- Email: [email protected]
New answers to an old question in the theory of differential–algebraic equations: Essential underlying ODE versus inherent ODE
Journal of Computational and Applied Mathematics (
)
–
Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
New answers to an old question in the theory of differential–algebraic equations: Essential underlying ODE versus inherent ODE Roswitha März Humboldt University of Berlin, Institute of Mathematics, D-10099 Berlin, Germany
article
info
Article history: Received 15 December 2015 Received in revised form 29 August 2016 Keywords: Differential–algebraic equation Essential underlying ODE Inherent ODE Adjoint pairs Lagrange identity Lyapunov spectrum Lyapunov regularity
abstract In the context of linear differential–algebraic equations (DAEs) one finds different associated explicit ordinary differential equations (ODEs), among them essential underlying and inherent explicit regular ones, abbreviated: EUODEs and IERODEs. EUODEs have been introduced in 1991 for index-2 DAEs in Hessenberg form by means of special transformations. IERODEs result within the framework of the projector based decoupling. Each such explicit ODE is occasionally considered to rule the flow of the DAE. The question to which extend EUODEs and IERODEs are related to each other has been asked promptly after 1991. For index-2 Hessenberg-form DAEs, answers have been given in 2005, saying that EUODEs represent somehow condensed IERODEs. Recently, EUODEs have been indicated for general arbitrary-index DAEs and it has been proved that they are condensed IERODEs. The understanding of the relation between the IERODE and the EUODEs enables to uncover the stability behavior of the DAE flow. In the present paper we show that both, the IERODEs and EUODEs of a DAE with arbitrary high index do not at all depend on derivatives of the right-hand side. We consider adjoint pairs of DAEs and provide generalizations of the classical Lagrange identity. Furthermore, we address Lyapunov spectra and Lyapunov regularity. © 2016 Elsevier B.V. All rights reserved.
1. Introduction In the context of linear differential–algebraic equations (DAEs) one finds different associated explicit ordinary differential equations (ODEs), among them essential underlying ODEs (EUODEs) and inherent explicit regular ODEs (IERODEs). EUODEs have been distinguished by Ascher and Petzold [1] for index-2 DAEs in Hessenberg form by means of special transformations. IERODEs result within the framework of the projector based decoupling. Both EUODEs and IERODEs are occasionally considered to rule the flow of the DAE. How are they related to each other? This question has been asked promptly after 1991. First answers have been given by Balla and Vu Hoang Linh [2,3] pointing out that, for index-2 Hessenberg-form DAEs and general index-1 DAEs, an EUODE represents a condensed IERODE. Recently, EUODEs associated with general arbitrary-index DAEs have been distinguished in [4]. Also in the general case, the EUODEs can be seen as condensed IERODEs. Any regular linear differential–algebraic equation with properly stated leading term features a unique IERODE living in the given space. In contrast, there are several EUODES living in a transformed space with possible minimal dimension. We show that the IERODEs and EUODEs are the only associated explicit ODEs which do not at all depend on derivatives of
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.cam.2016.09.039 0377-0427/© 2016 Elsevier B.V. All rights reserved.
2
R. März / Journal of Computational and Applied Mathematics (
)
–
the right-hand side. This offers the background of a preciser sensitivity analysis and allows to state boundary conditions accurately. The understanding of the relation between the IERODE and the EUODEs enables to uncover stability properties of the DAE flow such as Lyapunov regularity etc. We particularly concentrate on relevant versions of the Lagrange identity. This should be helpful in the context of numerical boundary value problems, sensitivity analysis, optimization, and control problems, see [1,5–10]. We investigate DAEs with properly involved derivative A(t )(Dx)′ (t ) + B(t )x(t ) = q(t ),
t ∈I
(1)
and standard form DAEs E (t )x′ (t ) + F (t )x(t ) = q(t ),
t ∈ I,
(2)
with coefficients E , F ∈ C (I, L(Km , Km )), A ∈ C (I, L(Kn , Km )),
D ∈ C (I, L(Km , Kn )),
B ∈ C (I, L(Km , Km )),
with K = R and K = C in the real and complex versions, respectively. The interval I ⊆ R is arbitrary, possibly infinite. We drop the argument t whenever reasonable. Then the given relations are meant pointwise for all t ∈ I. The paper is arranged as follows. In Section 2 we recap known results for index-2 DAEs in Hessenberg form already by means of the notation of the general approach in the following part in order to motivate this approach and to explicate the questions to be considered. In Section 3 we describe the structure of a regular DAE (1). In particular, we deal with its IERODE and EUODEs and show that these are the only associated q-derivative-free explicit ODEs. Adjoint pairs of DAEs are considered in Section 4. We show that, even though the IERODEs of the original DAE and its adjoint equation are not necessarily adjoint to each other, the solutions associated to the DAEs satisfy a Lagrange identity. We specify the results for standard form DAEs in Section 5. General conclusions are drawn in Section 6. 2. Recapping index-2 Hessenberg systems The system comprising the m = m1 + m2 equations
= q1 , = q2 ,
x′1 +B11 x1 + B12 x2 B21 x1
(3) (4)
is said to be a DAE in Hessenberg form of size 2, if the product B21 B12 remains everywhere nonsingular. This DAE is known to have differentiation index 2 as well as tractability index 2. Writing
′
I 0
B11 I 0 + x B21
B12 x = q, 0
(5)
=D
=A
one puts the DAE in the form (1). Owing to the nonsingularity of the product B21 B12 , the direct sum decomposition ker B21 ⊕ im B12 = Km1
(6)
is valid and the projector-valued function Ω given by
Ω := B12 B− 12 ,
−1 B− B21 , 12 := (B21 B12 )
(7)
projects pointwise onto im B12 along ker B21 . The usual approach to DAEs consists in extracting explicit ODEs with respect to x or x1 from a derivative array system. The resulting so-called underlying ODEs depend on the special way they are provided. For instance, differentiating Eq. (4) and then replacing the derivative x′1 by (3) results in the equation
Ax1 − Bx2 = q′2 − B21 q1 ,
(8)
with A := −B21 B11 + B′21 , B := B21 B12 . Differentiating now (8), replacing again the derivative x′1 by (3), multiplying by B−1 and rearranging terms finally leads to the ODE x′1 = −B11 x1 − B12 x2 + q1 , x′2 = −B−1 (AB11 − A′ )x1 − B−1 (AB12 + B′ )x2 + B−1 (A q1 − (q′2 − B21 q1 )′ ). The DAE flow is embedded into the m-dimensional flow of the underlying ODE. Observe that derivatives of components of q encroach on this ODE.
R. März / Journal of Computational and Applied Mathematics (
)
–
3
Furthermore, using (8) to replace x2 in Eq. (3), we obtain the ODE with respect to x1 , x′1 = −((I − Ω )B11 + B12 B−1 B′21 )x1 + (I − Ω )q1 + B12 B−1 q′2 , which is affected by the derivative q′2 . In contrast, an associated explicit ODE of minimal size, which is not at all affected by derivatives of q, has been obtained by special transformations in [1]. We follow the lines of [1], but use quite different notations. Put d := m1 − m2 . One finds a matrix function Γd ∈ C 1 (I, L(Km1 , Kd )) such that the columns of Γd∗ form a basis of ker B∗12 = (im B12 )⊥ . This implies ker Γd = im B12 , and the matrix function
Γd
remains nonsingular.
B21
We denote and verify
Γd := −
Γd
−1
Id , 0
B21
Γ d Γ d = Id −
0
Γd B21
ker Γd− Γd = im B12 ,
Γd− = Id ,
Γd Γd− Γd = Γd ,
Γd− Γd Γd− = Γd− ,
im Γd− Γd = ker B21 .
This actually means that Γd− is a generalized inverse of Γd and Γd− Γd = I − Ω . Furthermore, one can check that
Γd
−1
B21
0 = B12 (B21 B12 )−1 . I
Defining new variables
η = Γd x1 ,
x1 =
Γd
−1
B21
−1 η Γ 0 − = Γd η − d = Γd− η − B12 (B21 B12 )−1 q2 , −q2 B21 q2
one obtains an explicit ODE with respect to η,
η′ + Γd B11 Γd− η − Γd′ Γd− η = Γd q1 + (Γd′ − Γd B11 )B12 (B21 B12 )−1 q2 ,
(9)
which is referred to as essential underlying ODE (EUODE) in order to distinguish it from other underlying ODEs, emphasizing that it has minimal size d and yields a well-posed boundary-value problem with respect to x1 , see [1]. Below we will indicate such EUODEs for each arbitrary regular DAE. Having x1 one computes the component x2 by x2 = (B21 B12 )−1 B21 {q1 − (Γd− η)′ + (B12 (B21 B12 )−1 q2 )′ − B11 (Γd− η − B12 (B21 B12 )−1 q2 )}, which uncovers the structure of the DAE flow. The dynamical degree of freedom equals d. Since there are different possibilities to choose a basis of ker B∗12 , the essential underlying ODE depends on the specially chosen basis. As suggested in [1], one can restrict the options to Γd with orthonormal rows. The projector based approach (e.g. [11]) relies on projector functions such as Ω and decouples the original DAE as it is, devoid of any transformations. Regarding the properties B21 = B21 Ω and Ω B12 = B12 , we immediately derive from Eq. (3) that
(I − Ω )x′1 + (I − Ω )B11 x1 = (I − Ω )q1 ,
(10)
and from (4) that
Ω x1 = B12 (B21 B12 )−1 q2 , which leads to x1 = (I − Ω )x1 + B12 (B21 B12 )−1 q2 , x2 = (B21 B12 )−1 B21 {q1 − ((I − Ω )x1 )′ + (B12 (B21 B12 )−1 q2 )′ − B11 (I − Ω )x1 − B11 B12 (B21 B12 )−1 q2 }. Inserting the expression for x1 into the relation (10) we arrive at the explicit ODE with respect to the only component u = ( I − Ω ) x1 , u′ − (I − Ω )′ u + (I − Ω )B11 (I − Ω )u = (I − Ω )q1 + (I − Ω )′ B12 (B21 B12 )−1 q2 − (I − Ω )B11 B12 (B21 B12 )−1 q2 , (11) which is called inherent explicit regular ODE (IERODE) and which resides in Km1 . The varying d-dimensional subspace im (I − Ω ) ⊂ Km1 is an invariant subspace of the IERODE, that is, if u∗ is a solution of (11) and there is a t¯ ∈ I with u(t¯) ∈ im (I − Ω (t¯)), then one has u(t ) ∈ im (I − Ω (t )) for all t ∈ I. This is the basic property of the IERODE in matters of the DAE structure. The IERODE is uniquely defined by the DAE coefficients. It has size m1 ≥ d, but solely its flow within the d-dimensional subspace im (I − Ω ) is relevant for the DAE flow.
4
R. März / Journal of Computational and Applied Mathematics (
)
–
The so-called canonical projector function of the DAE (3), (4) reads (cf. [11, p. 107])
Πcan =
I −Ω
′ −B− 12 (B11 − Ω )(I − Ω )
0 0
and it has constant rank d together with the projector function (I − Ω ). One can check that all solutions of the homogeneous DAE with q = 0 have the form
(I − Ω )x1 = Πcan x, ′ −B12 (B11 − Ω )(I − Ω )x1
x=
−
and im Πcan (t ) represents the d-dimensional time-varying subspace of consistent values or solution space of the homogeneous DAE. In the light of our notation the relations
η = Γd x1 = Γd Γd− Γd x1 = Γd (I − Ω )x1 = Γd u, u = (I − Ω )x1 = Γd− η, become obvious. Multiplying the EUODE by Γd− results in the IERODE and, conversely, multiplying the IERODE by Γd yields the EUODE. In this sense, each EUODE is nothing else than a condensed IERODE. This fact was first pointed out in [3] for index-2 DAEs in Hessenberg form and recently verified for general regular DAEs in [4]. The adjoint DAE to the index-2 Hessenberg form DAE (3), (4),
= p1 , = p2 ,
−y′1 +B∗11 y1 + B∗21 y2 B∗12 y1 or written in the form (1),
′ ∗ B11 I I 0 y + − B∗12 0 =A∗
B∗21 y = p, 0
(12)
=D ∗
has also index 2 and the dynamical degree of freedom d. Its canonical projector function reads
Π∗ can =
I − Ω∗
B∗21− (−B∗11 − Ω ∗ )(I − Ω ∗ ) ′
0 . 0
It holds that DΠcan D− = I − Ω ,
A∗ Π∗ can A∗ − = I − Ω ∗ = (DΠcan D− )∗ ,
∗ however, the canonical projector function Π∗ can differs from Πcan , except for quite artificial special cases. Letting q = 0 and p = 0, the associated homogeneous IERODEs of the DAE and its adjoint become
u′ − (I − Ω )′ u + (I − Ω )B11 (I − Ω )u = 0
(13)
− v ′ + (I − Ω ∗ )′ v + (I − Ω ∗ )B∗11 (I − Ω ∗ )v = 0.
(14)
and As first pointed out in [2,3], the IERODEs (13) and (14) form a classical adjoint pair, only if the projector function Ω is constant, or, equivalently, if the associated subspaces im B12 and ker B21 do not vary with t. It seems, this fact has been considered as drawback of the IERODE, but wrongly. As it is observed above, only solution components u = (I − Ω )u and v = (I − Ω ∗ )v are relevant for the DAEs, and nevertheless for these components the classical Lagrange identity takes place1 : d dt
⟨u, v⟩ = ⟨u′ , v⟩ + ⟨u, v ′ ⟩ = 2⟨(I − Ω )′ u, v⟩ = 2⟨(I − Ω )′ (I − Ω )u, (I − Ω ∗ )v⟩ = 2⟨(I − Ω )(I − Ω )′ (I − Ω )u, v⟩ = 0.
Below we will show a corresponding general property for each arbitrary pair of adjoint regular DAEs and their IERODEs. On the other hand, choosing a condensing matrix function Γd ∈ C 1 (I, L(Km1 , Kd )), such that im Γd∗ = ker B∗12 , and determining the generalized inverse Γd− so that
Γd− Γd = I − Ω ,
Γd Γd− = Id ,
1 Here and later the following argument is used: If P is a differentiable projector function on an interval I, then P (I − P ) = 0 implies P ′ (I − P ) − PP ′ = 0, thus PP ′ P = P ′ (I − P )P = 0.
R. März / Journal of Computational and Applied Mathematics (
)
–
5
∗∗ we have at the same time im Γd− = ker B21 , i.e., Γd− serves as a basis of ker B21 = ker B21 . Then we may simultaneously condense both IERODEs to EUODEs, that is, ′
η′ + (Γd B11 Γd− + Γd Γd− )η = 0 and ′
−ζ ′ + (Γd− ∗ B∗11 Γd∗ + Γd− ∗ Γd∗ )ζ = 0, which are actually adjoint each to other, without any additional restrictions. Hereby, the IERODE of the adjoint is condensed by means of the special basis Γd− of ker B21 , and
v = (I − Ω ∗ )v = Γd∗ Γd− ∗ v = Γd∗ Γd ζ ,
ζ = Γd− ∗ v.
This result is consistent with the claim (cf. [7, Section 4.3]) that the EUODE of the adjoint system is the same as the adjoint of the EUODE of the original system. However, emphasize once again that the EUODEs depend on the specially chosen Γd and Γ∗ d . In particular, one obtains an EUODE of the adjoint DAE by means of any basis Γ∗∗d of ker B21 . As pointed out in [3], the above assertion is valid only, if the basis applied for the EUODE of the adjoint DAE is consistent with the basis chosen for the original DAE, that means
Γ∗ d = Γd− ∗ . 3. The structure of regular DAEs with properly involved derivative In this section we deal with DAEs of the form (1), that is, A(Dx)′ + Bx = q.
(15)
Let the time-varying subspaces ker A(t ) ⊆ Kn
im D(t ) ⊆ Kn ,
and
t ∈ I,
1
be C - subspaces. and let the transversality condition ker A(t ) ⊕ im D(t ) = Kn ,
t ∈ I,
(16)
be valid, which means that the DAE shows a properly stated leading term. The decomposition (16) determines the so-called border projector function R ∈ C 1 (I, L(Kn , Kn )) by ker R(t ) = ker A(t ),
im R(t ) = im D(t ),
t ∈ I.
