Theorem 2.7. Let the matrix pair (E,A) as in (2.1) be regular. Furthermore, let f ∈ Cν(I,Cn)withνis the Kronecker index of the matrix pair(E,A). Then every solution x of the DAE(2.1)has the form
x(t)=eEeDAte EeDE x(0)e + Z t
0
eEeDA(te −s)EeDEef˜(s)d s−(I−EeDE)e
ν−1X
j=0
(EeAeD)jAeDf˜(j)(t), for all t∈I.
Proof. For the proof see Theorem 3.1.3, [26].
In the case that the matrix pair (E,A) is not regular, it is well-known that the cor-responding IVP for the DDAE (2.1) either has more than one solution or there are ar-bitrarily smooth inhomogeneities for which there is no solution at all, as presented in the following theorem.
Theorem 2.8. Let(E,A)be as in(2.1)and suppose that(E,A)is a singular matrix pair.
1. Ifrank(λE−A)<n for allλ∈C, then the homogeneous initial value problem Ex(t)˙ =Ax(t), x(0)=0,
has a nontrivial solution.
2. If rank(λE−A)=n for some λ∈C and hence m> n, then there exist arbitrarily smooth inhomogeneities f for which the corresponding differential-algebraic equation is not solvable.
Proof. For the proof see Theorem 2.14, [75].
2.2. Time Varying Differential-Algebraic Equations 14 An initial vectorx0is calledconsistentto system (2.3a) if the IVP (2.3) has a solution.
System (2.3a) is calledsolvableif it has at least one solution. It is calledregular if in addition, for any consistent initial vectorx0, the corresponding IVP (2.3) has a unique solution.
We will make frequent use of the following results, compare Theorems 3.9, 3.25 in [75].
Theorem 2.10. Let E∈C`(I,Cm,n),`∈N0∪{∞}, with constantrankE(t)=r for all t∈I. Then there exist pointwise unitary functions U ∈C`(I,Cm,m)and V ∈C`(I,Cn,n), such that
UHEV =
•Σ 0 0 0
‚
, or UHE=
•E1
0
‚ ,
with pointwise nonsingularΣ∈C`(I,Cr,r), and E1has full row rank r .
Theorem 2.11. LetI⊂Rbe a closed interval and M∈C(I,Cm,n). Then there exist open intervalsIj ⊂I, j∈N, with
[
j∈NIj=I, Ii∩Ij= ; for i 6=j, and integers rj∈N0, j∈Nsuch that
rankM(t)=rj for all t∈Ij.
Lemma 2.12. For the pair(P,Q)with P ∈C`(I,Cp,n), Q∈Ck(I,Cq,n),`, k∈N0∪{∞}, as-sume that there exist two integers rQ6r[P;Q]such thatrankQ(t)=rQ andrank
•P(t) Q(t)
‚
= r[P;Q]for all t ∈I. Then, there exists
•S 0 Z1 Z2
‚
∈Cmin{`,k}(I,Cp,p+q)that satisfies the fol-lowing conditions.
i)
•S Z1
‚
∈C(I,Cp,p)is pointwise unitary, ii) Z1P+Z2Q=0,
iii) the function SP has pointwise full row rank, and the pair(SP,Q)satisfies rank
•SP Q
‚¶
=rank(SP)+rank(Q).
Proof. SinceQhas constant rank onI, one can apply Theorem2.10to factorizeQ, and then partitionPconformably to get
2 4
Ip 0 0 U11H 0 U12H 3 5
•P Q
‚
£V11 V12⁄
= 2 4
P1 P2 Σ 0
0 0
3 5
p rQ q−rQ
, (2.4)
whereU1=£
U11 U12⁄
∈Ck(I,Cq,q),V1=£
V11 V12⁄
∈Ck(I,Cn,n) are pointwise unitary functions, andΣ∈Ck(I,CrQ,rQ) is pointwise nonsingular. The sizes of the block rows in
(2.4) arep, rQ, q−rQ. Moreover, note that in (2.4),P2also has constant rank due to rank(P2)=rank
•Ip 0 0 U1H
‚ •P Q
‚
£V11 V12⁄
¶
−rank(Σ)=r[P;Q]−rQ.
