Proof. The proof follows directly from the previous discussion. IfE is positive semidefinite then the matrix
E11 E12
E12T E22
in (4.12) is positive definite and thus E22 is positive definite and therefore nonsingular.
In conclusion, we can see that we can obtain a structure preserving condensed form (4.11) for symmetric systems only under the strong Assumption 4.8 and a strangeness-free sys-tem (4.16) only for symmetric differential-algebraic syssys-tems of s-index µ ≤ 1, also under Assumption 4.8. This is not very convenient, since too many restrictions and assumptions have to be made in order to preserve the structure using global congruence transforma-tions (4.6). Other index reduction techniques can be applied that allow to preserve the symmetry of the system as will be seen in Section 4.4.
4.3 Condensed Forms for Self-Adjoint Differential-Algebraic
Definition 4.14(Sesquilinear form). Consider two real vector spacesX,X⋆. A mapping (., .) :X×X⋆ →R is called a sesquilinear formif
(a) (x, x⋆+y⋆) = (x, x⋆) + (x, y⋆), (b) (x, αx⋆) =α(x, x⋆),
(c) (x+y, x⋆) = (x, x⋆) + (y, x⋆), (d) (αx, x⋆) =α(x, x⋆),
for all x, y ∈X, x⋆, y⋆ ∈X⋆ and α∈R.
Then, we can define a conjugate operator corresponding to the differential-algebraic oper-ator D, see also [82].
Definition 4.15 (Conjugate operator). Given a linear differential-algebraic operator D:X→Y as in (4.17). Then aconjugate operator is defined as D⋆ :Y⋆ →X⋆ such that
(Dx, y) = (x, D⋆y) for all x∈X, y ∈Y⋆ with sesquilinear form (., .) as in (4.19) and function spaces
X⋆ =C0(I,Rn),
Y⋆ ={y∈C0(I,Rn)|EE+y ∈C1(I,Rn), EE+y(t1) = 0}. (4.20) Theorem 4.16. The differential-algebraic operator D : X → Y defined in (4.17) with regular and strangeness-free pair (E, A) has a unique conjugate operator D⋆ : Y⋆ → X⋆ with function spaces defined as in (4.20) that is given by
D⋆y =−d
dt(ETy)−ATy.
Proof. We have
(Dx, y) = Z
I
(Ex˙ −Ax)Ty dt
= Z
I
( ˙xTETy−xTATy)dt
= Z
I
d
dt(xTETy)−xTE˙Ty−xTETy˙−xTATy
dt
=
xTETyt1
t0 + Z
I
xT
−d
dt(ETy)−ATy
dt
= (x, D⋆y), where
xTETyt1
t0 = 0, since
xTETy=xT(EE+E)Ty=xTE+EETy=xTETEE+y
due to the properties (2.1) of the Moore-Penrose pseudo-inverse (see Definition 2.17) and x(t1)TE(t1)Ty(t1) = xTETEE+y(t1) = 0,
x(t0)TE(t0)Ty(t0) = xT(t0)E+EETy(t0) = 0,
due to the homogenous initial conditions inXandY⋆. To show uniqueness we assume that there exists another conjugate ˜D⋆ for D. Then it holds that
(Dx, y) = (x, D⋆y) for all x∈X, y ∈Y⋆, as well as
(Dx, y) = (x,D˜⋆y) for all x∈X, y ∈Y⋆. Thus,
0 = (x,D˜⋆y)−(x, D⋆y) for all x∈X, y ∈Y⋆, and therefore
D˜⋆y−D⋆y= 0 for all y∈Y⋆.
Due to Theorem 4.16, the differential-algebraic operator D belonging to a regular and strangeness-free pair of matrix-valued functions (E, A) has a unique conjugate operator D⋆ that can be described by the pair (−ET,(A + ˙E)T). For arbitrary pairs of matrix-valued functions we can now introduce the following terminology.
Definition 4.17 (Adjoint). For a pair of matrix-valued functions (E, A) with E ∈ C1(I,Rm,n) andA∈C0(I,Rm,n), the pair (−ET,(A+ ˙E)T) is called theadjointof (E, A).
Due to Definition 4.17, the linear differential-algebraic equation
−ETy˙ = (A+ ˙E)Ty+g(t), (4.21) with g ∈C(I,Rm) is also called theadjoint differential-algebraic equation of (2.5), see also [11, 83].