(17)
1
Since both involved subspaces are C -subspaces, the projector function R is actually continuously differentiable. We use the projector based analysis to uncover the basic structure of a regular DAE with arbitrarily high tractability index and refer to [11] for details and for general relations with other index notions as well. Let the DAE (15) be regular in the sense of [11, Definition 2.25]. Recall that regularity is supported by several constantrank requirements yielding the tractability index µ ∈ N and the characteristic values 0 ≤ r0 ≤ · · · ≤ rµ−1 < rµ = m,
d=m−
µ−1 (m − ri ),
(18)
i =0
of the DAE. Regularity is formally determined by means of admissible projector functions P0 , . . . , Pµ−1 ∈ C (I, L(Km , Km )) associated with the construction of admissible matrix functions sequences starting from G0 := AD and ending up with a nonsingular Gµ , see [11, Definition 2.6]. The tractability index is consistent with the Kronecker index of regular matrix pencils. For a regular matrix pencil, the characteristic values ri provide a complete description of the formal structure of the corresponding Weierstraß–Kronecker form. In particular, d is the dimension of the dynamical part [11, Section 1]. We use the further matrix functions Q0 := I − P0 ,
Π0 := P0 ,
Qi := I − Pi ,
Πi := Πi−1 Pi ∈ C (I, L(Km , Km )),
DΠ0 D , . . . , DΠµ−1 D −
−
i = 1, . . . , µ − 1,
∈ C (I, L(K , K )), 1
n
n
with the pointwise determined generalized inverse D− such that DD− D = D,
D− DD− = D− ,
DD− = R,
D− D = P0 .
The products Πi and DΠ iD are projector-valued functions again (see [11, Proposition 2.7]). −
(19)
6
R. März / Journal of Computational and Applied Mathematics (
)
–
The sequence of admissible projector functions serves as tool for the decoupling of the DAE itself and the decomposition of the solution x into their characteristic parts, see [11, Section 2.4]. In particular, the component u = DΠµ−1 x satisfies the explicit regular ODE residing in Kn , 1 − −1 u′ − (DΠµ−1 D− )′ u + DΠµ−1 G− µ BΠµ−1 D u = DΠµ−1 Gµ q.
(20)
The components v0 = Q0 x, v1 = Π0 Q1 x, . . . , vµ−1 = Πµ−2 Qµ−1 x satisfy the triangular subsystem involving several differentiations for µ > 1,
0
N01
0
··· .. . .. .
I
M01 I
+
N0,µ−1
0 (Dv1 )′ .. . Nµ−2,µ−1 ′ (Dvµ−1 ) 0 ··· M0,µ−1 v H L0 0 0 .. .. v1 H 1 L1 . . D− u = + .. .. .. q. .. . . . . Mµ−2,µ−1 Lµ−1 Hµ−1 vµ−1 I .. .
(21)
The subspace im DΠµ−1 is an invariant subspace for the ODE (20). The components v0 , v1 , . . . , vµ−1 remain within their subspaces im Q0 , im Πµ−2 Q1 , . . . , im Π0 Qµ−1 , respectively. The structural decoupling is associated with the decomposition x = D− u + v0 + v1 + · · · + vµ−1 . All coefficients are continuous and explicitly given in terms of an admissible matrix function sequence as
N01 := −Q0 Q1 D− N0j := −Q0 P1 · · · Pj−1 Qj D− ,
j = 2, . . . , µ − 1,
Ni,i+1 := −Πi−1 Qi Qi+1 D , −
Nij := −Πi−1 Qi Pi+1 · · · Pj−1 Qj D− ,
j = i + 2, . . . , µ − 1, i = 1, . . . , µ − 2,
M0j := Q0 P1 · · · Pµ−1 Mj DΠj−1 Qj ,
j = 1, . . . , µ − 1,
Mij := Πi−1 Qi Pi+1 · · · Pµ−1 Mj DΠj−1 Qj ,
j = i + 1, . . . , µ − 1, i = 1, . . . , µ − 2,
1 L0 := Q0 P1 · · · Pµ−1 G− µ , 1 Li := Πi−1 Qi Pi+1 · · · Pµ−1 G− µ ,
Lµ−1 :=
i = 1, . . . , µ − 2,
1 Πµ−2 Qµ−1 G− µ ,
H0 := Q0 P1 · · · Pµ−1 K Πµ−1 , Hi := Πi−1 Qi Pi+1 · · · Pµ−1 K Πµ−1 ,
i = 1, . . . , µ − 2, v
Hµ−1 := Πµ−2 Qµ−1 K Πµ−1 , in which 1 K := (I − Πµ−1 )G− µ Bµ−1 Πµ−1 +
µ−1 (I − Πl−1 )(Pl − Ql )(DΠl D− )′ DΠµ−1 , l =1
Mj :=
j −1
(I − Πk ){Pk D− (DΠk D− )′ − Qk+1 D− (DΠk+1 D− )′ }DΠj−1 Ql D− ,
k=0
l = 1, . . . , µ − 1. The ODE (20) is always uncoupled of the second subsystem, but the latter is tied to the ODE if among the coefficients H0 , . . . , Hµ−1 is at least one who does not vanish. One speaks about a fine decoupling, if H1 = · · · = Hµ−1 = 0, and about a complete decoupling, if H0 = 0, additionally. A complete decoupling is given, exactly if the coefficient K vanishes identically. For a regular DAE with sufficiently smooth original data, fine and complete decouplings always exist and can be constructed, see [11, Subsection 2.4.3]. More precisely, to each arbitrarily fixed P0 there are appropriate subsequent P1 , . . . , Pµ−1 achieving a fine decoupling. Adjusting also P0 in a specific way one accomplishes a complete decoupling. It should be added at this point, that the coefficients of the ODE (20) depend on the special choice of admissible projector functions. However, they are uniquely determined in the scope of fine decouplings. This justifies to speak about the inherent explicit regular ODE (IERODE) of the given DAE.
R. März / Journal of Computational and Applied Mathematics (
)
–
7
The so-called canonical projector function Πcan of a regular DAE (see [11, Definition 2.37]) is actually a generalization of the spectral projector of a regular matrix pencil onto the finite eigenspace along the infinite eigenspace (cf. [11, Section 1.4]). The subspace im Πcan (t ) ⊆ Km consists precisely of all consistent values of the homogeneous DAE at time t, that is, it represents the corresponding solution space. By means of fine decoupling projector functions P0 , . . . , Pµ−1 , the canonical projector function of the DAE can be represented as
Πcan = (I − H0 )Πµ−1 . It follows that DΠµ−1 = DΠcan ,
DΠµ−1 D− = DΠcan D− .
(22)
We emphasize that Πcan itself is independent of the choice of the projector functions. Therefore, also DΠµ−1 = DΠcan does not depend of the construction. Since fine decoupling projector functions P0 , . . . , Pµ−1 allow an arbitrarily fixed projector function P0 along ker D and the accordingly prescribing generalized inverse D− , it may happen that Πcan ̸= Πµ−1 . In contrast, completely decoupling projector functions P0 , . . . , Pµ−1 yield the representation Πcan = Πµ−1 , which is very useful in theory, but less comfortable in practice since then P0 and D− are very special. The IERODE (20) resulting from a fine decoupling is independent of the special choice of the fine decoupling. We emphasize this fact by rewriting it as 1 − −1 u′ − (DΠcan D− )′ u + DΠcan G− µ BΠcan D u = DΠcan Gµ q.
(23)
Obviously, in the IERODE (23), only inhomogeneities with values in 1 im DΠcan = im DΠcan G− µ
are responsible for the DAE (15). Theorem 3.1. Let the DAE (15) be regular with index µ and characteristic values (18). Let its coefficients be sufficiently smooth. Then, the following is valid: (1) The coefficients of the IERODE (23) are uniquely determined by fine decoupling projectors and its inhomogeneity does not inherit any derivative of q. (2) The component u = DΠcan D− Dx = DΠcan x of each DAE solution x satisfies the IERODE. (3) d is the dynamical degree of freedom of the DAE and there is an inherent d-dimensional flow located in im DΠcan , which is not at all affected by derivatives of the inhomogeneity q. (4) Except for the component u = DΠcan x and possibly subcomponents of it, there is no other genuine solution component satisfying an immanent explicit ODE the inhomogeneity of which does not involve any derivative of q.2 Proof. These assertions (1)–(3) can immediately be concluded from [11, Theorem 2.39] regarding that DΠµ−1 = DΠcan , d = rank Πcan = rank DΠcan D− . To verify (4) we turn to the unfolded structure associated with a complete decoupling. The system (21) has exactly one solution v0 , . . . , vµ−1 . It can be solved successively for vµ−1 , . . . , v0 in terms of q and possibly its derivatives. If µ > 1, part of the solution components actually depend on derivatives of q. There are also solution components depending only on q such as vµ−1 = Lµ−1 q and Tµ−2 vµ−2 = Tµ−2 {Lµ−2 q − Mµ−2 µ−1 Lµ−1 q}, in which Tµ−2 denotes a projection along im Nµ−2 µ−1 . This makes clear, that, in any case, each derivative vi′ necessarily depends at least on one component of a derivative of q. This extends also to any linear combination of the vi′ and u, since these are living in separated subspaces. The same arguments apply to the components Dv1 , . . . , Dvµ−1 . Example 3.2. The constant coefficient DAE, x′1 − α x1 − x2 = q1 , x′3 + x2 = q2 , x′4 + x3 = q3 , x′5 + x4 = q4 , x5 = q5 ,
2 There are allied skew components u˜ = PDx = Pu adopting this property. Namely, if P : I → L(Kn ) is a differentiable projector-valued function and ker P = ker DΠcan D− , then it results that P = PDΠcan D− , DΠcan D− = DΠcan D− P, and further u˜ = Pu, u = DΠcan D− u˜ . Inserting into (23) leads to the corresponding ODE for u˜ . A particular case of such a P is given by DΠµ−1 D− built with arbitrary admissible projector functions, cf. (20).
8
R. März / Journal of Computational and Applied Mathematics (
)
–
written in the form (15) reads
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 1 0 0 0 0 1 0 0
0 0 0 0
0 1 0 0
0 0 1 0
−α
′
−1
0 0 0 x + 0 0 0 1 0
0 0 1 0 0
1 0 0 0
0 0 0 1 0
0 0 0 = q, 0 1
with m = 5, n = 4. The following associated matrix sequence (cf. [11, Subsection 1.2.3]) is admissible and provides µ = 4 and the characteristic values r0 = r1 = r2 = r3 = 4, r4 = 5, d = 1:
1 0 G0 = 0 0 0
−1
1 0 G1 = 0 0 0
1 0 G2 = 0 0 0
1 0 G3 = 0 0 0
−1
α
1 0 0 0
1 1 0 0
α
−α 2
1 0 0 0
1 1 0 0
0 1 1 0
1 0 G4 = 0 0 0
0 1 0 0 0
0 1 0 0 0
1 0 0 0
−1
0 0 0 0 0
−α
1 0 0 0
1 1 0 0
0 1 1 0
α
3
0 0 Q1 = 0 0 0 0 0 Q2 = 0 0 0
0 0 Q3 = 0 0 0 2
0 0 Q0 = 0 0 0
0 0 0 , 1 0
α
0 0 0 , 1 0
0 0 1 0 0
0 0 0 , 1 0
0 0 1 0 0
0 0 0 , 1 0
−1
0 0 1 0 0
0 0 , 1 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1 0 Π3 = 0 0 0
0 1 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0
−1 −1
0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 , 0 0
0 0 0 , 0 0
1 1 0 0 0 0
−α 0 0 0 0
α2
0 0 0 0 0
0 0 Π1 Q2 = 0 0 0
0 0 Π2 Q3 = 0 0 0
0 0 0 1 0
0 0 0 , 0 1
−1
0 0 0 0 0
0 0 0 , 0 0
α
0 0 0 , 0 0
0 1 0 0 0 0 0 0 0
0 0 0 0 0
0 0 1 0
0 0 0 0 0
0 0 0 0 0
1 0 − DΠ3 D = 0 0
1 0 0 0
−α
0 0 1 0 0
0 0 0 0 0
0 0 , 0 0
1 0 0 0
0 0 0 Π0 Q1 = 0 0 0
1+α 1 −1 1 0
−1
0 0 0 , 0 0
−1 − α − α 2 −1 1 , −1 0 0 0 0 0
−α 0 B0 = 0 0
0 0 0
−α 2
0 0 0 1
,
α2
0 . 0 0
Additionally, it follows that 1 Q3 G− 4 B0 Π3 = 0,
1 Q2 P3 G− 4 B0 Π3 = 0,
1 Q1 P2 P3 G− 4 B0 Π3 = 0,
1 Q0 P1 P2 P3 G− 4 B0 Π3 = 0,
and 1 Π3 G− 4 B0 Π3 = −α Π3 .
(24)
Therefore, these projector functions P0 , P1 , P2 , P3 provide a complete decoupling of the given DAE and Π3 coincides with the canonical projector Πcan . The projector functions Q0 , Π0 Q1 , Π1 Q2 and Π2 Q3 represent the variables x2 , x3 , x4 and x5 , respectively. The projector DΠ3 D− and the coefficient (24) determine the IERODE with respect to the component u = DΠ3 x, namely 1 u′ − α u = DΠ3 G− 4 q.
Dropping the three zero rows it results that
(x1 + x3 − α x4 + α 2 x5 )′ − α(x1 + x3 − α x4 + α 2 x5 ) = q1 + q2 − α q3 + α 2 q4 − α 3 q5 .
(25)
Obviously, no derivative of the inhomogeneity q encroaches into the IERODE. In contrast, each associated explicit ODE with respect to the original components xi depends on derivatives of q. In particular, the component x1 satisfies the ODE x′1 − α x1 = q1 + q2 − (q3 − (q4 − q′5 )′ )′ .
(26)
The property of the IERODE to be free of all derivatives of q allows to state boundary conditions for the DAE accurately, see [9], which seizes the suggestion of [1] in essence. However, recall that though the respective flow resides in the d-dimensional
R. März / Journal of Computational and Applied Mathematics (
)
–
9
subspace im DΠcan ⊆ Kn the IERODE itself lives in Kn . Next we will see that the IERODE can be condensed to minimal size d, which results in a so-called essential underlying ODE. Each regular DAE can be transformed into a form with decoupled fast and slow parts, similar to the Weierstraß–Kronecker form of a regular matrix pencil. We quote the respective result from [4,11]: Theorem 3.3. Each regular DAE (15) with index µ and characteristic values (18) can be transformed by pointwise nonsingular matrix functions L ∈ C (I, L(Km , Km )),
K ∈ C (I, L(Km , Km )),
and a refactorization of the leading term by H with H ∈ C 1 (I, L(Ks , Kn )), HH H = H , −
−
H HH
H − ∈ C 1 (I, L(Kn , Ks )), −
= H −,
n, s ≥ r := rank D,
RHH − R = R,
into the structured form
˜ x˜ )′ + B˜ x˜ = Lq, A˜ (D
(27)
with A˜ = LAH =
Id 0
0 , N
˜ = H − DK = D
B˜ = LBK − LAH (H − R)′ DK = 0
N =
N0 1
..
.
W 0
··· .. .
N0 µ−1
0
Nµ−2 µ−1 0
,
0 Im−d
Id 0
0 , PN
, 0
m−r0
Im−r1
PN =
..
