conditions the formal adjoint equation −D ∗ A∗ x∗ + B ∗ x∗ = g, (2) g : C([a, b], Cm ), shares the basic properties with (1) In the paper, we intend to study two kinds of ordinary 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 homogeneous, 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 iD ∗ (Dx) + Bx = 0, with B = B ∗ (3) 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 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 is addressed to INHODEs Section deals with the construction of the bases and pairs of bases in subspaces connected with (1) and (2) Section derives the EUODEs and the pairs of EUODEs Section 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 we point out the practical importance of associated ODEs 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 on I := [a, b], first we assume that the leading coefficients A and D are “well-matched”, i.e., they fulfill as follows Condition C1: For each t ∈ I, ker A(t) ⊕ im D(t) = Cm , (4) and there exist functions a1 , , am−r , d1 , , dr ∈ C (I, Cm ) such that for all t ∈ [a, b], ker A(t) = span a1 (t), , am−r (t) , im D(t) = span d1 (t), , dr (t) (5) K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 133 We define the projector function R ∈ C (I, L(Cm )) by R = 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 ∗ ∈ C (I, L(Cm )) With Eq (1), the following chain is associated: G0 := AD, For i = 0, 1, B0 := B; Qi , Pi , Wi ∈ C(I, L(Cm )): Wi2 = Wi , Q2i = Qi , Pi = I − Qi , Ni := ker Gi = im Qi , ker Wi = im Gi , Bi+1 = Bi Pi , Gi+1 := Gi + Bi Qi , m Si := {z : I → C , Bi z ∈ im Gi } = ker Wi Bi (6) Similarly, Eq (2) gives rise to the chain B∗0 := B ∗ ; G∗0 := −D ∗ A∗ , For i = 0, 1, Q∗i , P∗i , W∗i ∈ C(I, L(Cm )): W∗i2 = W∗i , Q2∗i = Q∗i , P∗i = I − Q∗i , N∗i := ker G∗i = im Q∗i , ker W∗i = im G∗i , B∗i+1 = B∗i P∗i , G∗i+1 := G∗i + B∗i Q∗i , m S∗i := {z : I → C , B∗i z ∈ im G∗i } = ker W∗i B∗i (7) 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) D m and there exist functions s1D , , sρD , nD , , nν ∈ C (I, C ) such that for all t ∈ I, D(t)S1 (t) = span s1D (t), , sρD (t) , D D(t)N1 (t) = span nD (t), , nν (t) Let conditions C1 and C2 be valid Eq (1) is said to be an index-0 tractable DAE if N0 (t) = {0}, t ∈ I, (9) an index-1 tractable DAE if N0 (t) = {0}, N0 (t) ∩ S0 (t) = {0}, t ∈ I, (10) (11) an index-2 tractable DAE if dim N0 (t) ∩ S0 (t) = const > 0, t ∈ I N1 (t) ∩ S1 (t) = {0}, (12) (13) 134 K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 Denote the reflexive generalized inverses (RGIs) of D and A∗ by D − and A∗− if defined by D − D = P0 , DD − = R, A∗− A∗ = P∗0 , 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 Q0c (P0c ), if ker Q0 = S0 (im P0c = S0 ) In index-2 case, the subspace N1 depends on the special choice of Q0 (P0 ) while S1 does not Projector Q1 is marked by if ker Q1 = S1 holds All terms in chain (6) derived by the use of Q1 are marked by , too In index-2 case, canonical projector function Q0c onto N0 is defined as − − D, Q0c = Q0 P1 G−1 B + Q0 Q D D Q D where Q0 is an arbitrary projector function onto N0 ; Q0c does not depend on the special choice of Q0 All terms in chain (6) derived by the use of Q0c are marked with subscript c , too Q1c = Q1 holds 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 canonical projectors Q0c In index-2 case, decomposition D(t)S1 (t) ⊕ D(t)N1 (t) ⊕ ker A(t) = Cm induces projector functions D P1 D − , D Q1 D − , I − R ∈ C (I, L(Cm )) onto the subspaces in the decomposition along the other couple Similarly, projectors A∗ P∗1 A∗− , A∗ Q∗1 A∗− and I − R ∗ correspond to decomposition A∗ (t)S∗1 (t) ⊕ A∗ (t)N∗1 (t) ⊕ ker D ∗ (t) = Cm and identity (D P1 D − )∗ = A∗ P∗1 A∗− holds For an arbitrary V ∈ C(I, L(Cm )), CV1 denotes function space {v ∈ C(I, Cm ): V v ∈ C (I, Cm )} With Eq (1) we associate operator L : CD1 → CD1 Q −1 G2 Lx := A(Dx) + Bx, , x ∈ CD1 , and similarly, Eq (2) is related to operator L∗ : CA1 ∗ → CA1 ∗ Q −1 ∗1 G∗2 , L∗ x∗ := −D ∗ (A∗ x∗ ) + B ∗ x∗ , x∗ ∈ CA1 ∗ A function x ∈ CD1 is called the solution if it satisfies (1) pointwise For each t ∈ I, solutions to (1) form a linear subspace Sµ (t) ⊆ Cm , Sµ (t) := im Πcan µ (t), Πcan = I, Πcan = P0c , Πcan = P0c P1c = P0c P1 = U P0 P1 , here − − D, U = I − Q0 P1 G−1 B − Q0 P1 D D Q1 D U is invertible, Q0 is