Lemma 4.18. The adjoint of a pair of matrix-valued functions(E, A)withE ∈C1(I,Rm,n) and A∈C0(I,Rm,n) has itself an adjoint which corresponds to (E, A).
Proof. As −ET ∈ C1(I,Rn,m) and (A+ ˙E)T ∈C0(I,Rn,m) the adjoint of (E, A) has itself an adjoint, which is given by
(−(−ET)T,((A+ ˙E)T + (−E˙T))T) = (E, A).
Global equivalence transformations of the form (2.11) and the transfer to the adjoint are commutative. In particular, the adjoints of equivalent pairs of matrix-valued functions are again equivalent.
Theorem 4.19. Let (E, A) be a pair of matrix-valued functions with E ∈C1(I,Rm,n) and A∈C0(I,Rm,n) and consider the globally equivalent pair
( ˜E,A) = (P EQ, P AQ˜ −P EQ),˙
where P ∈C1(I,Rm,m) andQ∈C1(I,Rn,n) are pointwise nonsingular. Then, the adjoints of (E, A) and ( ˜E,A)˜ are again globally equivalent.
Proof. The adjoint of ( ˜E,A) is given by˜
(−E˜T,( ˜A+ ˙˜E)T) = (−(P EQ)T,(P AQ−P EQ˙ +dtd(P EQ))T)
= (−QTETPT, QTATPT +QTE˙TPT +QTETP˙T).
On the other hand, equivalence transformation of the adjoint (−ET,(A+ ˙E)T) of (E, A) with QT and PT yields
(−ET,(A+ ˙E)T)∼(−QTETPT, QT(A+ ˙E)TPT +QTETP˙T)
= (−QTETPT, QTATPT +QTE˙TPT +QTETP˙T).
Now, we can define self-adjointness for pairs of matrix-valued functions.
Definition 4.20 (Self-adjointness). A pair of matrix-valued functions (E, A) with E ∈ C1(I,Rn,n) and A∈C0(I,Rn,n) is called self-adjoint if it holds that
E =−ET, A= (A+ ˙E)T for all t∈I.
A linear differential-algebraic system (2.5) is calledself-adjointif the corresponding matrix pair (E, A) is self-adjoint.
Self-adjoint matrix pairs arise for example in linear-quadratic optimal control problems (1.3), or in gyroscopic mechanical systems.
Example 4.21. Consider a constraint gyroscopic mechanical system Mp¨+Cp˙+Kp=f(t) +GTλ,
Gp= 0,
with M, C, K ∈ Rn,n and M = MT positive definite, C = −CT and K = KT positive definite. Then, a structure preserving first order formulation introducing the new variable v = ˙pis given by the second companion form
C M 0
−M 0 0
0 0 0
˙ p
˙ v λ˙
=
−K 0 GT
0 −M 0
G 0 0
p v λ
+
f(t)
0 0
, and this system is self-adjoint.
In contrast to the symmetric structure global congruence transformations of the form (4.6) preserve the self-adjoint structure of a pair of matrix-valued functions.
Lemma 4.22. Let a pair of matrix-valued functions (E, A) with E, A ∈ C(I,Rn,n) be sufficiently smooth and self-adjoint. Then for each pointwise nonsingular matrix-valued function P ∈C1(I,Rn,n) the global congruent pair
( ˜E,A) = (P˜ TEP, PTAP −PTEP˙) is also self-adjoint.
Proof. It holds that
E˜T =PTETP =−PTEP =−E,˜ as well as
( ˜A+ ˙˜E)T = (PTAP −PTEP˙)T + dtd(PTEP)T
=PTATP −P˙TETP + ˙PTETP +PTE˙TP +PTETP˙
=PT(A+ ˙E)TP +PTETP˙
=PTAP −PTEP˙ = ˜A.
To derive a global condensed form for self-adjoint pairs of matrix-valued functions we use the following factorization for skew-symmetric matrix-valued functions.
Lemma 4.23. Let A ∈ Ck(I,Rn,n), k ∈ N0 ∪ {∞} be skew-symmetric, i.e., A = −AT, with rankA(t) = r for all t ∈ I. Then there exists a pointwise orthogonal matrix-valued function P ∈Ck(I,Rn,n) such that
PT(t)A(t)P(t) =
Σ(t) 0
0 0
, with pointwise nonsingular and skew-symmetric Σ∈Ck(I,Rr,r).
Proof. The Schur decomposition for skew-symmetric matrices is given in [54]. Then, the Lemma follows in the same way as Theorem 2.25 and Lemma 4.9.