Then, by Theorem 2.10, there exists a pointwise unitary function U2H=
•S Z1
‚
∈Cmin{`,k}(I,Cp,p) such that
U2HP2=
•S Z1
‚ P2=
•P12 0
‚
, (2.5)
whereP12∈Cmin{`,k}(I,Cr[P;Q]−rQ,n−rQ) has pointwise full row rank.
Combining (2.4) and (2.5), one obtains 2
6 6 6 4
S 0
Z1 0 0 U11H 0 U12H 3 7 7 7 5
•P Q
‚
£V11 V12⁄
= 2 6 6 6 4
P11 P12 P21 0
Σ 0
0 0
3 7 7 7 5
r[P;Q]−rQ p−r[P;Q]+rQ
rQ q−rQ
,
whereP12has pointwise full row rank andΣis pointwise nonsingular onI. Consequently,SP =£
P11 P12⁄
V1−1 has pointwise full row rank. Moreover, one sees that
rank
•SP Q
‚¶
= rank
•P11 P12
Σ 0
‚¶
= rank¡£
0 P12⁄¢
+rank¡£
Σ 0⁄¢
= rank(SP)+rank(Q).
SinceΣ∈Ck(I,CrQ,rQ) is pointwise nonsingular, it implies thatΣ−1∈Ck(I,CrQ,rQ). Fi-nally, settingZ2:= −P21Σ−1U11H∈Cmin{`,k}(I,Cp−r[P;Q]+rQ,q), we obtain
Z1P+Z2Q=¡
[P21 0]−P21Σ−1[Σ 0]¢
V1−1=0, which completes the proof.
Following the algebraic approach [75], we rewrite equation (2.3a) in the form E(t)d
dt−A(t)
¶
x(t)=f(t), (2.6)
for anyt∈I. For notational convenience, we will omit the time variabletin all matrix functions.
Making use of Theorem2.11and restricting ourselves if necessary to subintervals, we may assume that the following assumption holds.
Assumption 2.13. For the pair of matrix functions(E,A)of the DAE(2.3a), there exist integers r, a such that
rank(E)=r, rank¡£
E A⁄¢
=r+a for all t∈I.
2.2. Time Varying Differential-Algebraic Equations 16 Lemma 2.14. Consider the DAE(2.3a)and suppose that Assumption2.13holds. Then, there exists a pointwise unitary function P1∈C(I,Cm,m)such that by scaling system(2.6) with P1from the left one obtains a new system in the following form
2 4
M11dtd −M12
−M22
0
3 5x=
2 4
f1 f2 f3 3 5
r a v
, (2.7)
where the functions M11∈C(I,Cr,n), M22∈C(I,Ca,n)have pointwise full row rank. Here the sizes of the block row equations are r, a and v=m−r−a.
Proof. First we determine a pointwise unitary functionPE :I→Cm,mvia Theorem2.10 or a smooth QR-decomposition, see [39], that compresses the matrix functionE. This yields
PE E d dt −A
¶
=
•M11dtd −M12
−M˜22
‚ r m−r ,
such that M11 has full row rank. Continuing, by compressing the block ˜M22 with a pointwise unitary functionPA:I→Cm−r,m−r, this yields
•Ir 0 0 PA
‚
PE E d dt−A
¶
=
•Ir 0 0 PA
‚ •M11d dt−M12
−M˜22
‚
= 2 4
M11dtd −M12
−M22 0
3 5,
whereM11andM22have pointwise full row rank. SettingP1:=
•Ir 0 0 PA
‚
PE, we arrive at (2.7).
The formula (2.7) in Lemma2.14clearly shows that the number of scalar (nontriv-ial) differential equations in system (2.3a) isr, while the number of scalar (nontrivial) algebraic constraints isa.
In the following we suppose that the functionM22is continuously differentiable. Again, to be able to apply Lemma2.12, the following assumption is necessary.
Assumption 2.15. For the DAE(2.7), there existsmb ∈Nsuch that the functions M11,M22 satisfy
rank
•M11 M22
‚¶
=m,b for all t∈I.
