of the inhomogeneous problem. Thus, for an inhomogeneous problem (3.68) of indexν = 1 we get the solution representation
y(t) =eS(t−t0)Hy0+ Z t
t0
eS(t−s)(λH−S)ˆb(s)ds−(I−H) ˆADˆb(t), with
I−H =
0 0 0 Q3K33−1Kˆ3
, Hy0 =
v0
(I−Q3K33−1Kˆ3)x0
, as well as
(λH−S)(λE −A)−1b(s) =
H11 H12
H21 H22
QMˆ 1M2M3b(s)
=
M11−1( ˇf1−Cˇ12C22−1fˆ2+ ( ˇC12C22−1K23+K13)K33−1fˆ3) (Q2−Q3K33−1K32)C22−1( ˆf2−K23K33−1fˆ3)
, following from the proof of Lemma 3.39. Further, we have
(I−H) ˆADˆb(t) = ˆAD(I−H)ˆb(t) = ˆAD
0 0 0 Q3K33−1Kˆ3
(λE−A)−1b(t)
= ˆAD
0 0 0 Q3K33−1Kˆ3
QMˆ 1M2M3b(t)
= ˆAD
0
Q3K33−1fˆ3(t)
.
Therefore, for the solution x(t) of the original second order system (3.65) we have x(t) =
0 I
eS(t−t0)
v0
(I−Q3K33−1Kˆ3)x0
−AˆD
0 Q3K33−1fˆ3
+ Z t
t0
eS(t−s)
M11−1( ˇf1−Cˇ12C22−1fˆ2+ ( ˇC12C22−1K23+K13)K33−1fˆ3) (Q2−Q3K33−1K32)C22−1( ˆf2−K23K33−1fˆ3)
ds
.
(3.79)
Remark 3.40. For the solution representation (3.79) the Drazin inverse AˆD is required.
In most applications the matrix Kˆ might be nonsingular or Kˆ =I meaning that E and A in (3.69) commute such that AˆD is simply given by (3.72).
have been generalized to linear k-th order DAEs (3.2) in [102, 135], where in particular a strangeness-free canonical form as in Theorem 3.11 and a first order formulation as in Corollary 3.12 are derived. Also the derivative array approach presented in Section 3.1.2 can be generalized to linear k-th order DAEs, see also Remark 3.24, and by linearization along solution trajectories also nonlineark-th order systems of the form (3.1) can be handled in the same way as in Section 3.2. Further, given a strangeness-free lineark-th order system, the trimmed order reduction formalism derived in Section 3.3 can be applied successively to thek-th order system to reduce the order by one in each reduction step. In this process the derivative of order (k−1) of the transformation matrixQ, chosen similar as in (3.60), will occur. In the constant coefficient case that corresponds to the theory of matrix polynomials, structure preserving staircase forms for matrix tuples are given in [20], that allow trimmed linearizations for arbitrary high order systems in the context of matrix polynomials. For the variable coefficient case it is not clear if such structure preserving staircase forms exist and how trimmed first order formulations can be derived in this case.
Furthermore, it remains to prove Theorem 3.19, and consequently also Corollary 3.20, Theorem 3.21 and Theorem 3.22, for arbitrary strangeness indexµ > 2. To do this another global canonical form analogous to the form given in [82, Theorem 3.21] might be helpful.
Here, we only state the following conjecture.
Conjecture. Let the strangeness-index of (M, C, K) with M, C, K ∈ C(I,Rm,n) be well-defined. Then (M, C, K) is globally equivalent to a matrix triple of the form
Id(2)
µ 0 0 0
0 C 0 F
0 D 0 G
0 E 0 H
,
⋆ 0 ⋆ ⋆
0 Id(1)
µ 0 J
0 0 0 K
0 0 0 L
,
⋆ ⋆ ⋆ 0
⋆ ⋆ ⋆ 0
0 0 0 0
0 0 0 Iaµ
, with
X =
0 Xµ ⋆
. .. ...
. .. X1 0
, Y =
0 Yµ,µ ⋆
... ... . ..
0 Y1,µ . . . Y1,1 0 0 . . . 0
,
for X ∈ {C, D, E, J, K, L}, Y ∈ {F, G, H}, where the blocks Ki, Li, Ci, Di, Ei have sizes wi×ci−1, ci×ci−1, qi×qi−1, wi×qi−1, and ci×qi−1, respectively, and J =
0 ⋆ . . . ⋆ is partitioned accordingly. Further, the blocks Gi,j, Hi,j and Fi,j have sizeswi×cj−1, ci×cj−1
and qi×cj−1, respectively. In particular, we have the full row rank condition
rank
Cµ Fµ,µ Jµ
Dµ Gµ,µ Kµ
Eµ Hµ,µ Lµ
= 2s(M CK)µ−1 +s(M C)µ−1 +s(M K)µ−1 +s(CK)µ−1 =cµ+wµ+qµ.