,
. Im−rµ−1
in which the entries Ni−1 i have size (m − ri−1 ) × (m − ri ) and full rank m − ri , i = 1, . . . , µ − 1. We recall that the transformation matrices stated in Theorem 3.3 have the following explicit form (cf. [11, p. 146])
Γd DΠcan Γ0 Q0 −1 G , .. µ . Γµ−1 DΠµ−2 Qµ−1
L=
Id 0
0
(I + M˜ )−1
−1 Γd DΠcan Γ0 Q0 , K = .. . Γµ−1 DΠµ−2 Qµ−1
at which completely decoupling projector functions are used and the functions Γi , i = 0, 1, . . . , µ − 1, are defined as in [11, p. 143]. Though the matrix function Gµ and its inverse may depend on the special choice of the completely decoupling 1 projector functions, the expressions Πcan G− µ and Gµ Πcan = ADΠcan are invariant. The transformed DAE (27) comprises the explicit ODE 1 η′ + W η = Γd DΠcan G− µ q,
(28)
with size d. Its inhomogeneity does not comprise derivatives of q. The ODE (28) is said to be an essential underlying ODE (EUODE) of the DAE (15) in [4] after the respective notion introduced for index-2 Hessenberg form DAEs in [1,7,3]. Recall that the IERODE lives in Kn , n ≥ d. The IERODE is unique in the scope of fine decouplings. Its coefficients are expressed in terms of the original DAE. In contrast, the EUODE has minimal size d, but it is accessible by suitable transformations only. These transformations are not uniquely determined, so that several EUODEs accrue, cf. also Theorem 3.4. As described in Section 2 for index-2 DAEs in Hessenberg form (3), (4), and accordingly (5), each EUODE can be seen as condensed IERODE. Recall that in Section 2 the relations im Γd∗ = ker B∗12 = (im B12 )⊥ = (ker (I − Ω ))⊥ = im (I − Ω )∗ = im (DΠcan D− )∗ have been used to construct the condensing transform Γd . In the general case we can proceed similarly as described in [11,4, Section 2.8]. We take a closer look at the general condensing. Since the projector function DΠcan D− is continuously differentiable and has rank d, so is (DΠcan D− )∗ , and im (DΠcan D− )∗ is spanned by d continuously differentiable basis functions. This means that there is a matrix function Γd such that
Γd ∈ C 1 (I, L(Kn , Kd )),
im Γd∗ = im (DΠcan D− )∗ ,
ker Γd∗ = {0}.
(29)
10
R. März / Journal of Computational and Applied Mathematics (
)
–
Then we determine the pointwise generalized inverse Γd− ∈ C 1 (I, L(Kd , Kn )) by
Γd Γd− Γd = Γd ,
Γd− Γd Γd− = Γd− ,
Γd− Γd = DΠcan D− ,
Γd Γd− = Id .
(30)
Letting η = Γd u for the solutions u = DΠcan D− u = Γd Γd u of the IERODE leads to an EUODE (28) in which (cf. [11, Section 2.8]) −
1 − − ′ − −1 − − W = −Γd′ Γd− + Γd DΠcan G− µ BΠcan D Γd = −Γd Γd + Γd DΠcan Gµ BD Γd .
(31)
The next theorem records and accents the result obtained. Theorem 3.4. Each EUODE (28) of a regular DAE (15) with index µ and characteristic values (18) can be represented as a condensation of the IERODE (23) by means of a matrix function (29) and the generalized inverse (30). The coefficient W of the associated EUODE and the IERODE are related by (31). We emphasize once again that, though the IERODE is unique in the context of fine decouplings, the EUODE actually depends on the choice of the basis functions of im (DΠcan D− )∗ . Example 3.5. We continue to consider the index-4 DAE from Example 3.2. Seemingly, most authors would take the ODE (26) for an EUODE, but this is a mistake. Here Π3 coincides with the canonical projector Πcan and we have
Πcan
1 0 = 0 0 0
0 0 0 0 0
1 0 0 0 0
−α 0 0 0 0
α2
0 0 , 0 0
1 0 − DΠcan D = 0 0
1 0 0 0
−α 0 0 0
α2
0 , 0 0
1 1
im (DΠcan D− )∗ = im −α .
α2
1 Choosing Γd = [1 1 − α α 2 ] yields η = Γd u = x1 + x3 − α x4 + α 2 x5 . The associated EUODE η′ − αη = Γd DΠ3 G− 4 q reads in detail
(x1 + x3 − α x4 + α 2 x5 )′ − α(x1 + x3 − α x4 + α 2 x5 ) = q1 + q2 − α q3 + α 2 q4 − α 3 q5 , which coincides with (25). This EUODE is actually a condensed IERODE.
One can see the diversity of EUODEs as a drawback, however, as in [4, Section 6] concerning stability issues, one can even take advantage from the arbitrariness of the basis Γd∗ . Theorem 3.6. Let the DAE (15) be regular with index µ and characteristic values (18). Put q = 0. Let the condensing transformation Γd and Γd− be determined by (29), (30). Then the following is valid: (1) If the matrix functions DΠcan and Πcan D− remain bounded, the Lyapunov spectrum of the DAE coincides with that of the IERODE with respect to im DΠcan . (2) If the matrix functions Γd DΠcan and Πcan D− Γd− remain bounded, the Lyapunov spectra of the DAE and the EUODE coincide. (3) Γd can always be chosen so that the EUODE preserves the Lyapunov spectrum of the DAE. Proof. The assertions (1) and (2) can immediately be concluded from the relations given for q = 0: x = Πcan D− u,
u = DΠcan x,
x = Πcan D− Γd− η,
η = Γd DΠcan x.
(3) As shown in [4], the relations (29), (30) are fulfilled for Γd = U ∗ Πcan D− , Γd− = DU, in which U is an orthonormal basis of im Πcan , and one has x = U η. Note that Theorem 3.6(3) demands an unlimited further flexibility for the choice of the basis Γd∗ . If the choice is restricted ∗ U should be to orthonormal bases as originally proposed in [1], assertion (3) is no longer true. In general, Γd∗ = D− ∗ Πcan expected to fail to be orthonormal. 4. Adjoint pairs Pairs of explicit ODEs and their adjoints, x′ + Bx = 0,
−y′ + B∗ y = 0,
feature properties being useful in theory and practical treatment. For instance, the Lagrange identity means that the product ⟨x(t ), y(t )⟩ of each solution pair remains constant, and the Perron Theorem states that the original ODE is Lyapunov regular with Lyapunov exponents λ1 , . . . , λm exactly if the adjoint is Lyapunov regular with Lyapunov exponents −λ1 , . . . , −λm .
R. März / Journal of Computational and Applied Mathematics (
)
–
11
These properties are usually utilized when formulating and solving boundary-value problems, in sensibility analysis, optimization and control. Aiming at corresponding results we consider the DAE with properly stated leading term, A(Dx)′ + Bx = q,
(32)
together with its adjoint equation (cf. [12]),
− D∗ (A∗ y)′ + B∗ y = p.
(33)
The DAE (33) has a properly stated leading term at the same time as (32), with the associated border projector function R∗ . For any solution pair x ∈ CD1 (I, Km ) and y ∈ CA1∗ (I, Km ) of the DAEs (32) and (33), respectively, and a function w ∈ C 1 (I, K) satisfying the condition
w ′ (t ) = −⟨q(t ), y(t )⟩ + ⟨x(t ), p(t )⟩,
t ∈ I,
one obtains d dt
(⟨D(t )x(t ), A(t )∗ y(t )⟩ + w(t )) = ⟨(Dx)′ (t ), A(t )∗ y(t )⟩ + ⟨D(t )x(t ), (A∗ y)′ (t )⟩ + w ′ (t ) = ⟨A(t )(Dx)′ (t ), y(t )⟩ + ⟨x(t ), D(t )∗ (A∗ y)′ (t )⟩ + w′ (t ) = ⟨−B(t )x(t ) + q(t ), y(t )⟩ + ⟨x(t ), B(t )∗ y(t ) − p(t )⟩ + w′ (t ) = 0,
t ∈ I,
such that
⟨D(t )x(t ), A(t )∗ y(t )⟩ + w(t ) = constant,
t ∈ I.
This implies the Lagrange identity for the homogeneous DAEs (32) with q = 0 and (33) with p = 0:
⟨D(t )x(t ),
A(t )∗ y(t )⟩ = constant,
t ∈ I.
(34)
In contrast to the explicit ODEs, for DAEs and even for implicit ODEs, one can no longer expect a Lagrange identity with respect to ⟨x(t ), y(t )⟩ as the next example indicates. Example 4.1. The scalar homogeneous implicit ODE from [7, Section 4] et x′ (t ) +
1 2
et x(t ) = 0
can be written in the form (1) with m = n = 1, A(t ) = et , D(t ) = 1, and B(t ) =
1 t e. 2
The adjoint ODE is
1
−(et y(t ))′ + et y(t ) = 0. 2
The solutions are simply 1 x(t ) = e− 2 t cx ,
1 and y(t ) = e− 2 t cy ,
with constants cx , cy . The product x(t )y(t ) = e−t cx cy fails to be time-invariant, but 1
1
⟨D(t )x(t ), A(t )∗ y(t )⟩ = e− 2 t cx et e− 2 t cy = cx cy is so.
Recall that the IERODE and the EUODEs of the DAE (32) have the form 1 − −1 u′ − (DΠcan D− )′ u + DΠcan G− µ BΠcan D u = DΠcan Gµ q,
(35)
1 η′ + W η = Γd DΠcan G− µ q.
(36)
and
Applying the decoupling procedure to the adjoint DAE (33) accordingly we derive its IERODE as 1 ∗ ∗− 1 v ′ − (A∗ Π∗ can A∗ − )′ v + A∗ Π∗ can G− v = A∗ Π∗ can G− ∗ µ B Π∗ can A ∗ µ p,
(37)
in which the asterisk-index indicates the matrix functions associated with the coefficients of the DAE (33), i.e., A∗ := −D∗ , D∗ := A∗ , and B∗ := B∗ . Furthermore, the EUODEs of the adjoint DAE (33) have the form 1 ζ ′ + W∗ ζ = Γ∗ d A∗ Π∗ can G− ∗ µ p.
(38)
The next theorem generalizes and completes results from [2,3] concerning index-1 and index-2 DAEs. It clarifies the corresponding observations for index-2 DAEs in Hessenberg form outlined in Section 2.
12
R. März / Journal of Computational and Applied Mathematics (
)
–
Theorem 4.2. The following assertions are valid for each regular DAE (32) with index µ and characteristic values (18): (1) The adjoint DAE (33) is regular with the same index and characteristic values. (2) A∗ Π∗ can A∗ − = (DΠcan D− )∗ . (3) The IERODE (37) of the adjoint DAE can be described in terms of the original DAE as ′
1 − ∗ −∗ ∗ − v ′ + (DΠcan D− )∗ v + (DΠcan G− Πcan p, µ BΠcan D ) v = D
(39)
(4) The adjoint of the IERODE of the DAE (32) with q = 0 and the IERODE of the adjoint DAE (33) with p = 0 coincide precisely if DΠcan D− is time-invariant. (5) The EUODEs of the DAE (32) and the EUODEs of the adjoint DAE (33) built with consistent bases Γ∗ d = Γd− ∗ are adjoint each to other. It holds that W∗ = −W ∗ and the EUODE (38) can be rewritten as ∗ − ζ ′ + W ∗ ζ = Γd− ∗ D− ∗ Πcan p.
(40)
Proof. Assertion (1) is a special case of [4, Theorem 3(1)]. Assertion (2) is given by [4, Lemma 3]. Assertion (3) immediately implies (4). (5) can be derived from (3) by straightforward calculations. So it remains to check (3). We apply completely decoupling projector functions and regard that the IERODE is independent of the special choice of fine decoupling projector ∗ ∗ ∗ ∗ functions. By analogous arguments as used for the relation Π∗ can = G− µ Πcan D A in the proof of [4, Lemma 3] we find the relation ∗ ∗ ∗ ∗ −∗ ∗ Πcan = G− ∗ µ Π∗ can D∗ A∗ = −G∗ µ Π∗ can AD, ∗ 1 thus D− ∗ Πcan = −A∗ Π∗ can G− ∗ µ , and further 1 ∗ ∗− ∗ ∗ A∗ Π∗ can G− = −D− ∗ Πcan B Π∗ can A∗ − ∗ µ B Π∗ can A
∗ ∗ −∗ ∗ = −D− ∗ Πcan B Gµ Πcan D∗ A∗ A∗ − ∗ ∗ −∗ ∗ 1 − ∗ = −D− ∗ Πcan B Gµ Πcan D∗ = −(DΠcan G− µ BΠcan D ) .
This yields the equation ′
1 − ∗ −∗ ∗ v ′ − (DΠcan D− )∗ v − (DΠcan G− Πcan p. µ BΠcan D ) v = −D
Finally, multiplying this equation by −1 gives (39).
One might see in Theorem 4.2(4) a drawback of the IERODE, but this is unsubstantiated. One must regard that only the flows proceeding in the corresponding invariant subspaces im DΠcan D− and im A∗ Πcan A∗ − = im (DΠcan D− )∗ are relevant for the DAEs. Let u and v be solutions of the IERODEs (35) and (39) belonging to the relevant subspaces such that u = DΠcan D− u, v = (DΠcan D− )∗ v , and let the function ω satisfy the condition ∗ 1 ω′ (t ) = ⟨u(t ), (D− ∗ Πcan p)(t )⟩ − ⟨(DΠcan G− µ q)(t ), v(t )⟩,
t ∈ I.
We derive d dt
(⟨u(t ), v(t )⟩ + ω(t )) = ⟨u′ (t ), v(t )⟩ + ⟨u(t ), v ′ (t )⟩ + w′ (t ) = ⟨(DΠcan D− )′ (t )u(t ), v(t )⟩ + ⟨u(t ), (DΠcan D− )∗′ (t )v(t )⟩ = 2⟨(DΠcan D− )′ (t )u(t ), v(t )⟩ = 2⟨(DΠcan D− )′ (t )(DΠcan D− )(t )u(t ), (DΠcan D− )∗ (t )v(t )⟩ = 2⟨(DΠcan D− )(t )(DΠcan D− )′ (t )(DΠcan D− )(t ) u(t ), v(t )⟩ = 0,
t ∈ I,
=0
such that
⟨u(t ), v(t )⟩ + ω(t ) = constant,
t ∈ I.
(41)
This leads to the following theorem: Theorem 4.3. Let the DAE (32) be regular with index µ and characteristic values (18). Then, for each solution pair u and v of the corresponding IERODES (35) and (39), which proceed within the associated invariant subspaces im DΠcan D− and im A∗ Π∗ can A∗ − , respectively, the identity (41) is valid. In particular, for the homogeneous case with q = 0, p = 0, the classical Lagrange identity,
⟨u(t ), v(t )⟩ = constant , comes true.
t ∈I
(42)
R. März / Journal of Computational and Applied Mathematics (
)
–
13
Furthermore, if u, v are solutions of the IERODEs (35) and (37) residing in the associated invariant subspaces, and η = Γd u, ζ = Γ∗ d v are the corresponding solutions of the EUODEs (36) and (38) built by consistent bases, then the following identity becomes valid:
⟨η(t ), ζ (t )⟩ = ⟨Γd (t )u(t ), Γ∗ d (t )v(t )⟩ = ⟨Γd (t )− Γd (t )u(t ), v(t )⟩ = ⟨(DΠcan D− )(t )u(t ), v(t )⟩ = ⟨u(t ), v(t )⟩, t ∈ I. In [4], Lyapunov regularity of an arbitrary-index DAE is introduced and analyzed by means of its spectrum preserving EUODEs. In particular, the following statement is verified in [4]: Theorem 4.4. Let the DAE (32) be regular with index µ and characteristic values (18). Then, if ADΠcan , Πcan (AD)− , Γd DΠcan , and Πcan D− Γd− remain bounded, the DAE and its adjoint are Lyapunov regular at the same time and the Perron identity is valid. It is demonstrated in [4] by examples that, in essence, here one cannot do unless those strong boundedness conditions hold. Even though the statement Theorem 3.6(3) sounds promising it is not really helpful in this respect, since, in general, the spectrum preserving bases Γd and Γ∗ d do not fulfill the consistency condition of Theorem 4.2(5). 5. Standard form DAEs Now we turn to the standard form DAE and its adjoint (cf. [6]), Ex′ + Fx = q,
(43)
−(E y) + F y = p. ∗
′
∗
(44)
We assume that the time-varying subspace ker E (t ) ⊆ K , t ∈ I, is a C - subspace, which means that the orthoprojector function E + E is continuously differentiable. Then, E has constant rank. We refer to [11,13] for elaborated discussions concerning this constant-rank condition and the possibilities of avoiding this condition in the class of quasi-regular DAEs. Here we deal with regular DAEs only. Regularity, the tractability index and the characteristic values of a standard form DAE (43) are given by means of any proper factorization E =: AD, A ∈ C (I, L(Kn , Km )), D ∈ C 1 (I, L(Km , Kn )) such that ker E (t ) = ker D(t ), t ∈ I, and the transversality condition (16) is valid, see [11, Section 2.7]. At this place, it should be noted, that refactorizations do not change the index and the characteristic values, thus regularity. The canonical projector function Πcan is also invariant with respect to refactorizations. Since ker E is a C 1 -subspace, there are proper factorizations E =: AD. One can simply use A = E, D = P, with a projector function P along ker E, for instance, P = E + E, as applied already in [14]. Then we rewrite the standard form DAE (43) as DAE with properly stated leading term, m
1
A(Dx)′ + (F − AD′ )x = q.