an arbitrary projector function onto N0 Solutions x∗ ∈ CA1 ∗ 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(Cn ) 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(Cn ), then H − (t) := [H (t)]− , H ∗ (t) := [H (t)]∗ Inherent ODEs for adjoint pairs We remind that the equations under consideration are homogeneous K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 135 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 + A−1 BD −1 u = 0, −u∗ + D ∗−1 B ∗ A∗−1 u∗ = 0, (14) (15) 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 ∈ CD1 (x∗ ∈ CA1 ∗ ) is a solution of (1) ((2)) if and only if Dx (A∗ x∗ ) is a solution of (14) ((15)) Practically, we 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 connected by the following proposition Lemma Let (1) (and/or (2)) be an index-1 DAE and let Q0 and Q∗0 be arbitrary projector functions −1 ∗ ∗ onto N0 and N∗0 , respectively Then, P0c = −(D ∗ A∗ G−1 ∗1 ) and P∗0c = (ADG1 ) ∗ ∗ Proof The representation Q0c = G∗−1 ∗1 Q∗0 G∗1 is a simple consequence of the trivial identities ∗−1 ∗ ∗−1 ∗ −1 ∗ ∗ ∗ ker Q0 G−1 B = ker W0 B and Q∗0 G1 = G∗1 Q0 Indeed, G∗1 Q∗0 G∗1 = G∗1 Q∗0 B = Q0 G1 B We im∗−1 ∗ ∗−1 ∗ ∗−1 ∗ ∗ mediately get P0c = I − G∗1 Q∗0 G∗1 = G∗1 P∗0 G∗1 = −G∗1 AD The second representation can be checked in a similar way ✷ The lemma results in identities Q∗∗0c BP0c = and Q∗0c B ∗ P∗0c = 0, which, in turn, yield BP0c = ∗ ∗ ∗ = (P0c B P∗0c )∗ = (B ∗ P∗0c )∗ = P∗0c B Therefore, (1) can be rewritten as ∗ BP0c P∗0c ∗ BP0c x + Q∗∗0c BQ0c x = A(Dx) + BP0c x + BQ0c x = A(Dx) + P∗0c (16) − ∗− ∗ − Take the projections onto im Q∗∗0c and ker Q∗∗0c and use RGIs A∗− c , Dc : P∗0c = Ac A , P0c = Dc D and ∗−∗ − m Ac A = R = DDc , R ∈ C (I, L(C )): Q∗∗0c BQ0c x = 0, − (Dx) − R Dx + A∗−∗ c BDc Dx = (17) (18) ∗ ∗ ∗ A particular case of the above lemma is representation Q∗0c = G−∗ 1c Q0c G1c Thus, Q∗0c (BQ0c ) = −∗ ∗ (G1c Q0c G∗1c )∗ G1c Q0c = G1c Q0c , and therefore, Q0c x = 0, is an equivalent to (17) Set u = DP0c x = Dx and get the INHODE for (1) from (18): − u − R u + A∗−∗ c BDc u = The next “invariance” gives importance to the INHODE: (19) 136 K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 (I1) If u(t0 ) ∈ im D(t0 ) holds for an arbitrary t0 , then u ∈ im D ∗ Indeed, A∗− c (I − R ) = and (19) involve that function w := (I − R)u satisfies the ODE w = (I − R) w, therefore, w(t0 ) = for an arbitrary t0 yields w = The similar procedure applied to (2) with the same RGIs yields the decomposition Q∗0c x∗ = 0, ∗ − A∗ x∗ + R ∗ A∗ x∗ + Dc−∗ B ∗ A∗− c A x∗ = 0, ∗ (20) ∗ while setting u∗ = A P∗0c x∗ = A x∗ results in the inherent ODE for (2) −u∗ + R ∗ u∗ + Dc−∗ B ∗ A∗− c u∗ = (21) Now, the invariance of the inherent ODE reads as (I1∗ ) If u∗ (t0 ) ∈ im A∗ (t0 ) holds for an arbitrary t0 , then u∗ ∈ im A∗ Let us return to arbitrary projector functions Q0 and Q∗0 Due to the properties of RGIs, the identities ∗− ∗ ∗− ∗ ∗ ∗− A∗ A∗− DD − A∗− c = P∗0c Ac R = P∗0c Ac R R = P∗0c Ac ∗ ∗− = P∗0c A∗− DD − c A A ∗ = P∗0c P∗0c A∗− DD − ∗ ∗ ∗ = P∗0c A∗− DD − , Dc− = P0c Dc− R = P0c Dc− DD − = P0c D − = P0c D − , are valid Therefore, − − ∗−∗ ∗ ∗ − P∗0c BP0c D − = DD − A∗−∗ P∗0c BD − = DD − A∗−∗ G1 P0 G−1 A∗−∗ c BDc = DD A BD −1 −1 − − − − = DD − A∗−∗ ADG−1 BD = DD RDG1 BD = DG1 BD , and the inherent equation turns to be identical to that in [4] Similarly, ∗ −1 ∗ ∗− Dc−∗ B ∗ A∗− c = A G∗1 B 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, P0c , P∗0c are defined uniquely by them and RGIs are induced uniquely by the latter Furthermore, the following statement becomes transparent Theorem 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 = and vice versa ✷ 3.3 Index-2 equations In index-2 case, we use again canonical projectors Q0c , Q1 , Q∗0c , Q∗1 , Πcan , Π∗can and the decompositions relying upon them In the analysis we also utilize the identity Q1c = Q1 , i.e., Q1 = Q1 P0c K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 137 and its counterpart marked by ∗ We take the projections of the equations onto im G2c Πcan G−1 2c = −1 −1 im AD P1 G−1 and im G (I − Π ) G = im G (Q + P ) G In the solution x part Π Q 2c can 2c 0c 0c can x = 2c 2c 2c − − P0c P1 x is segregated Terms D P1 x = DP0c P1 x = D P1 Dc Dx and D Q1 x = D Q1 Dc Dx are differentiable for x ∈ CD1 and DQ0c P1 x = 0, so we may