Structured Differential-Algebraic Systems
In many technical applications the arising differential-algebraic equations exhibit certain structures as e.g. the equations of motions of multibody systems (1.1), the circuit equations (1.4), or linear systems as in (1.2) or (1.3), where the coefficient matrices are structured, see also Chapter 1. In the numerical solution of these systems the structural information can be used to develop efficient index reduction and solution methods. The equations of motion of multibody systems (1.1) have been an important research topic for many years and efficient methods for the index reduction and for the numerical solution have been developed, see e.g. [14, 34, 50, 137]. Also index reduction methods for electrical circuit equations (1.4) have been studied [6, 7, 8, 36]. But, the development of structure preserving index reduction methods for linear time-variant systems with symmetries in the coefficient matrices has remained open.
In general, the structure of a system reflects a physical property of the system that should be preserved during the numerical solution. In the case of linear DAEs with constant coefficient of the form (2.6), for example, the algebraic structure of the problem forces the eigenvalues of the corresponding eigenvalue problem to lie in certain regions in the complex plane (e.g., on the unit circle or the real axis) or to occur in different kind of pairings. If such a system is solved numerically without considering the structure then these physical properties are obscured and we might get physically meaningless results as rounding errors can cause eigenvalues to wander out of their required region, see e.g. [37].
In this field, mainly from the point of view of generalized eigenvalue problems, structure preserving canonical forms as well as structure preserving solution methods for matrix pairs have been investigated, see e.g. [20, 70, 132]. These results can be applied for differential-algebraic systems with constant coefficients, but they do not allow the treatment of time-variant differential-algebraic systems. Another important aspect in the numerical solution of structured differential-algebraic systems is that the structure of the system can be used for an efficient the solution of the linear systems arising in each integration step, which usually has the highest computational effort during the numerical integration.
In this chapter we consider linear differential-algebraic systems with variable coefficients of the form (2.5) where the coefficient matrices E(t) and A(t) are symmetric, e.g. as in the linearized equations of motions of mechanical systems (1.2), see also [53, 155, 156], or in the semidiscretization of the Stokes equation and the linearized Navier-Stokes equation [149]. On the other hand, we consider linear systems of the form (2.5) that have a self-adjoint structure as in linear-quadratic optimal control problems (1.3), see also [10, 83], or in gyroscopic mechanical systems [65, 89]. In Section 4.1 we review structured condensed
113
forms for symmetric matrix pairs that are extended to the case of pairs of symmetric matrix-valued functions in Section 4.2. Analogous structure preserving condensed forms for pairs of Hermitian matrix-valued function have also been derived in [153]. In Section 4.3 we derive a structure preserving condensed form and a strangeness-free formulation for self-adjoint pairs of matrix-valued functions. Finally, in Section 4.4, we present a structure preserving index reduction method for self-adjoint systems based on index reduction by minimal extension.
4.1 Condensed Forms for Symmetric Matrix Pairs
To derive structure preserving condensed forms for linear DAEs we start with linear time-invariant systems of the form
Ex˙ =Ax+b(t), t∈I, (4.1)
where E, A ∈ Rn×n are symmetric, i.e., E =ET and A = AT and b ∈ C(I,Rn). In order to obtain a structure preserving condensed form for the symmetric matrix pair (E, A) we cannot use general equivalence transformations but have to restrict to congruence trans-formations.
Definition 4.1 (Strong congruence). Two pairs of matrices (Ei, Ai), i = 1,2, with Ei, Ai ∈ Rn,n are called strongly congruent if there exists a nonsingular matrix P ∈ Rn,n such that
E2 =PTE1P, A2 =PTA1P. (4.2) This congruence transformation defines an equivalence relation. The canonical form for matrix pairs under general equivalence transformations is the well-known Kronecker canon-ical form, see e.g. [47]. In the symmetric case there also exists a symmetric version of the Kronecker canonical form under congruence transformations (4.2), see [140], but the nu-merical computation of this canonical form is an ill-conditioned problem as small rounding errors can radically change the kind and number of Kronecker blocks. A numerically com-putable structured staircase form for symmetric matrix pairs that displays the invariants of the structured Kronecker form is given e.g. in [20]. For the analysis of existence and uniqueness of solutions of DAEs, however, we do not need the complete information of gen-eralized eigenvalues and eigenspaces provided by the invariants of the Kronecker canonical form, but only the infomation about the eigenvalues at infinity. Therefore, it is sufficient to consider condensed forms for pairs of symmetric matrices that can be computed nu-merically using rank decisions based on orthogonal transformations and allow to analyze the index of a DAE as well as existence and uniqueness of solutions. To derive such a condensed form we use the symmetric Schur decomposition of a symmetric matrix, see e.g.