(45)
Each solution x of the DAE (43) is a solution of (45) and vice versa, and the Lyapunov spectra of the DAE (43) and the reformulated version (45) coincide.3 On the other hand, the IERODE of (45) and its solutions actually depend on the chosen factorization. Therefore, the corresponding representation of the DAE solutions x also depends on the factorization. The IERODEs of a standard form DAE (43) are given via proper factorizations of the leading coefficient E. Each factorization generates a special IERODE. Owing to Theorem 3.4, the EUODEs arise as condensed IERODEs in the form (28), (31). In turn, various bases Γd∗ of im (DΠcan D− )∗ can be chosen for that purposes. We emphasize that all these IERODEs and EUODEs are not affected by derivatives of the homogeneity q. Let U be a continuously differentiable matrix function such that its columns form an orthonormal basis of the solution space im Πcan , i.e., im U = im Πcan ,
U ∗ U = Id .
(46)
Choosing Γd = U ∗ Πcan D− and Γd− = DU one accomplishes a spectrum preserving EUODE, see Theorem 3.6 and [4, Theorem 6]. In this case, straightforward computations lead to the form 1 ∗ −1 η′ + { U ∗ Πcan D− D′ P0 U + U ∗ Πcan (P0 U )′ + U ∗ Πcan G− µ BU } η = U Πcan Gµ q.
(47)
Each solution x of the DAE (43) with q = 0 satisfies x = Πcan x and the corresponding solution of the associated EUODE (47) is η = Ux. x and η show the same Lyapunov exponents. 1 Owing to [11, Theorem 2.21] the expression Πcan G− µ remains unchanged under refactorizations of the leading coefficient E. P0 is a projector function onto ker E and also independent of the factorization. It turns out that the EUODE (47) associated
3 We suppose classical solutions as introduced e.g., in [14,11,13]. Note that further smoothness of the solutions can be ensured by sufficiently smooth coefficients.
14
R. März / Journal of Computational and Applied Mathematics (
)
–
with the standard form DAE (43) depends on the chosen basis U, but it appears to be widely independent of the special factorization E = AD. In particular, if ker E (t ) is time-invariant, then one can choose a constant factor D so that the term containing D′ disappears, and the EUODE (47) becomes actually independent of the special factorization. This may also happen if ker E (t ) varies with t, as the following example shows. Example 5.1. We consider the DAE in so-called reduced form E1 x′ +F1 x = q1 ,
(48)
F2 x = q2 ,
(49)
with E1 and F2 having constant ranks r and m − r, respectively, and
E1 F2
remains nonsingular. We look at the nontrivial
case 0 < r < m. Then the DAE is regular with index one, and its dynamical degree of freedom equals d = r. The canonical projector function Πcan projects pointwise onto ker F2 along ker E1 . Various representations can be used, e.g.,
Πcan =
−1 E1 0
E1 F2
= U (E1 U )−1 E1 = E1− E1 = Im−d − F2− F2 ,
with an orthonormal basis U of im Πcan = ker F2 and the generalized inverses E1− , F2− which satisfy the properties E1 E1− = Id , E1− E1 = Πcan , F2 F2− = Im−d , F2− F2 = Im − Πcan . Here E1 U remains nonsingular. We apply the simple proper factorization
E=
E1 0
=
Ir E =: AD 0 1
leading to DΠcan D− = E1 Πcan E1− = E1 E1− = Ir and the IERODE, u′ + (F1 − E1′ )E1− u = q1 − (F1 − E1′ )F2− q2 ,
(50)
for the component u = E1 x. We emphasize again that no derivatives of q encroach on (50). With the standard orthonormal basis Γd∗ = Id of im (DΠcan D− )∗ and η = Γd u = u, the corresponding EUODE coincides with the IERODE (50). In contrast, the special choice Γd = U ∗ Πcan E1− , Γd− = E1 U, cf., (46), results in the EUODE
η′ + U ∗ Πcan U ′ η + U ∗ Πcan E1− F1 U η = U ∗ Πcan E1− (q1 − (F1 − E1′ )F2− q2 ),
(51)
which preserves for q = 0 the spectrum of the DAE (48), (49). Regarding that U ∗ Πcan E1 = (E1 U )−1 and U ∗ Πcan = (E1 U )−1 E1 the homogeneous version of this EUODE reads simply −
η′ + (E1 U )−1 E1 U ′ η + (E1 U )−1 F1 U η = 0.
(52)
Remark 5.2. We point to a differing use of the abbreviation EUODE in [15] for strangeness-free DAEs in reduced form (48), (49). Though in [15] the main focus is directed to the homogeneous case, in the context of EUODEs the inhomogeneous case should not be disregarded. As in Example 5.1, let U be an orthonormal basis of ker F2 such that UU ∗ represents the orthoprojector onto ker F2 . Applying the decomposition x = UU ∗ x + (I − UU ∗ )x = Uz + (I − UU ∗ )x, with z = U ∗ x, one can extract from the DAE an implicit ODE with respect to the new variable z, namely E1 Uz ′ + E1 U ′ z + F1 Uz = q1 − E1 F2+ F2 (F2+ q2 )′ − E1 (F2+ F2 )′ F2+ q2 − F1 F2+ q2 .
(53)
In [15] the ODE (53) is further scaled by a nonsingular matrix function H such that the product HE1 U becomes upper triangular. The scaled version is then said to be an implicit essential underlying ODE, and the abbreviation EUODE is applied, too. In general, this is no longer consistent with the basic idea of an explicit ODE of minimal size remaining unaffected by derivatives of q, cf. [1–3]. Namely, the regular implicit ODE (53) as well as its scaled versions are affected by the derivative term (F2+ q2 )′ . The responsible term E1 F2+ F2 disappears only if the original DAE features the very special property ker E1 = (ker F2 )⊥ . If so, one has UU ∗ = Πcan , and (51) and (53) are equivalent. Solely in this special case, an EUODE in the context of [15] features the basic EUODE property to be unaffected by derivatives of q. Apart from that, we observe that for q = 0, the implicit ODE (53) differs from the explicit ODE (52) by multiplication by E1 U only. Thus these homogeneous ODEs feature the same solutions and Lyapunov spectra. In [15], Lyapunov regularity is traced back to the mentioned implicit essential ODE with upper triangular leading coefficient. Though the basic notions are differing, owing to the various boundedness conditions the resulting assertions concerning the Perron identity seem to be consistent. The DAE (44) is obviously out of the scope of a standard form DAE. Often, additionally supposing that E and y are continuously differentiable, one turns to the standard form DAE
− E ∗ y′ + (F ∗ − E ∗ ′ )y = p.
(54)
R. März / Journal of Computational and Applied Mathematics (
)
–
15
Having said that, applying the proper factorization E ∗ = (AD)∗ = D∗ A∗ , to Eq. (54) or to Eq. (44) leads to
− D∗ (A∗ y)′ + (F ∗ − D∗ ′ A∗ )y = p,
(55)
which is the correct adjoint counterpart of the DAE (45). Replacing both DAEs (43) and (44) by proper versions (45) and (55), one can apply all results mentioned in Sections 3 and 4. For adjoint pairs (43) and (44), the material from Section 4 naturally applies, if consistent factorizations yielding (45) and (55) are used. The next statement represents a consequence of [4, Theorem 3(1)] concerning so-called factorization-adjoint pairs of DAEs. It specifies and extends [8, Theorem 3.5]. Theorem 5.3. For the standard form DAE (43) with sufficiently smooth coefficients the following comes true: (1) If the DAE (43) is regular with index µ and characteristic values (18), then so is its adjoint (44) and vice versa. In particular, an adjoint pair shares in the dynamical degree of freedom d. (2) If the DAEs (43) and (44) are regular, then they possess EUODEs being adjoint each to other in the classical sense. As already demonstrated by Example 4.1, the classical Lagrange identity does not come true for the solutions DAEs (43) and (44), however, we can take use of the identity (34) via factorizations. Let x and y be solutions of the homogeneous DAEs (43) with q = 0 and (44) with p = 0. Then we have
⟨E (t )x(t ), y(t )⟩ = ⟨A(t )D(t )x(t ), y(t )⟩ = ⟨D(t )x(t ), A(t )∗ y(t )⟩ = constant, which coincides with the generalized Lagrange identity obtained in [5,10]. Remark 5.4. Adjoint pairs of index-1 DAEs (43), (44) are studied in detail in [16] by means of classical factorizations E = EP, in which P is a continuously differentiable projector-valued function along ker E. Let λ1 ≤ · · · ≤ λd and λ∗ 1 ≥ · · · ≥ λ∗ d be the Lyapunov spectra of the DAE (43) and its adjoint (44). The index-1 DAE (43) is called Lyapunov regular if the Perron identity, i.e.,
λi + λ∗ i = 0 ,
i = 1, . . . , d,
(56)
1 ′ is valid, see [16, Definition 3.2]. If the coefficients E , F , further G− 1 , and P , P are bounded, the DAE (43) is shown to be Lyapunov regular at the same time as the associated IERODE is so.
The mentioned differing points of view concerning Lyapunov regularity of DAEs (43) in the literature call for further careful studies to clarify the matter and to reach a consensus on the most adequate notion and natural boundedness condition.
6. Conclusions Linear regular DAEs of arbitrary index, both with properly involved derivative and in standard form, have been analyzed by means of projector based decouplings generating IERODEs and also by transformations into a special form uncovering the EUODEs. EUODEs can be derived from IERODEs by special condensing transformations up to size d, if d denotes the dynamical degree of freedom of the DAE. In case of higher-index DAEs, the size of the IERODEs is larger than d, and only that component of the IERODE flow which is located within a certain d-dimensional time-varying invariant subspace of the IERODE is associated with the DAE. Regarding adjoint pairs of DAEs, the IERODE of the adjoint DAE has been fully described in terms of the IERODE of the original DAE. Generalized Lagrange identities have been derived from adjoint pairs of DAEs. They comprise relations of associated solutions of both IERODEs and EUODEs. Both concepts, the IERODEs and the EUODEs have been pointed out to be very useful for analyzing the stability behavior of the DAE flow. In particular, spectrum preserving EUODEs have been found beneficial. General conditions ensuring the Perron identity for DAEs have been given. However, owing to differing notions of Lyapunov regularity in the literature, some further effort is required to clarify the matter. It has been shown that the IERODEs of DAEs of arbitrary high index do not depend on derivatives of the right-handside of the DAE, which, in particular, allows to rely on well-posed inner boundary value problems although the overall boundary value problem for the DAE is ill-posed. The IERODEs are shown to be the only those genuine ODEs. The EUODEs as transformed versions of the IERODEs reflect this property. These facts provide the basis of a precise input–output sensitivity analysis.
16
R. März / Journal of Computational and Applied Mathematics (
)
–
List of symbols and abbreviations
K L(Ks , Kn ) C (I , X ) C 1 (I, X ) 1 CM (I, X ) ∗ K K− K+ K∗− K−∗ K−∗− ker K im K
⟨·, ·⟩ |·| ⊕ DAE ODE IVP IERODE EUODE
Set of real numbers R and set of complex numbers C Set of K-valued n × s—matrices and linear operators from Ks to Kn Space of continuous functions mapping I into X Space of continuously differentiable functions mapping I into X {x ∈ C (I, X ) : Mx ∈ C 1 (I, Y ), with M ∈ L(X , Y )} Adjoint matrix Generalized inverse, KK − K = K , K − KK − = K − Moore–Penrose inverse [K ∗ ]− [K − ]∗ [[K − ]∗ ]− Nullspace (kernel) of K Image (range) of K Scalar product in Km Vector and matrix norms Direct sum Differential–algebraic equation Ordinary differential equation Initial value problem Inherent explicit regular ODE Essential underlying ODE
References [1] U.M. Ascher, L.R. Petzold, Projected implicit runge-kutta methods for differential-algebraic equations, SIAM J. Numer. Anal. 28 (4) (1991) 1097–1120. [2] K. Balla, Differential algebraic equations and their adjoints (Dissertation), Doctor of The Hungarian Academy of Sciences, Hungarian Academy of Sciences, Budapest, 2004. [3] K. Balla, Vu Hoang Linh, Adjoint pairs of differential-algebraic equations and Hamiltonian systems, Appl. Numer. Math. 53 (2005) 131–148. [4] Vu Hoang Linh, R. März, Adjoint pairs of differential-algebraic equation and their Lyapunov exponents, J. Dynam. Differential Equations (2015). http://dx.doi.org/10.1007/s10884-015-9474-6 (in press). [5] K. Balla, R. März, Transfer of boundary conditions for DAEs of index 1, SIAM J. Numer. Anal. 33 (6) (1996) 2318–2332. [6] S.L. Campbell, N.K. Nichols, W.I. Terell, Duality, observability, and controllability for linear time-varying descriptor systems, Circuits Systems Signal Process. 10 (4) (1991) 455–470. [7] 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. [8] P. Kunkel, V. Mehrmann, Formal adjoints of linear DAE operators and their role in optimal control, Electron. J. Linear Algebra 22 (2011) 672–693. [9] R. Lamour, R. März, E. Weinmüller, in: A. Ilchman, T. Reis (Eds.), Boundary-Value Problems for Differential-Algebraic Equations: A Survey, in: Surveys in Differential-Algebraic Equations III, Springer-Verlag, Berlin, Heidelberg, 2015, pp. 177–309. [10] T. Petry, On the stability of the Abramov transfer for differential-algebraic equations of index 1, SIAM J. Numer. Anal. 35 (1) (1998) 201–216. [11] R. Lamour, R. März, C. Tischendorf, Differential-Algebraic Equations: A Projector Based Analysis, in: A. Ilchman, T. Reis (Eds.), Differential-Algebraic Equations Forum, Springer-Verlag, Berlin, Heidelberg, 2013. [12] K. Balla, R. März, A unified approach to linear differential algebraic equations and their adjoints, Z. Anal. Anwend. 21 (3) (2002) 783–802. [13] R. März, in: A. Ilchman, T. Reis (Eds.), Differential-Algebraic Equations from a Functional-Analytic Viewpoint: A Survey, in: Surveys in DifferentialAlgebraic Equations II, Springer-Verlag, Berlin, Heidelberg, 2015, pp. 163–285. [14] E. Griepentrog, R. März, Differential-Algebraic Equations and their Numerical Treatment, in: Teubner-Texte zur Mathematik, vol. 88, BSB B.G. Teubner Verlagsgesellschaft, Leipzig, 1986. [15] Vu Hoang Linh, V. Mehrmann, Spectra and leading directions for linear DAEs, in: L.T. Biegler, S.L. Campbell, V. Mehrmann (Eds.), Control and Optimization with Differential-Algebraic Constraints, SIAM, Philadelphia, 2012, pp. 59–78. [16] Cong Nguyen Dinh, Nam. Hoang, Lyapunov regularity of linear differential algebraic equations of index 1, Acta Math. Vietnam. 29 (1) (2004) 1–21.