split their sum Dx Note that due to condition C1, − ker AD = ker D holds Relation G−1 2c A = P1 Dc also will be useful After partial differentiation, the first projection simplifies to − − D P1 x − D P1 Dc− D P1 x − D P1 Dc− D Q1 x + D P1 G−1 2c BDc D P1 Dc D P1 x = (22) We undertake the second projection to scaling by G−1 2c that yields = Q0c P1 Dc− (Dx) + Q0c + P0c Q1 G−1 2c Bx = −Q0c Q1 Dc− (Dx) + Q0c P1 + Q1 G−1 2c Bx + P0c Q1 x = −Q0c Q1 Dc− (Dx) − Q0c Q1 Dc− D Q1 Dc− Dx + Q0c Q1 Dc− D Q1 Dc− D + Q0c P1 G−1 2c B x + Q1 x = −Q0c Q1 Dc− D Q1 x + Q0c x + Q1 x (23) 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: P0c Q1 x = 0, Q0c x + Q1 x = (24) (25) 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 P1 x − − u − D P1 Dc− u + D P1 G−1 2c BDc D P1 Dc u = (26) This is the same equation we got in [4] Indeed, the multiplier D P1 Dc− of u in the last term could be inserted in [4] Subscript c was not present there However, it was shown that the equation is independent of the choice of Q0 − − − ∗−∗ − − Now, we check that D P1 G−1 2c BDc D P1 Dc = D P1 Dc Ac BDc D P1 Dc − − − ∗−∗ − − D P1 G−1 2c BDc D P1 Dc − D P1 Dc Ac BDc D P1 Dc −1 − − = D P1 Dc− D − A∗−∗ c G2c G2c BDc D P1 Dc −1 − − − − = −D P1 Dc− A∗−∗ BQ0c P1 + Q1 G−1 c 2c BDc D P1 Dc + BP0c Q1 G2c BDc D P1 Dc − − − − BQ0c P1 G−1 = −D P1 Dc− A∗−∗ c 2c BDc D P1 Dc + BQ0c Q1 P0c P1 Dc + BP0c Q1 P0c P1 Dc − − − = −D P1 Dc− A∗−∗ c B Q0c − Q0c Q1 Dc D Q1 Dc D P0c P1 Dc − − − = −D P1 Dc− A∗−∗ c BQ0c Q1 Dc D Q1 Dc DP0c P1 Dc − − − = +D P1 Dc− A∗−∗ c AD Q1 Dc D Q1 Dc DP0c P1 Dc = +D P1 Q1 Dc− D Q1 Dc− DP0c P1 Dc− = Thus, the transparent form of (26) is − − u − D P1 Dc− u + D P1 Dc− A∗−∗ c BDc D P1 Dc u = (27) 138 K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 We can treat the adjoint equation (2) in a quite similar way The final result for u∗ = A∗ P∗1 x∗ is Dc−∗ B ∗ A∗− A∗ P∗1 A∗− u∗ = 0, −u∗ + A∗ P∗1 A∗− u∗ + A∗ P∗1 A∗− c c c c (28) while the constrains are P∗0c Q∗1 x∗ = 0, Recall that D P1 Dc− Q∗0c x∗ + Q∗1 x∗ = − ∗ = D P1 D and A P∗1 A∗− c (29) ∗ ∗− = A P∗1 A 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∗ (t0 )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 P1 D − − − ∗ ∗ ∗− −∗ ∗ ∗− ∗ ∗− connecting the last terms in Relation (D P1 G−1 2c BDc D P1 Dc ) = A P∗1 Ac Dc B Ac A P∗1 Ac (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 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 P1 D − is constant Proof We again recall that (A∗ P∗1 A∗− )∗ = D P1 D − ✷ We formulate this result in terms of the basic subspaces Theorem 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 image 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 P1 D − and together with it, projector function A∗ P∗1 A∗− be constant Then, their images, DS1 and A∗ S∗1 are constant ✷ Remark In the proof we referred to identities A∗ S∗1 = (DN1 ⊕ ker A)⊥ and DS1 = (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 There is another way to show the connection between the last terms in INHODEs of the adjoint pairs of DAEs We provide it since the method used here is applicable in later material, as well The K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 139 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 = t0 and omit the argument We begin with a decomposition resulting block matrices of special form     I 0 B11 B12 G0 = AD = U  0  V , B0 = B = U  B21 0 V, 0 B33 0 where U and V are nonsingular The blocksizes in the decompositions of G0 and B0 correspond each to other Let the diagonal blocks be of dimension n1 , n2 , n3 , respectively One always may achieve either n3 = (no third block-row and no block-column appear) or B33 be nonsingular The special case n2 = indicates that the matrix pair has index (The decomposition trivially exists: first one should transform G0 to 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 = (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     0 I 0 Q0 = V −1  I  V , then P0 = V −1  0  V 0 I 0 We may choose   0 W0 = U  I  U −1 0 I An elementary computation yields     I B12 B11 0 G1 = G0 + B0 Q0 = U  0 B1 = B0 P0 = U  B21 0  V , V, 0 0 B33     z   N1 = ker G1 = z =  z2  ∈ Cm , zˆ + B12 zˆ = zˆ = 0, where zˆ = V z   z3     z1   S1 = z =  z2  ∈ Cm , B21 zˆ = 0, where zˆ = V z   z3 By definition in Section 2, if the index exists and it