We can derive a local condensed form for self-adjoint pairs of matrix-valued functions similar as in Theorem 4.6 with the corresponding invariant characteristic quantities.
Theorem 4.24. Let E, A∈Rn,n and (E, A) be self-adjoint. Further, 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 matrix P ∈ Rn,n and a matrix R ∈ Rn,n such that the matrix pair (E, A) is locally congruent to a self-adjoint 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
,
0 0 A13 Σs 0
0 0 A23 0 0
AT13 AT23 Σa 0 0
Σs 0 0 0 0
0 0 0 0 0
,
s d a s u
(4.22)
where the matrices Σa ∈ Ra,a and Σs ∈ Rs,s are nonsingular and diagonal, the matrix E11 E12
−E12T E22
∈ Rr,r is nonsingular, and the last block rows and block columns are of dimension 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 (4.8).
Proof. The proof is analogous to the proof of Theorem 4.6.
We can also derive a global condensed form for self-adjoint pairs of matrix-valued functions under the regularity assumptions (4.10).
Theorem 4.25. Let the pair (E, A) of matrix-valued functions E, A∈C(I,Rn,n) be suffi-ciently smooth and self-adjoint with rankE(t) = r for all t ∈I. Suppose that a regularity assumption as in (4.10) holds. Then the pair (E, A) is globally congruent to a self-adjoint pair of matrix-valued functions of the form
E11 E12 0 0 0 E21 E22 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
,
A11 A12 0 Σs 0
0 A22 0 0 0
0 0 Σa 0 0
ΣTs 0 0 0 0
0 0 0 0 0
,
s d a s u
(4.23)
where all blocks are again matrix-valued functions, the matrices Σa = ΣTa and Σs are pointwise nonsingular, and
E11 E12
E21 E22
is pointwise nonsingular and skew-symmetric.
Proof. Again, we give a constructive proof. First, we determine a pointwise orthogonal matrix-valued function P1 ∈C(I,Rn,n) such that
E1 :=P1TEP1 =
Σr 0 0 0
, A1 :=P1TAP1−P1TEP˙1 =
A11 A12 A21 A22
,
where Σr ∈ C(I,Rr,r) is skew-symmetric and pointwise nonsingular and the pair (E1, A1) is again self-adjoint. As rankA22=a is constant in I, there exists a pointwise orthogonal matrix-valued function Q∈C(I,Rn−r,n−r) such that
QTA22Q=
Σa 0 0 0
,
where Σa ∈ C(I,Ra,a) is pointwise nonsingular and symmetric. Defining P2 accordingly, we get
E2 :=P2TE1P2 =
Σr 0 0 0 0 0 0 0 0
, A2 :=P2TA1P2−P2TE1P˙2 =
A11 A12 A13
A21 Σa 0 A31 0 0
. We can now eliminate the blocks A12 and A21 with a nonsingular transformationP3 using the block Σa such that
E3 :=P3TE2P3 =
Σr 0 0
0 0 0
0 0 0
, A3 :=P3TA2P3−P3TE2P˙3 =
A11 0 A13 0 Σa 0 A31 0 0
. As rankA13 =sfor allt∈IandA13 =AT31, we can find pointwise orthogonal matrix-valued functions ˆP4 ∈C(I,Rn−r−a,n−r−a) and ˆQ4 ∈C(I,Rr,r) such that ˆQ4A13Pˆ4T =
Σs 0 0 0
with Σs∈C(I,Rs,s) pointwise nonsingular. Setting P4 accordingly, we get
E4 := P4TE3P4 =
E11 E12 0 0 0 E21 E22 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
,
A4 := P4TA3P4−P4TE3P˙4 =
A˜11 A˜12 0 Σs 0 A˜21 A˜22 0 0 0
0 0 Σa 0 0
ΣTs 0 0 0 0
0 0 0 0 0
,
where ˆQ4ΣrQˆT4 =
E11 E12
E21 E22
is pointwise nonsingular and skew-symmetric and ˆQ4A11QˆT4− Qˆ4ΣrQ˙ˆT4 =
A˜11 A˜12
A˜21 A˜22
. Further, we can eliminate the block ˜A21 with a nonsingular trans-formationP5 to get
E5 := P5TE4P5 =
E11 E12 0 0 0 E21 E22 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
,
A5 := P5TA4P5−P5TE4P˙5 =
A˜11 A˜12+ ˜AT21 0 Σs 0
0 A˜22 0 0 0
0 0 Σa 0 0
ΣTs 0 0 0 0
0 0 0 0 0
,
where ˜A12+ ˜AT21 =−E˙12. The pair (E5, A5) is still self-adjoint as all congruence transfor-mations preserve the structure.