)
–
Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
New answers to an old question in the theory of differential–algebraic equations: Essential underlying ODE versus inherent ODE Roswitha März Humboldt University of Berlin, Institute of Mathematics, D-10099 Berlin, Germany
article
info
Article history: Received 15 December 2015 Received in revised form 29 August 2016 Keywords: Differential–algebraic equation Essential underlying ODE Inherent ODE Adjoint pairs Lagrange identity Lyapunov spectrum Lyapunov regularity
abstract In the context of linear differential–algebraic equations (DAEs) one finds different associated explicit ordinary differential equations (ODEs), among them essential underlying and inherent explicit regular ones, abbreviated: EUODEs and IERODEs. EUODEs have been introduced in 1991 for index-2 DAEs in Hessenberg form by means of special transformations. IERODEs result within the framework of the projector based decoupling. Each such explicit ODE is occasionally considered to rule the flow of the DAE. The question to which extend EUODEs and IERODEs are related to each other has been asked promptly after 1991. For index-2 Hessenberg-form DAEs, answers have been given in 2005, saying that EUODEs represent somehow condensed IERODEs. Recently, EUODEs have been indicated for general arbitrary-index DAEs and it has been proved that they are condensed IERODEs. The understanding of the relation between the IERODE and the EUODEs enables to uncover the stability behavior of the DAE flow. In the present paper we show that both, the IERODEs and EUODEs of a DAE with arbitrary high index do not at all depend on derivatives of the right-hand side. We consider adjoint pairs of DAEs and provide generalizations of the classical Lagrange identity. Furthermore, we address Lyapunov spectra and Lyapunov regularity. © 2016 Elsevier B.V. All rights reserved.
1. Introduction In the context of linear differential–algebraic equations (DAEs) one finds different associated explicit ordinary differential equations (ODEs), among them essential underlying ODEs (EUODEs) and inherent explicit regular ODEs (IERODEs). EUODEs have been distinguished by Ascher and Petzold [1] for index-2 DAEs in Hessenberg form by means of special transformations. IERODEs result within the framework of the projector based decoupling. Both EUODEs and IERODEs are occasionally considered to rule the flow of the DAE. How are they related to each other? This question has been asked promptly after 1991. First answers have been given by Balla and Vu Hoang Linh [2,3] pointing out that, for index-2 Hessenberg-form DAEs and general index-1 DAEs, an EUODE represents a condensed IERODE. Recently, EUODEs associated with general arbitrary-index DAEs have been distinguished in [4]. Also in the general case, the EUODEs can be seen as condensed IERODEs. Any regular linear differential–algebraic equation with properly stated leading term features a unique IERODE living in the given space. In contrast, there are several EUODES living in a transformed space with possible minimal dimension. We show that the IERODEs and EUODEs are the only associated explicit ODEs which do not at all depend on derivatives of
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.cam.2016.09.039 0377-0427/© 2016 Elsevier B.V. All rights reserved.
2
R. März / Journal of Computational and Applied Mathematics (
)
–
the right-hand side. This offers the background of a preciser sensitivity analysis and allows to state boundary conditions accurately. The understanding of the relation between the IERODE and the EUODEs enables to uncover stability properties of the DAE flow such as Lyapunov regularity etc. We particularly concentrate on relevant versions of the Lagrange identity. This should be helpful in the context of numerical boundary value problems, sensitivity analysis, optimization, and control problems, see [1,5–10]. We investigate DAEs with properly involved derivative A(t )(Dx)′ (t ) + B(t )x(t ) = q(t ),
t ∈I
(1)
and standard form DAEs E (t )x′ (t ) + F (t )x(t ) = q(t ),
t ∈ I,
(2)
with coefficients E , F ∈ C (I, L(Km , Km )), A ∈ C (I, L(Kn , Km )),
D ∈ C (I, L(Km , Kn )),
B ∈ C (I, L(Km , Km )),
with K = R and K = C in the real and complex versions, respectively. The interval I ⊆ R is arbitrary, possibly infinite. We drop the argument t whenever reasonable. Then the given relations are meant pointwise for all t ∈ I. The paper is arranged as follows. In Section 2 we recap known results for index-2 DAEs in Hessenberg form already by means of the notation of the general approach in the following part in order to motivate this approach and to explicate the questions to be considered. In Section 3 we describe the structure of a regular DAE (1). In particular, we deal with its IERODE and EUODEs and show that these are the only associated q-derivative-free explicit ODEs. Adjoint pairs of DAEs are considered in Section 4. We show that, even though the IERODEs of the original DAE and its adjoint equation are not necessarily adjoint to each other, the solutions associated to the DAEs satisfy a Lagrange identity. We specify the results for standard form DAEs in Section 5. General conclusions are drawn in Section 6. 2. Recapping index-2 Hessenberg systems The system comprising the m = m1 + m2 equations
= q1 , = q2 ,
x′1 +B11 x1 + B12 x2 B21 x1
(3) (4)
is said to be a DAE in Hessenberg form of size 2, if the product B21 B12 remains everywhere nonsingular. This DAE is known to have differentiation index 2 as well as tractability index 2. Writing
′
I 0
B11 I 0 + x B21
B12 x = q, 0
(5)
=D
=A
one puts the DAE in the form (1). Owing to the nonsingularity of the product B21 B12 , the direct sum decomposition ker B21 ⊕ im B12 = Km1
(6)
is valid and the projector-valued function Ω given by
Ω := B12 B− 12 ,
−1 B− B21 , 12 := (B21 B12 )
(7)
projects pointwise onto im B12 along ker B21 . The usual approach to DAEs consists in extracting explicit ODEs with respect to x or x1 from a derivative array system. The resulting so-called underlying ODEs depend on the special way they are provided. For instance, differentiating Eq. (4) and then replacing the derivative x′1 by (3) results in the equation
Ax1 − Bx2 = q′2 − B21 q1 ,
(8)
with A := −B21 B11 + B′21 , B := B21 B12 . Differentiating now (8), replacing again the derivative x′1 by (3), multiplying by B−1 and rearranging terms finally leads to the ODE x′1 = −B11 x1 − B12 x2 + q1 , x′2 = −B−1 (AB11 − A′ )x1 − B−1 (AB12 + B′ )x2 + B−1 (A q1 − (q′2 − B21 q1 )′ ). The DAE flow is embedded into the m-dimensional flow of the underlying ODE. Observe that derivatives of components of q encroach on this ODE.
R. März / Journal of Computational and Applied Mathematics (
)
–
3
Furthermore, using (8) to replace x2 in Eq. (3), we obtain the ODE with respect to x1 , x′1 = −((I − Ω )B11 + B12 B−1 B′21 )x1 + (I − Ω )q1 + B12 B−1 q′2 , which is affected by the derivative q′2 . In contrast, an associated explicit ODE of minimal size, which is not at all affected by derivatives of q, has been obtained by special transformations in [1]. We follow the lines of [1], but use quite different notations. Put d := m1 − m2 . One finds a matrix function Γd ∈ C 1 (I, L(Km1 , Kd )) such that the columns of Γd∗ form a basis of ker B∗12 = (im B12 )⊥ . This implies ker Γd = im B12 , and the matrix function
Γd
remains nonsingular.
B21
We denote and verify
Γd := −
Γd
−1
Id , 0
B21
Γ d Γ d = Id −
0
Γd B21
ker Γd− Γd = im B12 ,
Γd− = Id ,
Γd Γd− Γd = Γd ,
Γd− Γd Γd− = Γd− ,
im Γd− Γd = ker B21 .
This actually means that Γd− is a generalized inverse of Γd and Γd− Γd = I − Ω . Furthermore, one can check that
Γd
−1
B21
0 = B12 (B21 B12 )−1 . I
Defining new variables
η = Γd x1 ,
x1 =
Γd
−1
B21
−1 η Γ 0 − = Γd η − d = Γd− η − B12 (B21 B12 )−1 q2 , −q2 B21 q2
one obtains an explicit ODE with respect to η,
η′ + Γd B11 Γd− η − Γd′ Γd− η = Γd q1 + (Γd′ − Γd B11 )B12 (B21 B12 )−1 q2 ,
(9)
which is referred to as essential underlying ODE (EUODE) in order to distinguish it from other underlying ODEs, emphasizing that it has minimal size d and yields a well-posed boundary-value problem with respect to x1 , see [1]. Below we will indicate such EUODEs for each arbitrary regular DAE. Having x1 one computes the component x2 by x2 = (B21 B12 )−1 B21 {q1 − (Γd− η)′ + (B12 (B21 B12 )−1 q2 )′ − B11 (Γd− η − B12 (B21 B12 )−1 q2 )}, which uncovers the structure of the DAE flow. The dynamical degree of freedom equals d. Since there are different possibilities to choose a basis of ker B∗12 , the essential underlying ODE depends on the specially chosen basis. As suggested in [1], one can restrict the options to Γd with orthonormal rows. The projector based approach (e.g. [11]) relies on projector functions such as Ω and decouples the original DAE as it is, devoid of any transformations. Regarding the properties B21 = B21 Ω and Ω B12 = B12 , we immediately derive from Eq. (3) that
(I − Ω )x′1 + (I − Ω )B11 x1 = (I − Ω )q1 ,
(10)
and from (4) that
Ω x1 = B12 (B21 B12 )−1 q2 , which leads to x1 = (I − Ω )x1 + B12 (B21 B12 )−1 q2 , x2 = (B21 B12 )−1 B21 {q1 − ((I − Ω )x1 )′ + (B12 (B21 B12 )−1 q2 )′ − B11 (I − Ω )x1 − B11 B12 (B21 B12 )−1 q2 }. Inserting the expression for x1 into the relation (10) we arrive at the explicit ODE with respect to the only component u = ( I − Ω ) x1 , u′ − (I − Ω )′ u + (I − Ω )B11 (I − Ω )u = (I − Ω )q1 + (I − Ω )′ B12 (B21 B12 )−1 q2 − (I − Ω )B11 B12 (B21 B12 )−1 q2 , (11) which is called inherent explicit regular ODE (IERODE) and which resides in Km1 . The varying d-dimensional subspace im (I − Ω ) ⊂ Km1 is an invariant subspace of the IERODE, that is, if u∗ is a solution of (11) and there is a t¯ ∈ I with u(t¯) ∈ im (I − Ω (t¯)), then one has u(t ) ∈ im (I − Ω (t )) for all t ∈ I. This is the basic property of the IERODE in matters of the DAE structure. The IERODE is uniquely defined by the DAE coefficients. It has size m1 ≥ d, but solely its flow within the d-dimensional subspace im (I − Ω ) is relevant for the DAE flow.
4
R. März / Journal of Computational and Applied Mathematics (
)
–
The so-called canonical projector function of the DAE (3), (4) reads (cf. [11, p. 107])
Πcan =
I −Ω
′ −B− 12 (B11 − Ω )(I − Ω )
0 0
and it has constant rank d together with the projector function (I − Ω ). One can check that all solutions of the homogeneous DAE with q = 0 have the form
(I − Ω )x1 = Πcan x, ′ −B12 (B11 − Ω )(I − Ω )x1
x=
−
and im Πcan (t ) represents the d-dimensional time-varying subspace of consistent values or solution space of the homogeneous DAE. In the light of our notation the relations
η = Γd x1 = Γd Γd− Γd x1 = Γd (I − Ω )x1 = Γd u, u = (I − Ω )x1 = Γd− η, become obvious. Multiplying the EUODE by Γd− results in the IERODE and, conversely, multiplying the IERODE by Γd yields the EUODE. In this sense, each EUODE is nothing else than a condensed IERODE. This fact was first pointed out in [3] for index-2 DAEs in Hessenberg form and recently verified for general regular DAEs in [4]. The adjoint DAE to the index-2 Hessenberg form DAE (3), (4),
= p1 , = p2 ,
−y′1 +B∗11 y1 + B∗21 y2 B∗12 y1 or written in the form (1),
′ ∗ B11 I I 0 y + − B∗12 0 =A∗
B∗21 y = p, 0
(12)
=D ∗
has also index 2 and the dynamical degree of freedom d. Its canonical projector function reads
Π∗ can =
I − Ω∗
B∗21− (−B∗11 − Ω ∗ )(I − Ω ∗ ) ′
0 . 0
It holds that DΠcan D− = I − Ω ,
A∗ Π∗ can A∗ − = I − Ω ∗ = (DΠcan D− )∗ ,
∗ however, the canonical projector function Π∗ can differs from Πcan , except for quite artificial special cases. Letting q = 0 and p = 0, the associated homogeneous IERODEs of the DAE and its adjoint become
u′ − (I − Ω )′ u + (I − Ω )B11 (I − Ω )u = 0
(13)
− v ′ + (I − Ω ∗ )′ v + (I − Ω ∗ )B∗11 (I − Ω ∗ )v = 0.
(14)
and As first pointed out in [2,3], the IERODEs (13) and (14) form a classical adjoint pair, only if the projector function Ω is constant, or, equivalently, if the associated subspaces im B12 and ker B21 do not vary with t. It seems, this fact has been considered as drawback of the IERODE, but wrongly. As it is observed above, only solution components u = (I − Ω )u and v = (I − Ω ∗ )v are relevant for the DAEs, and nevertheless for these components the classical Lagrange identity takes place1 : d dt
⟨u, v⟩ = ⟨u′ , v⟩ + ⟨u, v ′ ⟩ = 2⟨(I − Ω )′ u, v⟩ = 2⟨(I − Ω )′ (I − Ω )u, (I − Ω ∗ )v⟩ = 2⟨(I − Ω )(I − Ω )′ (I − Ω )u, v⟩ = 0.
Below we will show a corresponding general property for each arbitrary pair of adjoint regular DAEs and their IERODEs. On the other hand, choosing a condensing matrix function Γd ∈ C 1 (I, L(Km1 , Kd )), such that im Γd∗ = ker B∗12 , and determining the generalized inverse Γd− so that
Γd− Γd = I − Ω ,
Γd Γd− = Id ,
1 Here and later the following argument is used: If P is a differentiable projector function on an interval I, then P (I − P ) = 0 implies P ′ (I − P ) − PP ′ = 0, thus PP ′ P = P ′ (I − P )P = 0.
R. März / Journal of Computational and Applied Mathematics (
)
–
5
∗∗ we have at the same time im Γd− = ker B21 , i.e., Γd− serves as a basis of ker B21 = ker B21 . Then we may simultaneously condense both IERODEs to EUODEs, that is, ′
η′ + (Γd B11 Γd− + Γd Γd− )η = 0 and ′
−ζ ′ + (Γd− ∗ B∗11 Γd∗ + Γd− ∗ Γd∗ )ζ = 0, which are actually adjoint each to other, without any additional restrictions. Hereby, the IERODE of the adjoint is condensed by means of the special basis Γd− of ker B21 , and
v = (I − Ω ∗ )v = Γd∗ Γd− ∗ v = Γd∗ Γd ζ ,
ζ = Γd− ∗ v.
This result is consistent with the claim (cf. [7, Section 4.3]) that the EUODE of the adjoint system is the same as the adjoint of the EUODE of the original system. However, emphasize once again that the EUODEs depend on the specially chosen Γd and Γ∗ d . In particular, one obtains an EUODE of the adjoint DAE by means of any basis Γ∗∗d of ker B21 . As pointed out in [3], the above assertion is valid only, if the basis applied for the EUODE of the adjoint DAE is consistent with the basis chosen for the original DAE, that means
Γ∗ d = Γd− ∗ . 3. The structure of regular DAEs with properly involved derivative In this section we deal with DAEs of the form (1), that is, A(Dx)′ + Bx = q.
(15)
Let the time-varying subspaces ker A(t ) ⊆ Kn
im D(t ) ⊆ Kn ,
and
t ∈ I,
1
be C - subspaces. and let the transversality condition ker A(t ) ⊕ im D(t ) = Kn ,
t ∈ I,
(16)
be valid, which means that the DAE shows a properly stated leading term. The decomposition (16) determines the so-called border projector function R ∈ C 1 (I, L(Kn , Kn )) by ker R(t ) = ker A(t ),
im R(t ) = im D(t ),
t ∈ I.