equals then N1 ∩ S1 = {0} One can verify directly that this is the case if and only if det B21 B12 = Let this assumption hold For brevity, let us introduce the notations C = B12 (B21 B12 )−1 B21 , M = (B21 B12 )−1 B21 140 K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 If we choose  C −1  Q1 = V −M 0 0  0V, then Q1 is the projector onto N1 along S1 We compute    I + B11 C I −C 0 −1  P1 = V G2 = G1 + B1 Q1 = U  B21 M I 0V, 0 I Elementary calculations give us  (I − C)B11 (I − C) + C −1 −1  G2 B = V M[B11 (I − C) − I ] 0 I B12 0  0 V B33  0V I − Let us denote the expression DP1 G−1 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 = P1 and, therefore, DP1 = DP0 P1 = DP0 P1 P0 = DP1 P0 , i.e.,   (I − C)B11 (I − C) 0 − −1  H = DP1 P0 G−1 0  V D− BD = DV 0 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   ∗ (I − C ∗ )B11 (I − C ∗ ) 0 H ∗ = D −∗ V ∗  0  V −∗ D ∗ A∗ A∗− 0   ∗ (I − C ∗ ) 0 (I − C ∗ )B11 = D −∗ V ∗  (30) 0  U ∗ A∗− 0 ∗ ∗− Let H∗ = A∗ P∗1 G−1 ∗2 B A Similarly to (30), we obtain   ∗ (I − C ∗ ) 0 (I − C ∗ )B11 H∗ = A∗ U −∗  0  U ∗ A∗− 0   ∗ (I − C ∗ ) 0 (I − C ∗ )B11 = D −∗ D ∗ A∗ U −∗  0  U ∗ A∗− = H ∗ (31) 0 It remains to recall that H and H∗ were shown to be independent of the choice of P0 and P∗0 [4] and H = H DP1 D − and H∗ = H∗ A∗ P∗1 A∗− By this the proof is completed K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 141 Invariant subspaces and their bases This section deals with the existence and the construction of smooth bases in certain subspaces Suppose that for each t, E(t) ⊆ Cn is a subspace of the same dimension and assume that it is defined by the values of a continuously differentiable projector function P, that is, E(t) = im P(t), t ∈ I The existence of a continuously differentiable basis function for E was proven in [10] Based on the idea proposed in [1], we describe a construction that results in a basis function T satisfying a prescribed normalization condition First, let k denote dim E, k n and R an arbitrary continuously differentiable projector function onto E(t), i.e., im R(t) = im P(t) = E(t), t ∈ I Dissimilarly to [1], here R is not necessarily an orthoprojector Let Q ∈ L(Cn ), S ∈ L(Ck ) and rank Q k = rank S Assume that either both Q and S are Hermitian or both of them are skew-Hermitian Additionally, we assume that v ∈ E(t) implies Q∗ v ∈ E∗ (t) := im R∗ (t) Obviously, if R is an orthoprojector, i.e., R = R∗ , the simplest matrix pair that satisfies the assumptions is Q = In , S = Ik , where I , ∈ N, is the identity matrix of dimension Let T ∈ C (I, L(Ck , Cn )) be the solution to the initial value problem T =RT, T (t0 ) = T0 , (32) where the columns of T0 form a basis in E(t0 ) = im R(t0 ) = im P(t0 ) for some fixed t0 ∈ I Lemma If T ∈ C (I, L(Ck , Cn )) is the solution to the initial value problem (32), then, for each t im T (t) = E(t) Moreover, if T0∗ QT0 = S, (33) holds, then T (t)∗ QT (t) = S, (34) is valid for all t ∈ I Proof For the first part of the Lemma define V = RT − T Then, V = R T + RT − T = RR T Since R2 = R, we have RR = R − R R Hence, V = R − R R T = R T − R (V + T ) = −R V Since V(t0 ) = 0, we have V(t) = at any t, i.e., the columns of T belong to E Since the columns of T (t0 ) are linearly independent, the linear independence of the columns of T (t) for each t follows from the elementary ODE theory This completes the proof of the first part For the second part, assume that (33) holds with some appropriate pair (Q, S) and for each t, v ∈ E(t) implies Q∗ v ∈ E∗ (t) = im R∗ We derived RT = T , it involves RT = Then, T ∗ QT = Q∗ T ∗ T = R∗ Q∗ T ∗ T = T ∗ QRT = Finally, when Q is Hermitian, we have T ∗ QT = T ∗ QT + T ∗ QT ∗ = (35) 142 K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 When Q is skew-Hermitian, T ∗ QT ∗ = T ∗ QT − T ∗ QT = Therefore, in both cases T (t)∗ QT (t) ≡ T (0)∗ QT (0) = S ✷ In the next section, we may apply Lemma in its simplest form by setting R = R in the index-1 case and R = D P1 D − in the index-2 case Or, in the index-1 case we may set R = RR + and (Q, S) = (Im , Ir ), while in the index-2 case R = D P1 D − D P1 D − + (Q, S) = (Im , Iρ ) and + Here, denotes the Moore–Penrose generalized inverse Clearly, R defined as above is an orthoprojector Proceeding from an arbitrary basis in im D(t0 ) or im D(t0 )P1 (t0 ), respectively, the Gram–Schmidt orthonormalization procedure provides us the required orthonormal initial value, i.e., one that satisfies (33) Consequently, the solution of the initial value problem (32) yields a continuously differentiable function the values of which form an orthonormal basis in im R(t) = im D(t) or D(t)S1 (t) = im D(t)P1 (t)D − (t), respectively for each t ∈ I Essentially underlying ODEs for adjoint pairs In this section, we work with the adjoint pair of DAE-s of index-2, only The result for the index-1 case follows directly, by setting P1 = P∗1 = Im We recall the INHODEs (27) and (28) for the original equation (1) and its adjoint (2) We also refer to the properties (I2) and (I2∗ ) of the “invariant subspaces” im D P1 D − and im A∗ P∗1 A∗− and recall that dim im D P1 D − = dim im A∗ P∗1 A∗− = ρ Let u ∈ im D P1 D − be a solution of (27) and T ∈ C (I, L(Cρ , Cm )) an arbitrary basis function in im D P1 D − Then there exists v : I → Cρ such that u = T v Moreover, v ∈ C (I, Cρ ), since v = Iρ v = T ∗ T −1 T ∗T v = T ∗T −1 T∗ Tv = T ∗T −1 T ∗ u, and both terms in the latter product are differentiable Partial differentiation and (27) yields v = T ∗T −1 T ∗ u + T ∗T −1 = T ∗T −1 T ∗ T v + T ∗T T∗ u −1 T∗ − − D P1 D − − D P1 D − A∗−∗ T v c BDc D P1 D It is worth noting that (T ∗ T )−1 T ∗ = T + Since D P1 D − T = T , we may simplify to get − v + T + D P1 D − T + A∗−∗ c BDc T v = 0, (36) Eq (36) is called the essentially underlying ODE (EUODE) for Eq (1) It is a lower dimensional ODE than (27) is However, unlike (1), it is not uniquely defined since the basis function is not defined uniquely, either In a similar way, we could derive the EUODE for (2): Set u∗ = T∗ v∗ where u∗ ∈ im A∗ P∗1 A∗− is the solution of (28) and T∗ is a smooth basis function in im A∗ P∗1 A∗− The result is −v∗ + T∗+ A∗ P∗1 A∗− −T∗ + Dc−∗ B ∗ A∗− c T∗ v∗ = (37) K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 143 Corollary (of Theorems and 3) Let D P1 D − be constant Then, there exist constant basis functions T and T∗ in DS1 and A∗ S∗1 , respectively, and with them, the EUODE pair takes the form − v + T + D P1 D − A∗−∗ c BDc T v = 0, −v∗ + T∗+ ∗ ∗− A P∗1 A Dc−∗ B ∗ A∗− c T∗ v∗ (38) = (39) Now our aim is to find appropriate basis pairs allowing us to see the connection between the flows in general case That is we look for specific pairs of EUODEs for pair (1), (2) Lemma Let T ∈ C (I, L(Cρ , Cm )) be an arbitrary function such that im T (t) = im D(t)P1 (t)D − (t), t ∈ I, and E ∈ L(Cρ ) be invertible There exists a unique function T∗ ∈ C (I, L(Cρ , Cm )) such that im T∗ (t) = im A∗ (t)P∗1 (t)A∗− (t), t ∈ I, and T∗∗ (t)T (t) ≡ E, t ∈ I Remark 10 The existence of a function T ∈ C (I, L(Cρ , Cm )) with the required property is ensured by Lemma However, the construction of Lemma does not exhaust the set of all basis functions applicable in Lemma Proof Take an arbitrary smooth matrix function S such that the columns of S(t) constitute a basis of ker D(t)P1 (t)D − (t) Since D P1 D − is a projector function, we have im D(t)P1 (t)D − (t) ⊕ ker D(t)P1 (t)D − (t) = Cm , t ∈ I Hence, the columns of T (t) and S(t) form a basis of C Therefore, there exists a unique solution T∗ to the system of equations T∗∗ T = E, (40) T∗∗ S = m This solution is obviously smooth The second equation means that the columns of T∗ (t) belong to ker D(t)P1 (t)D − (t) ⊥ = im D(t)P1 (t)D − (t) ∗ = im A∗ (t)P∗1 (t)A∗− (t) On the other hand, since E is nonsingular, the first equation implies that rank T∗ (t) = rank T (t) = dim im D(t)P1 (t)D − (t) = dim im A∗ (t)P∗1 (t)A∗− (t) = ρ Consequently, the columns of T∗ (t) form a basis of im A∗ (t)P∗1 (t)A∗− (t) The solution T∗ of (40) depends purely on T (and E) Indeed, if S = S, and im S = ker D(t)P1 (t)D − (t), then S = Sη, η(t) ∈ L(Cρ ), η(t) is nonsingular The corresponding system is equivalent to (40) ✷ Obviously, a lemma similar to Lemma is valid if T and T∗ and the relevant subspaces change their role 144 K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 Now, let us apply Lemma and for the given T find the associated T∗ such that T∗∗ (t)T (t) ≡ T ∗ (t)T∗ (t) ≡ Iρ (41) hold We show that with this choice of T∗ , Eqs (36) and (37) are adjoint each to other First, we check that T∗ = A∗ P∗1 A∗− T (T ∗ T )−1 Indeed, let V := A∗ P∗1 A∗− T (T ∗ T )−1 Obviously, V = T∗ η with some η Further, η = T ∗ T∗ η = T ∗ V = T ∗ A∗ P∗1 A∗− T T ∗ T ∗ = D P1 D − T T T ∗ T −1 = T ∗T T ∗T −1 −1 = Iρ Analogously, one can verify that T = D P1 D − T∗ (T∗∗ T∗ )−1 holds Thus, the pair (36) and (37) may be rewritten as follows − v + T∗∗ T v + T∗∗ A∗−∗ c BDc T v = 0, (42) −v∗ − T T∗ v∗ + T (43) ∗ ∗ (T∗∗ T Dc−∗ B ∗ A∗− T∗ v∗ = c ∗ ∗ ) = −T T∗ since T ∗ T∗ = Iρ In order to formulate the result, it remains to Finally, note that recall that we could proceed from a fixed T∗ and use its counterpart T uniquely defined by (41) Theorem 11 Let (T , T∗ ) be a continuously differentiable basis pair satisfying (41) Then, the adjoint of the EUODE (36) of the DAE (1) coincides with the EUODE (37) of the adjoint DAE (2) Example 12 Consider the adjoint pair of index-2 DAE-s of upper Hessenberg form x1 + B11 x1 + B12 x2 = 0, (44) B21 x1 = 0, and ∗ ∗ −x∗1 + B11 x∗1 + B21 x∗2 = 0, (45) ∗ B12 x∗1 = Here, one may set I B11 B12 A=D= , B= B21 0 Matrix chains for (44) and its adjoint (45) can be constructed similarly to those in Remark (This is the case with n3 = 0.) The index-2 property of (44)–(45) is equivalent to the condition det B21 B12 = We obtain the INHODE for (44) (I − C) (I − C)B11 u − u+ u = 0, (46) 0 0 where C = B12 (B21 B12 )−1 B21 and I −C , D P1 D − = 0 u= (I − C)x1 Choose an arbitrary smooth basis T = T0 in im D P1 , T ∈ im(I − C), and the corresponding T∗ = in im A∗ P∗1 , T∗ ∈ im(I − C ∗ ) The EUODE of (44) is of the form v + T∗∗ T + T∗∗ B11 T v = 0, T∗ (47) K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 145 where v is defined by (I − C)x1 = T v Similarly, we have the EUODE of (45) ∗ T∗ v∗ = 0, −v∗ + −T ∗ T∗ + T ∗ B11 (48) ∗ where v∗ is defined by (I − C )x∗1 = T∗ v∗ Eqs (47) and (48) illustrate the claim in Theorem 11 Furthermore, one can observe that im(I − C) = ker B21 , ∗ im(I − C ) = ∗ ker B12 , ker(I − C) = im B12 , ∗ ker(I − C ∗ ) = im B21 That is, the EUODE pairs in [8] are particular cases of (47) and (48) Here, we not require the basis T to be orthonormalized Moreover, the smoothness condition of the coefficients is relaxed somewhat: functions B12 and B21 are not necessarily continuously differentiable, but function C, only Self-adjoint DAEs and self-adjoint BVPs In this section, we consider the special DAE D T J (Dx) + Bx = 0, (49) with real coefficients D, J, B, the superscript T denotes transposition Here J is a constant skewsymmetric matrix, the leading pair (D T J, D) is well matched, B is symmetric If not confusing, the argument t is omitted Without loss of generality, we may assume that either J = J2n , m = 2n or J = diag(J2n , 0m−2n ), n, 2n < m, where −In J2n = , In and 0m−2n is a zero matrix of dimension m − 2n In the second case, the condition C1 gives D= D1 D2 , D1 (t) ∈ L R2n , D2 (t) ∈ L Rm−2n , R2n Indeed, let J be the given arbitrary skew-symmetric constant matrix and the pair (D T J , D) be wellmatched It follows from [12] that there exists a decomposition J = U T J U with nonsingular constant U Then, dim ker J = m − 2n and for D := U D, ker D T J ⊇ ker J , that is, im D ∩ ker J = {0} holds, i.e., in, the second case D(t) has the last m − 2n rows filled with zeros, only and dim im D 2n With the notation of the previous sections, here A = D T J Since AT = (D T J )T = −J D, we immediately have CD1 = CA1 T and for each x ∈ CD1 , D T J (Dx) + Bx and −D T ((D T J )T x) + B T x coincide We show now, that im D P1 is of even dimension, i.e., ρ = im D P1 = 2l, with some l n Note that one may choose projectors Q0 and Q∗0 to be equal in chains (6) and (7) Then, P∗1 = P1 This leads to AT P∗1 AT− = −J D P1 D − J (50) Let T be an arbitrary smooth basis function T ∈ C (I, L(Rρ , Rm )) in DS1 Due to Lemma 9, there exists a unique basis function T∗ ∈ C (I, L(Rρ , Rm )) in AT S∗1 such that T∗T (t)T (t) = Iρ , while (50) 146 K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 gives im T∗ ⊆ im J T Since dim im T∗ = dim im T , there exists an invertible function ωˆ such that T∗ = J T ω ˆ Thus, ωˆ T T T J T T = Iρ , or, equivalently, T T J T = ω, with a differentiable, skew-symmetric and invertible ω If the rank of a skew-symmetric real matrix is odd, then it is singular [12] Thus rank ω(t) is even, ρ = 2l with some l n It follows from [12] and the eigenstructure