[54, 90].
Lemma 4.2. LetA∈Rn,n be symmetric with rankA=r. Then there exists an orthogonal matrix P ∈Rn,n such that
PTAP =
Σr 0 0 0
, (4.3)
with nonsingular and diagonal Σr ∈Rr,r. Proof. See e.g. [54, Theorem 8.1.1].
Now, we can derive a condensed form for pairs of symmetric matrices using orthogonal congruence transformations.
Theorem 4.3. Let E, A∈Rn,n be symmetric and let T be a basis of kernelE,
T′ be a basis of cokernelE = rangeE, V be a basis of corange (TTAT).
Then there exists an orthogonal matrixP ∈Rn,n such that the matrix pair(E, A)is strongly congruent to a symmetric matrix pair of the form
E11 E12 0 0 0 E12T E22 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
,
A11 A12 A13 Σs 0 AT12 A22 A23 0 0 AT13 AT23 Σa 0 0
Σs 0 0 0 0
0 0 0 0 0
,
s d a s u
(4.4)
where the matrices Σa ∈ Ra,a, and Σs ∈ Rs,s are nonsingular and diagonal, the block E11 E12
E12T E22
∈Rr,r is nonsingular, and the last block rows and block columns are of dimen-sion u. Further, the quantities
(a) r = rankE, (rank)
(b) a= rank (TTAT), (algebraic part) (c) s= rank (VTTTAT′), (strangeness) (d) d=r−s, (differential part)
(e) u=n−r−a−s (undetermined unknowns/vanishing equations) are invariant under the congruence relation (4.2).
Proof. To derive the condensed form (4.4) we use the following sequence of congruence transformations with orthogonal transformation matrices
(E, A)∼
Σr 0 0 0
,
A11 A12
AT12 A22
∼
Σr 0 0 0 0 0 0 0 0
,
A11 A12 A13 AT12 Σa 0 AT13 0 0
,
∼
E11 E12 0 0 0 E12T E22 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
,
A11 A12 A13 Σs 0 AT12 A22 A23 0 0 AT13 AT23 Σa 0 0
Σs 0 0 0 0
0 0 0 0 0
.
To show the invariance of the quantitiesr, a, s, d, uunder congruence transformations (4.2), we consider two matrix pairs (Ei, Ai), i = 1,2 that are congruent, i.e., there exits a nonsingular matrix P such that
E2 =PTE1P, A2 =PTA1P.
Since
rankE2 = rank(PTE1P) = rankE1,
it follows that r is invariant under congruence transformation. The quantities a and s are well-defined as they do not depend on the choice of the bases. Let T2, T2′ and V2 be the bases associated with (E2, A2), i.e.,
rank(E2T2) = 0, T2TT2 nonsingular, rank(T2TT2) =n−r, rank(E2T2′) =r, T2′TT2′ nonsingular, rank(T2′TT2′) =r, rank(V2TT2TA2T2) = 0, V2TV2 nonsingular, rank(V2TV2) = k,
with k = dim corange(T2TA2T2). Inserting the congruence relation (4.2) and defining T1 =P T2, T1′ =P T2′, V1T =V2T,
we obtain the same relations for (E1, A1) with the matricesT1 andT1′. Hence, T1 is a basis of kernelE1 and T1′ is a basis of rangeE1. Because of
k= dim corange(T2TA2T2) = dim corange(T2TPTA1P T2) = dim corange(T1TA1T1), this also applies toV1. With
rank(T2TA2T2) = rank(T2TPTA1P T2) = rank(T1TA1T1), and
rank(V2TT2TA2T2′) = rank(V2TT2TPTA1P T2′) = rank(V1TT1TA1T1′), we finally get the invariance of a and s and therefore also of d and u.
Note, that the matrix pair (E, A) can be reduced further if we also allow non-orthogonal transformations, see e.g. [73].
4.2 Condensed Forms for Pairs of Symmetric Matrix-Valued Functions