(17)
1
Since both involved subspaces are C -subspaces, the projector function R is actually continuously differentiable. We use the projector based analysis to uncover the basic structure of a regular DAE with arbitrarily high tractability index and refer to [11] for details and for general relations with other index notions as well. Let the DAE (15) be regular in the sense of [11, Definition 2.25]. Recall that regularity is supported by several constantrank requirements yielding the tractability index µ ∈ N and the characteristic values 0 ≤ r0 ≤ · · · ≤ rµ−1 < rµ = m,
d=m−
µ−1 (m − ri ),
(18)
i =0
of the DAE. Regularity is formally determined by means of admissible projector functions P0 , . . . , Pµ−1 ∈ C (I, L(Km , Km )) associated with the construction of admissible matrix functions sequences starting from G0 := AD and ending up with a nonsingular Gµ , see [11, Definition 2.6]. The tractability index is consistent with the Kronecker index of regular matrix pencils. For a regular matrix pencil, the characteristic values ri provide a complete description of the formal structure of the corresponding Weierstraß–Kronecker form. In particular, d is the dimension of the dynamical part [11, Section 1]. We use the further matrix functions Q0 := I − P0 ,
Π0 := P0 ,
Qi := I − Pi ,
Πi := Πi−1 Pi ∈ C (I, L(Km , Km )),
DΠ0 D , . . . , DΠµ−1 D −
−
i = 1, . . . , µ − 1,
∈ C (I, L(K , K )), 1
n
n
with the pointwise determined generalized inverse D− such that DD− D = D,
D− DD− = D− ,
DD− = R,
D− D = P0 .
The products Πi and DΠ iD are projector-valued functions again (see [11, Proposition 2.7]). −
(19)
6
R. März / Journal of Computational and Applied Mathematics (
)
–
The sequence of admissible projector functions serves as tool for the decoupling of the DAE itself and the decomposition of the solution x into their characteristic parts, see [11, Section 2.4]. In particular, the component u = DΠµ−1 x satisfies the explicit regular ODE residing in Kn , 1 − −1 u′ − (DΠµ−1 D− )′ u + DΠµ−1 G− µ BΠµ−1 D u = DΠµ−1 Gµ q.
(20)
The components v0 = Q0 x, v1 = Π0 Q1 x, . . . , vµ−1 = Πµ−2 Qµ−1 x satisfy the triangular subsystem involving several differentiations for µ > 1,
0
N01
0
··· .. . .. .
I
M01 I
+
N0,µ−1
0 (Dv1 )′ .. . Nµ−2,µ−1 ′ (Dvµ−1 ) 0 ··· M0,µ−1 v H L0 0 0 .. .. v1 H 1 L1 . . D− u = + .. .. .. q. .. . . . . Mµ−2,µ−1 Lµ−1 Hµ−1 vµ−1 I .. .
(21)
The subspace im DΠµ−1 is an invariant subspace for the ODE (20). The components v0 , v1 , . . . , vµ−1 remain within their subspaces im Q0 , im Πµ−2 Q1 , . . . , im Π0 Qµ−1 , respectively. The structural decoupling is associated with the decomposition x = D− u + v0 + v1 + · · · + vµ−1 . All coefficients are continuous and explicitly given in terms of an admissible matrix function sequence as
N01 := −Q0 Q1 D− N0j := −Q0 P1 · · · Pj−1 Qj D− ,
j = 2, . . . , µ − 1,
Ni,i+1 := −Πi−1 Qi Qi+1 D , −
Nij := −Πi−1 Qi Pi+1 · · · Pj−1 Qj D− ,
j = i + 2, . . . , µ − 1, i = 1, . . . , µ − 2,
M0j := Q0 P1 · · · Pµ−1 Mj DΠj−1 Qj ,
j = 1, . . . , µ − 1,
Mij := Πi−1 Qi Pi+1 · · · Pµ−1 Mj DΠj−1 Qj ,
j = i + 1, . . . , µ − 1, i = 1, . . . , µ − 2,
1 L0 := Q0 P1 · · · Pµ−1 G− µ , 1 Li := Πi−1 Qi Pi+1 · · · Pµ−1 G− µ ,
Lµ−1 :=
i = 1, . . . , µ − 2,
1 Πµ−2 Qµ−1 G− µ ,
H0 := Q0 P1 · · · Pµ−1 K Πµ−1 , Hi := Πi−1 Qi Pi+1 · · · Pµ−1 K Πµ−1 ,
i = 1, . . . , µ − 2, v
Hµ−1 := Πµ−2 Qµ−1 K Πµ−1 , in which 1 K := (I − Πµ−1 )G− µ Bµ−1 Πµ−1 +
µ−1 (I − Πl−1 )(Pl − Ql )(DΠl D− )′ DΠµ−1 , l =1
Mj :=
j −1
(I − Πk ){Pk D− (DΠk D− )′ − Qk+1 D− (DΠk+1 D− )′ }DΠj−1 Ql D− ,
k=0
l = 1, . . . , µ − 1. The ODE (20) is always uncoupled of the second subsystem, but the latter is tied to the ODE if among the coefficients H0 , . . . , Hµ−1 is at least one who does not vanish. One speaks about a fine decoupling, if H1 = · · · = Hµ−1 = 0, and about a complete decoupling, if H0 = 0, additionally. A complete decoupling is given, exactly if the coefficient K vanishes identically. For a regular DAE with sufficiently smooth original data, fine and complete decouplings always exist and can be constructed, see [11, Subsection 2.4.3]. More precisely, to each arbitrarily fixed P0 there are appropriate subsequent P1 , . . . , Pµ−1 achieving a fine decoupling. Adjusting also P0 in a specific way one accomplishes a complete decoupling. It should be added at this point, that the coefficients of the ODE (20) depend on the special choice of admissible projector functions. However, they are uniquely determined in the scope of fine decouplings. This justifies to speak about the inherent explicit regular ODE (IERODE) of the given DAE.
R. März / Journal of Computational and Applied Mathematics (
)
–
7
The so-called canonical projector function Πcan of a regular DAE (see [11, Definition 2.37]) is actually a generalization of the spectral projector of a regular matrix pencil onto the finite eigenspace along the infinite eigenspace (cf. [11, Section 1.4]). The subspace im Πcan (t ) ⊆ Km consists precisely of all consistent values of the homogeneous DAE at time t, that is, it represents the corresponding solution space. By means of fine decoupling projector functions P0 , . . . , Pµ−1 , the canonical projector function of the DAE can be represented as
Πcan = (I − H0 )Πµ−1 . It follows that DΠµ−1 = DΠcan ,
DΠµ−1 D− = DΠcan D− .
(22)
We emphasize that Πcan itself is independent of the choice of the projector functions. Therefore, also DΠµ−1 = DΠcan does not depend of the construction. Since fine decoupling projector functions P0 , . . . , Pµ−1 allow an arbitrarily fixed projector function P0 along ker D and the accordingly prescribing generalized inverse D− , it may happen that Πcan ̸= Πµ−1 . In contrast, completely decoupling projector functions P0 , . . . , Pµ−1 yield the representation Πcan = Πµ−1 , which is very useful in theory, but less comfortable in practice since then P0 and D− are very special. The IERODE (20) resulting from a fine decoupling is independent of the special choice of the fine decoupling. We emphasize this fact by rewriting it as 1 − −1 u′ − (DΠcan D− )′ u + DΠcan G− µ BΠcan D u = DΠcan Gµ q.
(23)
Obviously, in the IERODE (23), only inhomogeneities with values in 1 im DΠcan = im DΠcan G− µ
are responsible for the DAE (15). Theorem 3.1. Let the DAE (15) be regular with index µ and characteristic values (18). Let its coefficients be sufficiently smooth. Then, the following is valid: (1) The coefficients of the IERODE (23) are uniquely determined by fine decoupling projectors and its inhomogeneity does not inherit any derivative of q. (2) The component u = DΠcan D− Dx = DΠcan x of each DAE solution x satisfies the IERODE. (3) d is the dynamical degree of freedom of the DAE and there is an inherent d-dimensional flow located in im DΠcan , which is not at all affected by derivatives of the inhomogeneity q. (4) Except for the component u = DΠcan x and possibly subcomponents of it, there is no other genuine solution component satisfying an immanent explicit ODE the inhomogeneity of which does not involve any derivative of q.2 Proof. These assertions (1)–(3) can immediately be concluded from [11, Theorem 2.39] regarding that DΠµ−1 = DΠcan , d = rank Πcan = rank DΠcan D− . To verify (4) we turn to the unfolded structure associated with a complete decoupling. The system (21) has exactly one solution v0 , . . . , vµ−1 . It can be solved successively for vµ−1 , . . . , v0 in terms of q and possibly its derivatives. If µ > 1, part of the solution components actually depend on derivatives of q. There are also solution components depending only on q such as vµ−1 = Lµ−1 q and Tµ−2 vµ−2 = Tµ−2 {Lµ−2 q − Mµ−2 µ−1 Lµ−1 q}, in which Tµ−2 denotes a projection along im Nµ−2 µ−1 . This makes clear, that, in any case, each derivative vi′ necessarily depends at least on one component of a derivative of q. This extends also to any linear combination of the vi′ and u, since these are living in separated subspaces. The same arguments apply to the components Dv1 , . . . , Dvµ−1 . Example 3.2. The constant coefficient DAE, x′1 − α x1 − x2 = q1 , x′3 + x2 = q2 , x′4 + x3 = q3 , x′5 + x4 = q4 , x5 = q5 ,
2 There are allied skew components u˜ = PDx = Pu adopting this property. Namely, if P : I → L(Kn ) is a differentiable projector-valued function and ker P = ker DΠcan D− , then it results that P = PDΠcan D− , DΠcan D− = DΠcan D− P, and further u˜ = Pu, u = DΠcan D− u˜ . Inserting into (23) leads to the corresponding ODE for u˜ . A particular case of such a P is given by DΠµ−1 D− built with arbitrary admissible projector functions, cf. (20).
8
R. März / Journal of Computational and Applied Mathematics (
)
–
written in the form (15) reads
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 1 0 0 0 0 1 0 0
0 0 0 0
0 1 0 0
0 0 1 0
−α
′
−1
0 0 0 x + 0 0 0 1 0
0 0 1 0 0
1 0 0 0
0 0 0 1 0
0 0 0 = q, 0 1
with m = 5, n = 4. The following associated matrix sequence (cf. [11, Subsection 1.2.3]) is admissible and provides µ = 4 and the characteristic values r0 = r1 = r2 = r3 = 4, r4 = 5, d = 1:
1 0 G0 = 0 0 0
−1
1 0 G1 = 0 0 0
1 0 G2 = 0 0 0
1 0 G3 = 0 0 0
−1
α
1 0 0 0
1 1 0 0
α
−α 2
1 0 0 0
1 1 0 0
0 1 1 0
1 0 G4 = 0 0 0
0 1 0 0 0
0 1 0 0 0
1 0 0 0
−1
0 0 0 0 0
−α
1 0 0 0
1 1 0 0
0 1 1 0
α
3
0 0 Q1 = 0 0 0 0 0 Q2 = 0 0 0
0 0 Q3 = 0 0 0 2
0 0 Q0 = 0 0 0
0 0 0 , 1 0
α
0 0 0 , 1 0
0 0 1 0 0
0 0 0 , 1 0
0 0 1 0 0
0 0 0 , 1 0
−1
0 0 1 0 0
0 0 , 1 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1 0 Π3 = 0 0 0
0 1 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0
−1 −1
0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 , 0 0
0 0 0 , 0 0
1 1 0 0 0 0
−α 0 0 0 0
α2
0 0 0 0 0
0 0 Π1 Q2 = 0 0 0
0 0 Π2 Q3 = 0 0 0
0 0 0 1 0
0 0 0 , 0 1
−1
0 0 0 0 0
0 0 0 , 0 0
α
0 0 0 , 0 0
0 1 0 0 0 0 0 0 0
0 0 0 0 0
0 0 1 0
0 0 0 0 0
0 0 0 0 0
1 0 − DΠ3 D = 0 0
1 0 0 0
−α
0 0 1 0 0
0 0 0 0 0
0 0 , 0 0
1 0 0 0
0 0 0 Π0 Q1 = 0 0 0
1+α 1 −1 1 0
−1
0 0 0 , 0 0
−1 − α − α 2 −1 1 , −1 0 0 0 0 0
−α 0 B0 = 0 0
0 0 0
−α 2
0 0 0 1
,
α2
0 . 0 0
Additionally, it follows that 1 Q3 G− 4 B0 Π3 = 0,
1 Q2 P3 G− 4 B0 Π3 = 0,
1 Q1 P2 P3 G− 4 B0 Π3 = 0,
1 Q0 P1 P2 P3 G− 4 B0 Π3 = 0,
and 1 Π3 G− 4 B0 Π3 = −α Π3 .
(24)
Therefore, these projector functions P0 , P1 , P2 , P3 provide a complete decoupling of the given DAE and Π3 coincides with the canonical projector Πcan . The projector functions Q0 , Π0 Q1 , Π1 Q2 and Π2 Q3 represent the variables x2 , x3 , x4 and x5 , respectively. The projector DΠ3 D− and the coefficient (24) determine the IERODE with respect to the component u = DΠ3 x, namely 1 u′ − α u = DΠ3 G− 4 q.
Dropping the three zero rows it results that
(x1 + x3 − α x4 + α 2 x5 )′ − α(x1 + x3 − α x4 + α 2 x5 ) = q1 + q2 − α q3 + α 2 q4 − α 3 q5 .
(25)
Obviously, no derivative of the inhomogeneity q encroaches into the IERODE. In contrast, each associated explicit ODE with respect to the original components xi depends on derivatives of q. In particular, the component x1 satisfies the ODE x′1 − α x1 = q1 + q2 − (q3 − (q4 − q′5 )′ )′ .
(26)
The property of the IERODE to be free of all derivatives of q allows to state boundary conditions for the DAE accurately, see [9], which seizes the suggestion of [1] in essence. However, recall that though the respective flow resides in the d-dimensional
R. März / Journal of Computational and Applied Mathematics (
)
–
9
subspace im DΠcan ⊆ Kn the IERODE itself lives in Kn . Next we will see that the IERODE can be condensed to minimal size d, which results in a so-called essential underlying ODE. Each regular DAE can be transformed into a form with decoupled fast and slow parts, similar to the Weierstraß–Kronecker form of a regular matrix pencil. We quote the respective result from [4,11]: Theorem 3.3. Each regular DAE (15) with index µ and characteristic values (18) can be transformed by pointwise nonsingular matrix functions L ∈ C (I, L(Km , Km )),
K ∈ C (I, L(Km , Km )),
and a refactorization of the leading term by H with H ∈ C 1 (I, L(Ks , Kn )), HH H = H , −
−
H HH
H − ∈ C 1 (I, L(Kn , Ks )), −
= H −,
n, s ≥ r := rank D,
RHH − R = R,
into the structured form
˜ x˜ )′ + B˜ x˜ = Lq, A˜ (D
(27)
with A˜ = LAH =
Id 0
0 , N
˜ = H − DK = D
B˜ = LBK − LAH (H − R)′ DK = 0
N =
N0 1
..
.
W 0
··· .. .
N0 µ−1
0
Nµ−2 µ−1 0
,
0 Im−d
Id 0
0 , PN
, 0
m−r0
Im−r1
PN =
..