of J , that there exists a nonsingular U0 ∈ L(R2l ) such that T T (0)J T (0) = U0T J2l U0 Then, Tˇ (t) = T (t)U0−1 is also a differentiable basis in DS1 , with Tˇ T (0)J Tˇ (0) = J2l Let T be a smooth basis in DS1 constructed by the help of the IVP (32) with R = D P1 D − and setting Q = J, S = J2l in (33) Clearly, the assumptions of Lemma are fulfilled Hence, we have T T J T = (see the proof of Lemma 7) and T T (t)J T (t) ≡ const = T T (0)J T (0) = J2l With a basis T indicated above, we derive the EUODE (36) for (49) In order to have the result in = a transparent form, we shall obtain the same equation in a different form Recall again that AT−T c −J Dc−T We could represent v as v = Iρ v = (T T J T )−1 (T T J T )v = J2l−1 T T J u Therefore, J2l v = T T J u = T T J u + T T J u = T T J T v + T T J D P1 D − T + D P1 D − J Dc−T BDc− D P1 D − T v = T TJ T T + T T J D P1 D − T − T T J D P1 D − T + T T J D P1 D − J Dc−T BDc− D P1 D − T v − (51) Here T J T = Due to (D P1 D )T = T , the second term also vanishes Since im J T ⊆ A S∗1 , we have T T T J D P1 D − T = − AT P1 AT− J T T T T = (−J T )T T = T T J T = In order to simplify the last term, additionally we take into account that im T = DS1 ⊆ im D, thus the last m − 2n rows in T (t) vanish Therefore J T = −Im T = −T T T J D P1 D − J Dc−T BDc− D P1 D − T v = − AT P1 AT− J T T J Dc−T BDc− T v = T T J Dc−T BDc− T v = −T T Dc−T BDc− T v The final result is J2l v + F v = 0, F = T T Dc−T BDc− T = F T (52) This equation is self-adjoint Now, we look for boundary conditions for (49) that yield a self-adjoint BVP for (52) Theorem 13 Let Ka , Kb ∈ L(Rm ) be given The boundary condition KaT D(a)x(a) + KbT D(b)x(b) = 0, (53) for Eq (1) induces a self-adjoint boundary condition for (52) if and only if the conditions KaT J Ka = KbT J Kb , im Ka ⊂ J D(a)S1 (a), im Kb ⊂ J D(b)S1 (b), T T rank Ka , Kb = dim DS1 , hold (54) (55) (56) K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 147 Proof Let Ka , Kb satisfy the assumptions It means that im Ka ⊂ J D(a)S1 (a) = im J T (a) = im J T T (a)J2l Therefore, there exists M ∈ L(R2l ) such that Ka = J T T (a)J2l M, (57) where T is the basis functions as above Similarly, there exists N ∈ L(R2l ) such that Kb = J T T (b)J2l N (58) If x is a solution of (1), then Dx = T v where v satisfies (52) Condition (53) turns into M T v(a) + N T v(b) = 0, (59) while the relation (54) takes the form M T J2l M = N T J2l N (60) The condition (56) means that rank(KaT Ka +KbT Kb ) = 2l In terms of M, N we obtain that, i.e., (M T , N T ) is of full rank 2l We recall [11, Theorem 3.2, Chapter 11], that the boundary condition (59) yields a selfadjoint boundary value problem for (52) if and only if (60) is valid and (M T , N T ) is of full rank In the opposite direction: If (52) and (60) form a self-adjoint BVP, then one can define Ka , Kb by (57) and (58) They satisfy the assumptions ✷ Numerical consequences The decoupling of the linear DAE into the INHODE and the algebraic constraints in Section shows that the dynamic properties of DAE are defined by the INHODE Any type of conditions, initial values or boundary conditions may be imposed only on the components involved into the INHODE The conditions should allow to exist a solution within the invariant subspace of the INHODE Theoretically, a more transparent picture of the freedom in imposing conditions arises if one turns to the description of the dynamics in minimal coordinates that is to an EUODE Practically, however, neither the reduction to an EUODE nor that to the INHODE is desirable Nevertheless, the stability properties of the DAE are inherited from the INHODE [13,14], or, respectively, from an EUODE [8,9] In [13,14], numerical integration of DAEs (IVPs) with well matched leading terms is considered A particular case is the linear equation (1) One of the questions concerned there is whether the decoupling of the DAE and the discretization (by a stiffly accurate Runge–Kutta method or a BDF method) commute In other words, the question is whether the numerical method properly reproduces the problem dynamics It is stated [13, Proposition 3] that in the index-1 case property im D = const is sufficient for that Therefore, DAEs of index-1 with this property are called numerically qualified In the index-2 case, a