Under the assumptions of Theorem 4.25, we can now transform a self-adjoint pair of matrix-valued functions (E, A) into the global condensed form (4.23). The DAE associated with the pair in global condensed form (4.23) can be written as
E11x˙1+E12x˙2 =A11x1+A12x2+ Σsx4+b1, E21x˙1+E22x˙2 =A22x2+b2,
0 = Σax3+b3, 0 = ΣTsx1+b4, 0 =b5.
In order to obtain a strangeness-free formulation we have to eliminate the strangeness parts. To do this, we use again the derivative of the fourth equation to eliminate the terms with ˙x1 in the first two equations and get
E12x˙2 =A11x1+A12x2+ Σsx4+ ˜b1, E22x˙2 =A22x2+ ˜b2,
0 = Σax3+b3, 0 = ΣTsx1+b4, 0 = b5,
(4.24)
where ˜b1 =b1+E11dtd (ΣTs)−1b4
, and ˜b2 =b2+E21dtd (ΣTs)−1b4
. Due to the occurrence of the block E12, the self-adjoint structure of the system is destroyed. In the same way as in Section 4.2, ifE22 is nonsingular, i.e., if the strangeness index isµ≤1, we can eliminate the blockE12 and get the equivalent system
0 =A11x1+ (A12−E12E22−1A22)x2+ Σsx4+ ˜b1−E12E22−1˜b2, E22x˙2 =A22x2+ ˜b2,
0 = Σax3+b3, 0 = ΣTsx1+b4, 0 =b5.
(4.25)
We can further eliminate the term A12−E12E22−1A22 via block Gaussian elimination using the invertible block Σs and get a strangeness-free system which is again self-adjoint.
Rear-anging and renaming the matrices and vector-valued functions finally yields the strange-ness-free self-adjoint differential-algebraic system
Eˆ11(t) ˙ˆx1 = ˆA11(t)ˆx1+ ˆb1(t),
0 = ˆA22(t)ˆx2+ ˆb2(t), (4.26) 0 = ˆb3(t),
consisting of dµ differential equations, aµ algebraic equations, anduµ vanishing equations, with
Eˆ11 =E22, Aˆ11 =A22, Aˆ22=
Σa 0 0
0 0 Σs
0 ΣTs 0
,
ˆ
x1 =x2, xˆ2 =
x3
x1
x4−Σ−s1E12E22−1A22x2
, xˆ3 =x5,
ˆb1 = ˜b2, ˆb2 =
b3
˜b1−E12E22−1˜b2 b4
, ˆb3 =b5.
In the same way as in the case of symmetric matrix pairs, a structure preserving strange-ness-free formulation (4.26) for self-adjoint differential-algebraic systems in general only exists if the strangeness index is µ ≤ 1. For systems with strangeness index µ > 1 in general we cannot preserve the self-adjointness of the strangeness-free system. Counter-examples similar to Example 4.12 can be found for self-adjoint matrix pairs.
Remark 4.26. For constant coefficient systems of the form (2.6) a matrix pair (E, A) is self-adjoint if it is skew-symmetric/symmetric, i.e., E = −ET and A = AT. Similar results as obtained in Section 4.1 can be derived analogously for skew-symmetric/symmetric matrix pairs as strong congruence (4.2) preserve this structure. A local condensed form as given in Theorem 4.6 can be obtained in the same way. Structured staircase forms for skew-symmetric/symmetric matrix pairs have also been considered in [20].
Remark 4.27. Linear-quadratic optimal control problems as in (1.3) have been the moti-vation for considering self-adjoint differential-algebraic systems. The solution strategies for these systems actually lead to boundary value problems of the form (1.3c) with two-point boundary conditions. The classical approach to solve these boundary value problems is the use of Riccati differential-algebraic equations, see e.g. [77, 83]. First, index reduction and feedback regularization are used to transform the system to a regular, strangeness-free con-trol problem and then the Riccati approach is used on the reduced system. If the reduced problem can be obtained in a structure preserving way, then the solution of the Riccati equations can be adapted to the self-adjoint structure.
4.4 Structure Preserving Index Reduction by Minimal Extension