,
. Im−rµ−1
in which the entries Ni−1 i have size (m − ri−1 ) × (m − ri ) and full rank m − ri , i = 1, . . . , µ − 1. We recall that the transformation matrices stated in Theorem 3.3 have the following explicit form (cf. [11, p. 146])
Γd DΠcan Γ0 Q0 −1 G , .. µ . Γµ−1 DΠµ−2 Qµ−1
L=
Id 0
0
(I + M˜ )−1
−1 Γd DΠcan Γ0 Q0 , K = .. . Γµ−1 DΠµ−2 Qµ−1
at which completely decoupling projector functions are used and the functions Γi , i = 0, 1, . . . , µ − 1, are defined as in [11, p. 143]. Though the matrix function Gµ and its inverse may depend on the special choice of the completely decoupling 1 projector functions, the expressions Πcan G− µ and Gµ Πcan = ADΠcan are invariant. The transformed DAE (27) comprises the explicit ODE 1 η′ + W η = Γd DΠcan G− µ q,
(28)
with size d. Its inhomogeneity does not comprise derivatives of q. The ODE (28) is said to be an essential underlying ODE (EUODE) of the DAE (15) in [4] after the respective notion introduced for index-2 Hessenberg form DAEs in [1,7,3]. Recall that the IERODE lives in Kn , n ≥ d. The IERODE is unique in the scope of fine decouplings. Its coefficients are expressed in terms of the original DAE. In contrast, the EUODE has minimal size d, but it is accessible by suitable transformations only. These transformations are not uniquely determined, so that several EUODEs accrue, cf. also Theorem 3.4. As described in Section 2 for index-2 DAEs in Hessenberg form (3), (4), and accordingly (5), each EUODE can be seen as condensed IERODE. Recall that in Section 2 the relations im Γd∗ = ker B∗12 = (im B12 )⊥ = (ker (I − Ω ))⊥ = im (I − Ω )∗ = im (DΠcan D− )∗ have been used to construct the condensing transform Γd . In the general case we can proceed similarly as described in [11,4, Section 2.8]. We take a closer look at the general condensing. Since the projector function DΠcan D− is continuously differentiable and has rank d, so is (DΠcan D− )∗ , and im (DΠcan D− )∗ is spanned by d continuously differentiable basis functions. This means that there is a matrix function Γd such that
Γd ∈ C 1 (I, L(Kn , Kd )),
im Γd∗ = im (DΠcan D− )∗ ,
ker Γd∗ = {0}.
(29)
10
R. März / Journal of Computational and Applied Mathematics (
)
–
Then we determine the pointwise generalized inverse Γd− ∈ C 1 (I, L(Kd , Kn )) by
Γd Γd− Γd = Γd ,
Γd− Γd Γd− = Γd− ,
Γd− Γd = DΠcan D− ,
Γd Γd− = Id .
(30)
Letting η = Γd u for the solutions u = DΠcan D− u = Γd Γd u of the IERODE leads to an EUODE (28) in which (cf. [11, Section 2.8]) −
1 − − ′ − −1 − − W = −Γd′ Γd− + Γd DΠcan G− µ BΠcan D Γd = −Γd Γd + Γd DΠcan Gµ BD Γd .
(31)
The next theorem records and accents the result obtained. Theorem 3.4. Each EUODE (28) of a regular DAE (15) with index µ and characteristic values (18) can be represented as a condensation of the IERODE (23) by means of a matrix function (29) and the generalized inverse (30). The coefficient W of the associated EUODE and the IERODE are related by (31). We emphasize once again that, though the IERODE is unique in the context of fine decouplings, the EUODE actually depends on the choice of the basis functions of im (DΠcan D− )∗ . Example 3.5. We continue to consider the index-4 DAE from Example 3.2. Seemingly, most authors would take the ODE (26) for an EUODE, but this is a mistake. Here Π3 coincides with the canonical projector Πcan and we have
Πcan
1 0 = 0 0 0
0 0 0 0 0
1 0 0 0 0
−α 0 0 0 0
α2
0 0 , 0 0
1 0 − DΠcan D = 0 0
1 0 0 0
−α 0 0 0
α2
0 , 0 0
1 1
im (DΠcan D− )∗ = im −α .
α2
1 Choosing Γd = [1 1 − α α 2 ] yields η = Γd u = x1 + x3 − α x4 + α 2 x5 . The associated EUODE η′ − αη = Γd DΠ3 G− 4 q reads in detail
(x1 + x3 − α x4 + α 2 x5 )′ − α(x1 + x3 − α x4 + α 2 x5 ) = q1 + q2 − α q3 + α 2 q4 − α 3 q5 , which coincides with (25). This EUODE is actually a condensed IERODE.
One can see the diversity of EUODEs as a drawback, however, as in [4, Section 6] concerning stability issues, one can even take advantage from the arbitrariness of the basis Γd∗ . Theorem 3.6. Let the DAE (15) be regular with index µ and characteristic values (18). Put q = 0. Let the condensing transformation Γd and Γd− be determined by (29), (30). Then the following is valid: (1) If the matrix functions DΠcan and Πcan D− remain bounded, the Lyapunov spectrum of the DAE coincides with that of the IERODE with respect to im DΠcan . (2) If the matrix functions Γd DΠcan and Πcan D− Γd− remain bounded, the Lyapunov spectra of the DAE and the EUODE coincide. (3) Γd can always be chosen so that the EUODE preserves the Lyapunov spectrum of the DAE. Proof. The assertions (1) and (2) can immediately be concluded from the relations given for q = 0: x = Πcan D− u,
u = DΠcan x,
x = Πcan D− Γd− η,
η = Γd DΠcan x.
(3) As shown in [4], the relations (29), (30) are fulfilled for Γd = U ∗ Πcan D− , Γd− = DU, in which U is an orthonormal basis of im Πcan , and one has x = U η. Note that Theorem 3.6(3) demands an unlimited further flexibility for the choice of the basis Γd∗ . If the choice is restricted ∗ U should be to orthonormal bases as originally proposed in [1], assertion (3) is no longer true. In general, Γd∗ = D− ∗ Πcan expected to fail to be orthonormal. 4. Adjoint pairs Pairs of explicit ODEs and their adjoints, x′ + Bx = 0,
−y′ + B∗ y = 0,
feature properties being useful in theory and practical treatment. For instance, the Lagrange identity means that the product ⟨x(t ), y(t )⟩ of each solution pair remains constant, and the Perron Theorem states that the original ODE is Lyapunov regular with Lyapunov exponents λ1 , . . . , λm exactly if the adjoint is Lyapunov regular with Lyapunov exponents −λ1 , . . . , −λm .
R. März / Journal of Computational and Applied Mathematics (
)
–
11
These properties are usually utilized when formulating and solving boundary-value problems, in sensibility analysis, optimization and control. Aiming at corresponding results we consider the DAE with properly stated leading term, A(Dx)′ + Bx = q,
(32)
together with its adjoint equation (cf. [12]),
− D∗ (A∗ y)′ + B∗ y = p.
(33)
The DAE (33) has a properly stated leading term at the same time as (32), with the associated border projector function R∗ . For any solution pair x ∈ CD1 (I, Km ) and y ∈ CA1∗ (I, Km ) of the DAEs (32) and (33), respectively, and a function w ∈ C 1 (I, K) satisfying the condition
w ′ (t ) = −⟨q(t ), y(t )⟩ + ⟨x(t ), p(t )⟩,
t ∈ I,
one obtains d dt
(⟨D(t )x(t ), A(t )∗ y(t )⟩ + w(t )) = ⟨(Dx)′ (t ), A(t )∗ y(t )⟩ + ⟨D(t )x(t ), (A∗ y)′ (t )⟩ + w ′ (t ) = ⟨A(t )(Dx)′ (t ), y(t )⟩ + ⟨x(t ), D(t )∗ (A∗ y)′ (t )⟩ + w′ (t ) = ⟨−B(t )x(t ) + q(t ), y(t )⟩ + ⟨x(t ), B(t )∗ y(t ) − p(t )⟩ + w′ (t ) = 0,
t ∈ I,
such that
⟨D(t )x(t ), A(t )∗ y(t )⟩ + w(t ) = constant,
t ∈ I.
This implies the Lagrange identity for the homogeneous DAEs (32) with q = 0 and (33) with p = 0:
⟨D(t )x(t ),
A(t )∗ y(t )⟩ = constant,
t ∈ I.
(34)
In contrast to the explicit ODEs, for DAEs and even for implicit ODEs, one can no longer expect a Lagrange identity with respect to ⟨x(t ), y(t )⟩ as the next example indicates. Example 4.1. The scalar homogeneous implicit ODE from [7, Section 4] et x′ (t ) +
1 2
et x(t ) = 0
can be written in the form (1) with m = n = 1, A(t ) = et , D(t ) = 1, and B(t ) =
1 t e. 2
The adjoint ODE is
1
−(et y(t ))′ + et y(t ) = 0. 2
The solutions are simply 1 x(t ) = e− 2 t cx ,
1 and y(t ) = e− 2 t cy ,
with constants cx , cy . The product x(t )y(t ) = e−t cx cy fails to be time-invariant, but 1
1
⟨D(t )x(t ), A(t )∗ y(t )⟩ = e− 2 t cx et e− 2 t cy = cx cy is so.
Recall that the IERODE and the EUODEs of the DAE (32) have the form 1 − −1 u′ − (DΠcan D− )′ u + DΠcan G− µ BΠcan D u = DΠcan Gµ q,
(35)
1 η′ + W η = Γd DΠcan G− µ q.
(36)
and
Applying the decoupling procedure to the adjoint DAE (33) accordingly we derive its IERODE as 1 ∗ ∗− 1 v ′ − (A∗ Π∗ can A∗ − )′ v + A∗ Π∗ can G− v = A∗ Π∗ can G− ∗ µ B Π∗ can A ∗ µ p,
(37)
in which the asterisk-index indicates the matrix functions associated with the coefficients of the DAE (33), i.e., A∗ := −D∗ , D∗ := A∗ , and B∗ := B∗ . Furthermore, the EUODEs of the adjoint DAE (33) have the form 1 ζ ′ + W∗ ζ = Γ∗ d A∗ Π∗ can G− ∗ µ p.
(38)
The next theorem generalizes and completes results from [2,3] concerning index-1 and index-2 DAEs. It clarifies the corresponding observations for index-2 DAEs in Hessenberg form outlined in Section 2.
12
R. März / Journal of Computational and Applied Mathematics (
)
–
Theorem 4.2. The following assertions are valid for each regular DAE (32) with index µ and characteristic values (18): (1) The adjoint DAE (33) is regular with the same index and characteristic values. (2) A∗ Π∗ can A∗ − = (DΠcan D− )∗ . (3) The IERODE (37) of the adjoint DAE can be described in terms of the original DAE as ′
1 − ∗ −∗ ∗ − v ′ + (DΠcan D− )∗ v + (DΠcan G− Πcan p, µ BΠcan D ) v = D
(39)
(4) The adjoint of the IERODE of the DAE (32) with q = 0 and the IERODE of the adjoint DAE (33) with p = 0 coincide precisely if DΠcan D− is time-invariant. (5) The EUODEs of the DAE (32) and the EUODEs of the adjoint DAE (33) built with consistent bases Γ∗ d = Γd− ∗ are adjoint each to other. It holds that W∗ = −W ∗ and the EUODE (38) can be rewritten as ∗ − ζ ′ + W ∗ ζ = Γd− ∗ D− ∗ Πcan p.
(40)
Proof. Assertion (1) is a special case of [4, Theorem 3(1)]. Assertion (2) is given by [4, Lemma 3]. Assertion (3) immediately implies (4). (5) can be derived from (3) by straightforward calculations. So it remains to check (3). We apply completely decoupling projector functions and regard that the IERODE is independent of the special choice of fine decoupling projector ∗ ∗ ∗ ∗ functions. By analogous arguments as used for the relation Π∗ can = G− µ Πcan D A in the proof of [4, Lemma 3] we find the relation ∗ ∗ ∗ ∗ −∗ ∗ Πcan = G− ∗ µ Π∗ can D∗ A∗ = −G∗ µ Π∗ can AD, ∗ 1 thus D− ∗ Πcan = −A∗ Π∗ can G− ∗ µ , and further 1 ∗ ∗− ∗ ∗ A∗ Π∗ can G− = −D− ∗ Πcan B Π∗ can A∗ − ∗ µ B Π∗ can A
∗ ∗ −∗ ∗ = −D− ∗ Πcan B Gµ Πcan D∗ A∗ A∗ − ∗ ∗ −∗ ∗ 1 − ∗ = −D− ∗ Πcan B Gµ Πcan D∗ = −(DΠcan G− µ BΠcan D ) .
This yields the equation ′
1 − ∗ −∗ ∗ v ′ − (DΠcan D− )∗ v − (DΠcan G− Πcan p. µ BΠcan D ) v = −D
Finally, multiplying this equation by −1 gives (39).
One might see in Theorem 4.2(4) a drawback of the IERODE, but this is unsubstantiated. One must regard that only the flows proceeding in the corresponding invariant subspaces im DΠcan D− and im A∗ Πcan A∗ − = im (DΠcan D− )∗ are relevant for the DAEs. Let u and v be solutions of the IERODEs (35) and (39) belonging to the relevant subspaces such that u = DΠcan D− u, v = (DΠcan D− )∗ v , and let the function ω satisfy the condition ∗ 1 ω′ (t ) = ⟨u(t ), (D− ∗ Πcan p)(t )⟩ − ⟨(DΠcan G− µ q)(t ), v(t )⟩,
t ∈ I.
We derive d dt
(⟨u(t ), v(t )⟩ + ω(t )) = ⟨u′ (t ), v(t )⟩ + ⟨u(t ), v ′ (t )⟩ + w′ (t ) = ⟨(DΠcan D− )′ (t )u(t ), v(t )⟩ + ⟨u(t ), (DΠcan D− )∗′ (t )v(t )⟩ = 2⟨(DΠcan D− )′ (t )u(t ), v(t )⟩ = 2⟨(DΠcan D− )′ (t )(DΠcan D− )(t )u(t ), (DΠcan D− )∗ (t )v(t )⟩ = 2⟨(DΠcan D− )(t )(DΠcan D− )′ (t )(DΠcan D− )(t ) u(t ), v(t )⟩ = 0,
t ∈ I,
=0
such that
⟨u(t ), v(t )⟩ + ω(t ) = constant,
t ∈ I.
(41)
This leads to the following theorem: Theorem 4.3. Let the DAE (32) be regular with index µ and characteristic values (18). Then, for each solution pair u and v of the corresponding IERODES (35) and (39), which proceed within the associated invariant subspaces im DΠcan D− and im A∗ Π∗ can A∗ − , respectively, the identity (41) is valid. In particular, for the homogeneous case with q = 0, p = 0, the classical Lagrange identity,
⟨u(t ), v(t )⟩ = constant , comes true.
t ∈I
(42)
R. März / Journal of Computational and Applied Mathematics (
)
–
13
Furthermore, if u, v are solutions of the IERODEs (35) and (37) residing in the associated invariant subspaces, and η = Γd u, ζ = Γ∗ d v are the corresponding solutions of the EUODEs (36) and (38) built by consistent bases, then the following identity becomes valid:
⟨η(t ), ζ (t )⟩ = ⟨Γd (t )u(t ), Γ∗ d (t )v(t )⟩ = ⟨Γd (t )− Γd (t )u(t ), v(t )⟩ = ⟨(DΠcan D− )(t )u(t ), v(t )⟩ = ⟨u(t ), v(t )⟩, t ∈ I. In [4], Lyapunov regularity of an arbitrary-index DAE is introduced and analyzed by means of its spectrum preserving EUODEs. In particular, the following statement is verified in [4]: Theorem 4.4. Let the DAE (32) be regular with index µ and characteristic values (18). Then, if ADΠcan , Πcan (AD)− , Γd DΠcan , and Πcan D− Γd− remain bounded, the DAE and its adjoint are Lyapunov regular at the same time and the Perron identity is valid. It is demonstrated in [4] by examples that, in essence, here one cannot do unless those strong boundedness conditions hold. Even though the statement Theorem 3.6(3) sounds promising it is not really helpful in this respect, since, in general, the spectrum preserving bases Γd and Γ∗ d do not fulfill the consistency condition of Theorem 4.2(5). 5. Standard form DAEs Now we turn to the standard form DAE and its adjoint (cf. [6]), Ex′ + Fx = q,
(43)
−(E y) + F y = p. ∗
′
∗
(44)
We assume that the time-varying subspace ker E (t ) ⊆ K , t ∈ I, is a C - subspace, which means that the orthoprojector function E + E is continuously differentiable. Then, E has constant rank. We refer to [11,13] for elaborated discussions concerning this constant-rank condition and the possibilities of avoiding this condition in the class of quasi-regular DAEs. Here we deal with regular DAEs only. Regularity, the tractability index and the characteristic values of a standard form DAE (43) are given by means of any proper factorization E =: AD, A ∈ C (I, L(Kn , Km )), D ∈ C 1 (I, L(Km , Kn )) such that ker E (t ) = ker D(t ), t ∈ I, and the transversality condition (16) is valid, see [11, Section 2.7]. At this place, it should be noted, that refactorizations do not change the index and the characteristic values, thus regularity. The canonical projector function Πcan is also invariant with respect to refactorizations. Since ker E is a C 1 -subspace, there are proper factorizations E =: AD. One can simply use A = E, D = P, with a projector function P along ker E, for instance, P = E + E, as applied already in [14]. Then we rewrite the standard form DAE (43) as DAE with properly stated leading term, m
1
A(Dx)′ + (F − AD′ )x = q.