sufficient condition is DS1 = const and DN1 = const [14, Lemma 7] In [5], two-point boundary value problems for index-1 and index-2 DAEs (1) are reduced to initial value problems for DAE (2) Clearly, an index-1 DAE (2) is numerically qualified if im A∗ = const If compared with Theorem 2, we conclude that provided the INHODE of the index-1 DAE (1) coincides with the adjoint to the INHODE of the DAE (2), solutions of both the IVP and the BVP (solved by the help of the adjoint equation) are properly reproduced numerically and vice versa 148 K Balla, V.H Linh / Applied Numerical Mathematics 53 (2005) 131–148 Computation in Example 12 shows that for an index-2 Hessenberg system, DS1 = const and DN1 = const hold if and only if ker B21 = const and im B12 = const These two subspaces are constant simultaneously with their orthogonal complements It means that A∗ S∗1 = const and A∗ N∗1 = const Thus, on one hand, an index-2 Hessenberg system is numerically qualified if and only if so is its adjoint, and then, the numerical boundary value problem also well reproduces the dynamics On the other hand, combining this with Theorem 4, we can state that an index-2 Hessenberg DAE (1) is numerically qualified if and only if its INHODE coincides with the adjoint to the INHODE of the DAE (2) Acknowledgement The authors thank the referees offering suggestions for the improvement of the paper References [1] A Abramov, K Balla, V.I Ulyanova, L.F Yukhno, On a nonlinear self-adjoint eigenvalue problem for certain differential algebraic equations of index 1, Comput Math Math Phys 42 (7) (2002) 957–973, Translated from Zh Vychisl Mat Mat Fiz 42 (7) (2002) 996–1012 [2] A Abramov, K Balla, V.I Ulyanova, L.F Yukhno, On a nonlinear self-adjoint eigenvalue problem for a certain class of differential algebraic equations, Comput Math Math Phys 43 (3) (2003) 391–402, Translated from Zh Vychisl Mat Mat Fiz 43 (3) (2003) 410–421 [3] K Balla, G Kurina, R März, Index criteria for differential algebraic equations arising from linear-quadratic optimal control problems, Preprint 2003-14, Institut für Mathematik an der Mathematisch-Naturwissenschaftlichen Fakultät II der Humboldt-Universität zu Berlin, 2003, p 25 [4] K Balla, R März, A unified approach to linear differential algebraic equations and their adjoints, Z Anal Anwendungen 21 (3) (2002) 783–802 [5] K Balla, R März, Linear boundary value problems for differential algebraic equations, Math Notes (2004) 3–17 [6] G Kurina, R März, On linear quadratic optimal control problems for time-varying descriptor systems, SIAM J Control Optim 42 (6) (2004) 2062–2077 [7] R März, Adjoint equations of differential-algebraic systems and optimal control problems, in: Proceedings of the Institute of Mathematics, vol 7, NAS of Belarus, Minsk, 2001 [8] Y Cao, S.T Li, L.R Petzold, R Serban, Adjoint sensitivity analysis for differential-algebraic equations: The adjoint DAE system and its numerical solution, SIAM J Sci Comput 24 (3) (2003) 1076–1089 [9] Y Cao, S.T Li, L.R Petzold, Adjoint sensitivity analysis for differential-algebraic equations: Algorithms and software Scientific and engineering computations for the 21st century—methodologies and applications (Shizuoka, 2001), J Comput Appl Math 149 (1) (2002) 171–191 [10] V Dolezal, The existence of a continuous basis of a certain linear subspace of Er which depends on a parameter, Casopis P˘est Mat 89 (1964) 466–468 [11] E.A Coddington, N Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955 [12] R Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, 1960 [13] J Higueras, R März, C Tischendorf, Stability preserving integration of index-1 DAEs, Appl Numer Math 45 (2003) 175–200 [14] J Higueras, R März, C Tischendorf, Stability preserving integration of index-2 DAEs, Appl Numer Math 45 (2003) 201–229  ... construction of the bases and pairs of bases in subspaces connected with (1) and (2) Section derives the EUODEs and the pairs of EUODEs Section defines a class of (formally) self -adjoint DAEs In terms of. .. Proceedings of the Institute of Mathematics, vol 7, NAS of Belarus, Minsk, 2001 [8] Y Cao, S.T Li, L.R Petzold, R Serban, Adjoint sensitivity analysis for differential-algebraic equations: The adjoint. .. kinds of ordinary 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