(45)
Each solution x of the DAE (43) is a solution of (45) and vice versa, and the Lyapunov spectra of the DAE (43) and the reformulated version (45) coincide.3 On the other hand, the IERODE of (45) and its solutions actually depend on the chosen factorization. Therefore, the corresponding representation of the DAE solutions x also depends on the factorization. The IERODEs of a standard form DAE (43) are given via proper factorizations of the leading coefficient E. Each factorization generates a special IERODE. Owing to Theorem 3.4, the EUODEs arise as condensed IERODEs in the form (28), (31). In turn, various bases Γd∗ of im (DΠcan D− )∗ can be chosen for that purposes. We emphasize that all these IERODEs and EUODEs are not affected by derivatives of the homogeneity q. Let U be a continuously differentiable matrix function such that its columns form an orthonormal basis of the solution space im Πcan , i.e., im U = im Πcan ,
U ∗ U = Id .
(46)
Choosing Γd = U ∗ Πcan D− and Γd− = DU one accomplishes a spectrum preserving EUODE, see Theorem 3.6 and [4, Theorem 6]. In this case, straightforward computations lead to the form 1 ∗ −1 η′ + { U ∗ Πcan D− D′ P0 U + U ∗ Πcan (P0 U )′ + U ∗ Πcan G− µ BU } η = U Πcan Gµ q.
(47)
Each solution x of the DAE (43) with q = 0 satisfies x = Πcan x and the corresponding solution of the associated EUODE (47) is η = Ux. x and η show the same Lyapunov exponents. 1 Owing to [11, Theorem 2.21] the expression Πcan G− µ remains unchanged under refactorizations of the leading coefficient E. P0 is a projector function onto ker E and also independent of the factorization. It turns out that the EUODE (47) associated
3 We suppose classical solutions as introduced e.g., in [14,11,13]. Note that further smoothness of the solutions can be ensured by sufficiently smooth coefficients.
14
R. März / Journal of Computational and Applied Mathematics (
)
–
with the standard form DAE (43) depends on the chosen basis U, but it appears to be widely independent of the special factorization E = AD. In particular, if ker E (t ) is time-invariant, then one can choose a constant factor D so that the term containing D′ disappears, and the EUODE (47) becomes actually independent of the special factorization. This may also happen if ker E (t ) varies with t, as the following example shows. Example 5.1. We consider the DAE in so-called reduced form E1 x′ +F1 x = q1 ,
(48)
F2 x = q2 ,
(49)
with E1 and F2 having constant ranks r and m − r, respectively, and
E1 F2
remains nonsingular. We look at the nontrivial
case 0 < r < m. Then the DAE is regular with index one, and its dynamical degree of freedom equals d = r. The canonical projector function Πcan projects pointwise onto ker F2 along ker E1 . Various representations can be used, e.g.,
Πcan =
−1 E1 0
E1 F2
= U (E1 U )−1 E1 = E1− E1 = Im−d − F2− F2 ,
with an orthonormal basis U of im Πcan = ker F2 and the generalized inverses E1− , F2− which satisfy the properties E1 E1− = Id , E1− E1 = Πcan , F2 F2− = Im−d , F2− F2 = Im − Πcan . Here E1 U remains nonsingular. We apply the simple proper factorization
E=
E1 0
=
Ir E =: AD 0 1
leading to DΠcan D− = E1 Πcan E1− = E1 E1− = Ir and the IERODE, u′ + (F1 − E1′ )E1− u = q1 − (F1 − E1′ )F2− q2 ,
(50)
for the component u = E1 x. We emphasize again that no derivatives of q encroach on (50). With the standard orthonormal basis Γd∗ = Id of im (DΠcan D− )∗ and η = Γd u = u, the corresponding EUODE coincides with the IERODE (50). In contrast, the special choice Γd = U ∗ Πcan E1− , Γd− = E1 U, cf., (46), results in the EUODE
η′ + U ∗ Πcan U ′ η + U ∗ Πcan E1− F1 U η = U ∗ Πcan E1− (q1 − (F1 − E1′ )F2− q2 ),
(51)
which preserves for q = 0 the spectrum of the DAE (48), (49). Regarding that U ∗ Πcan E1 = (E1 U )−1 and U ∗ Πcan = (E1 U )−1 E1 the homogeneous version of this EUODE reads simply −
η′ + (E1 U )−1 E1 U ′ η + (E1 U )−1 F1 U η = 0.
(52)
Remark 5.2. We point to a differing use of the abbreviation EUODE in [15] for strangeness-free DAEs in reduced form (48), (49). Though in [15] the main focus is directed to the homogeneous case, in the context of EUODEs the inhomogeneous case should not be disregarded. As in Example 5.1, let U be an orthonormal basis of ker F2 such that UU ∗ represents the orthoprojector onto ker F2 . Applying the decomposition x = UU ∗ x + (I − UU ∗ )x = Uz + (I − UU ∗ )x, with z = U ∗ x, one can extract from the DAE an implicit ODE with respect to the new variable z, namely E1 Uz ′ + E1 U ′ z + F1 Uz = q1 − E1 F2+ F2 (F2+ q2 )′ − E1 (F2+ F2 )′ F2+ q2 − F1 F2+ q2 .
(53)
In [15] the ODE (53) is further scaled by a nonsingular matrix function H such that the product HE1 U becomes upper triangular. The scaled version is then said to be an implicit essential underlying ODE, and the abbreviation EUODE is applied, too. In general, this is no longer consistent with the basic idea of an explicit ODE of minimal size remaining unaffected by derivatives of q, cf. [1–3]. Namely, the regular implicit ODE (53) as well as its scaled versions are affected by the derivative term (F2+ q2 )′ . The responsible term E1 F2+ F2 disappears only if the original DAE features the very special property ker E1 = (ker F2 )⊥ . If so, one has UU ∗ = Πcan , and (51) and (53) are equivalent. Solely in this special case, an EUODE in the context of [15] features the basic EUODE property to be unaffected by derivatives of q. Apart from that, we observe that for q = 0, the implicit ODE (53) differs from the explicit ODE (52) by multiplication by E1 U only. Thus these homogeneous ODEs feature the same solutions and Lyapunov spectra. In [15], Lyapunov regularity is traced back to the mentioned implicit essential ODE with upper triangular leading coefficient. Though the basic notions are differing, owing to the various boundedness conditions the resulting assertions concerning the Perron identity seem to be consistent. The DAE (44) is obviously out of the scope of a standard form DAE. Often, additionally supposing that E and y are continuously differentiable, one turns to the standard form DAE
− E ∗ y′ + (F ∗ − E ∗ ′ )y = p.
(54)
R. März / Journal of Computational and Applied Mathematics (
)
–
15
Having said that, applying the proper factorization E ∗ = (AD)∗ = D∗ A∗ , to Eq. (54) or to Eq. (44) leads to
− D∗ (A∗ y)′ + (F ∗ − D∗ ′ A∗ )y = p,
(55)
which is the correct adjoint counterpart of the DAE (45). Replacing both DAEs (43) and (44) by proper versions (45) and (55), one can apply all results mentioned in Sections 3 and 4. For adjoint pairs (43) and (44), the material from Section 4 naturally applies, if consistent factorizations yielding (45) and (55) are used. The next statement represents a consequence of [4, Theorem 3(1)] concerning so-called factorization-adjoint pairs of DAEs. It specifies and extends [8, Theorem 3.5]. Theorem 5.3. For the standard form DAE (43) with sufficiently smooth coefficients the following comes true: (1) If the DAE (43) is regular with index µ and characteristic values (18), then so is its adjoint (44) and vice versa. In particular, an adjoint pair shares in the dynamical degree of freedom d. (2) If the DAEs (43) and (44) are regular, then they possess EUODEs being adjoint each to other in the classical sense. As already demonstrated by Example 4.1, the classical Lagrange identity does not come true for the solutions DAEs (43) and (44), however, we can take use of the identity (34) via factorizations. Let x and y be solutions of the homogeneous DAEs (43) with q = 0 and (44) with p = 0. Then we have
⟨E (t )x(t ), y(t )⟩ = ⟨A(t )D(t )x(t ), y(t )⟩ = ⟨D(t )x(t ), A(t )∗ y(t )⟩ = constant, which coincides with the generalized Lagrange identity obtained in [5,10]. Remark 5.4. Adjoint pairs of index-1 DAEs (43), (44) are studied in detail in [16] by means of classical factorizations E = EP, in which P is a continuously differentiable projector-valued function along ker E. Let λ1 ≤ · · · ≤ λd and λ∗ 1 ≥ · · · ≥ λ∗ d be the Lyapunov spectra of the DAE (43) and its adjoint (44). The index-1 DAE (43) is called Lyapunov regular if the Perron identity, i.e.,
λi + λ∗ i = 0 ,
i = 1, . . . , d,
(56)
1 ′ is valid, see [16, Definition 3.2]. If the coefficients E , F , further G− 1 , and P , P are bounded, the DAE (43) is shown to be Lyapunov regular at the same time as the associated IERODE is so.
The mentioned differing points of view concerning Lyapunov regularity of DAEs (43) in the literature call for further careful studies to clarify the matter and to reach a consensus on the most adequate notion and natural boundedness condition.
6. Conclusions Linear regular DAEs of arbitrary index, both with properly involved derivative and in standard form, have been analyzed by means of projector based decouplings generating IERODEs and also by transformations into a special form uncovering the EUODEs. EUODEs can be derived from IERODEs by special condensing transformations up to size d, if d denotes the dynamical degree of freedom of the DAE. In case of higher-index DAEs, the size of the IERODEs is larger than d, and only that component of the IERODE flow which is located within a certain d-dimensional time-varying invariant subspace of the IERODE is associated with the DAE. Regarding adjoint pairs of DAEs, the IERODE of the adjoint DAE has been fully described in terms of the IERODE of the original DAE. Generalized Lagrange identities have been derived from adjoint pairs of DAEs. They comprise relations of associated solutions of both IERODEs and EUODEs. Both concepts, the IERODEs and the EUODEs have been pointed out to be very useful for analyzing the stability behavior of the DAE flow. In particular, spectrum preserving EUODEs have been found beneficial. General conditions ensuring the Perron identity for DAEs have been given. However, owing to differing notions of Lyapunov regularity in the literature, some further effort is required to clarify the matter. It has been shown that the IERODEs of DAEs of arbitrary high index do not depend on derivatives of the right-handside of the DAE, which, in particular, allows to rely on well-posed inner boundary value problems although the overall boundary value problem for the DAE is ill-posed. The IERODEs are shown to be the only those genuine ODEs. The EUODEs as transformed versions of the IERODEs reflect this property. These facts provide the basis of a precise input–output sensitivity analysis.
16
R. März / Journal of Computational and Applied Mathematics (
)
–
List of symbols and abbreviations
K L(Ks , Kn ) C (I , X ) C 1 (I, X ) 1 CM (I, X ) ∗ K K− K+ K∗− K−∗ K−∗− ker K im K
⟨·, ·⟩ |·| ⊕ DAE ODE IVP IERODE EUODE
Set of real numbers R and set of complex numbers C Set of K-valued n × s—matrices and linear operators from Ks to Kn Space of continuous functions mapping I into X Space of continuously differentiable functions mapping I into X {x ∈ C (I, X ) : Mx ∈ C 1 (I, Y ), with M ∈ L(X , Y )} Adjoint matrix Generalized inverse, KK − K = K , K − KK − = K − Moore–Penrose inverse [K ∗ ]− [K − ]∗ [[K − ]∗ ]− Nullspace (kernel) of K Image (range) of K Scalar product in Km Vector and matrix norms Direct sum Differential–algebraic equation Ordinary differential equation Initial value problem Inherent explicit regular ODE Essential underlying ODE
References [1] U.M. Ascher, L.R. Petzold, Projected implicit runge-kutta methods for differential-algebraic equations, SIAM J. Numer. Anal. 28 (4) (1991) 1097–1120. [2] K. Balla, Differential algebraic equations and their adjoints (Dissertation), Doctor of The Hungarian Academy of Sciences, Hungarian Academy of Sciences, Budapest, 2004. [3] K. Balla, Vu Hoang Linh, Adjoint pairs of differential-algebraic equations and Hamiltonian systems, Appl. Numer. Math. 53 (2005) 131–148. [4] Vu Hoang Linh, R. März, Adjoint pairs of differential-algebraic equation and their Lyapunov exponents, J. Dynam. Differential Equations (2015). http://dx.doi.org/10.1007/s10884-015-9474-6 (in press). [5] K. Balla, R. März, Transfer of boundary conditions for DAEs of index 1, SIAM J. Numer. Anal. 33 (6) (1996) 2318–2332. [6] S.L. Campbell, N.K. Nichols, W.I. Terell, Duality, observability, and controllability for linear time-varying descriptor systems, Circuits Systems Signal Process. 10 (4) (1991) 455–470. [7] 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. [8] P. Kunkel, V. Mehrmann, Formal adjoints of linear DAE operators and their role in optimal control, Electron. J. Linear Algebra 22 (2011) 672–693. [9] R. Lamour, R. März, E. Weinmüller, in: A. Ilchman, T. Reis (Eds.), Boundary-Value Problems for Differential-Algebraic Equations: A Survey, in: Surveys in Differential-Algebraic Equations III, Springer-Verlag, Berlin, Heidelberg, 2015, pp. 177–309. [10] T. Petry, On the stability of the Abramov transfer for differential-algebraic equations of index 1, SIAM J. Numer. Anal. 35 (1) (1998) 201–216. [11] R. Lamour, R. März, C. Tischendorf, Differential-Algebraic Equations: A Projector Based Analysis, in: A. Ilchman, T. Reis (Eds.), Differential-Algebraic Equations Forum, Springer-Verlag, Berlin, Heidelberg, 2013. [12] K. Balla, R. März, A unified approach to linear differential algebraic equations and their adjoints, Z. Anal. Anwend. 21 (3) (2002) 783–802. [13] R. März, in: A. Ilchman, T. Reis (Eds.), Differential-Algebraic Equations from a Functional-Analytic Viewpoint: A Survey, in: Surveys in DifferentialAlgebraic Equations II, Springer-Verlag, Berlin, Heidelberg, 2015, pp. 163–285. [14] E. Griepentrog, R. März, Differential-Algebraic Equations and their Numerical Treatment, in: Teubner-Texte zur Mathematik, vol. 88, BSB B.G. Teubner Verlagsgesellschaft, Leipzig, 1986. [15] Vu Hoang Linh, V. Mehrmann, Spectra and leading directions for linear DAEs, in: L.T. Biegler, S.L. Campbell, V. Mehrmann (Eds.), Control and Optimization with Differential-Algebraic Constraints, SIAM, Philadelphia, 2012, pp. 59–78. [16] Cong Nguyen Dinh, Nam. Hoang, Lyapunov regularity of linear differential algebraic equations of index 1, Acta Math. Vietnam. 29 (1) (2004) 1–21.