Under Assumption 2.15, applying Lemma 2.12to the pair (M11,M22) implies the existence of matrix functions S, Z1, Z2 of appropriate sizes that have the following properties
i) the function
•S Z1
‚
∈C(I,Cr,r) is pointwise unitary,
ii) the functionSM11has pointwise full row rank and the following identities hold:
Z1M11=Z2M22, (2.8)
and
rank
•SM11 M22
‚¶
=rank(SM11)+rank(M22).
Define the operator
P2:=
2 6 6 6 4
S 0 0
Z1 Z2dtd 0
0 Ia 0
0 0 Iv
3 7 7 7 5
d s a v
, (2.9)
wherer =d+s, we see thatP2has a left-inverse given by the formula
P2−1= 2 6 6 6 4
•S Z1
‚−1
−
•S Z1
‚−1• 0 Z2dtd
‚ 0
0 Ia 0
0 0 Iv
3 7 7 7 5
d+s a v
.
Applying the operatorP2to system (2.7), we obtain 2
6 6 6 4
S 0 0
Z1 Z2dtd 0
0 Ia 0
0 0 Iv
3 7 7 7 5
2 4
M11dtd −M12
−M22
0
3 5x=
2 6 6 6 4
S 0 0
Z1 Z2dtd 0
0 Ia 0
0 0 Iv
3 7 7 7 5
2 4
f1 f2 f3 3 5
d s a v ,
or equivalently, 2 6 6 6 4
SM11dtd −SM12 Z1M11dtd −Z1M12−Z2dtd M22
−M22 0
3 7 7 7 5
x= 2 6 6 6 4
S f1 Z1f1+Z2f˙2
f2 f3
3 7 7 7 5
d s a v
. (2.10)
According to dtd M22=M˙22+M22dtd and (2.8), it follows that the second block equation of (2.10) becomes
¡−Z1M12−Z2M˙22¢
x=Z1f1+Z2f˙2. As a result, (2.10) becomes
2 6 6 6 4
SM11dtd −SM12
−Z1M12−Z2M˙22
−M22 0
3 7 7 7 5
x= 2 6 6 6 4
S f1 Z1f1+Z2f˙2
f2 f3
3 7 7 7 5
d s a v
. (2.11)
It is worth to note that the existence of the left-inverseP2−1ofP2guarantees that the step of transforming system (2.6) via (2.7) to (2.11) does not alter the solution set of sys-tem (2.6). Furthermore, the number of scalar differential equations has been reduced fromr tod. Continuing this reduction process leads us to the following algorithm.
2.2. Time Varying Differential-Algebraic Equations 18 Algorithm 2.1Reformulation algorithm for the DAE (2.3a)
1: Seti =0 and letE0=E, A0=A, f0=f,r0=r,a0=a.
2: Determine a pointwise unitary function P1 as in Lemma 2.14to bring the DAE
¡Eidtd −Ai¢
x=fito the form 2 4
M11 d dt −M12
−M22 0
3 5x=
2 4
f1 f2 f3 3 5
ri ai vi
, (2.12)
where the functionsM11,M22have pointwise full row rank.
3: ifrank£
M11T M22T ⁄T
=ri+aithen STOPwith the resulting system (2.12),
4: else proceed to 5.
5: Determine the operatorP2as in (2.9) and apply it to system (2.12) results in 2
6 6 6 4
SM11dtd −SM12
−Z1M12−Z2M˙22
−M22 0
3 7 7 7 5
x= 2 6 6 6 4
S f1 Z1f1+Z2f˙2
f2 f3
3 7 7 7 5
di si ai vi .
6: Increasei by 1, set
Ei:=
2 6 6 6 4
SM11 0 0 0
3 7 7 7 5
, Ai:=
2 6 6 6 4
SM12 Z1M12+Z2M˙22
M22 0
3 7 7 7 5
, fi= 2 6 6 6 4
S f1 Z1f1+Z2f˙2
f2 f3
3 7 7 7 5 ,
and repeat the process from 2.
7: end if
Sinceri+1=ri−si, Algorithm2.1terminates after a finite number of iterations. This guarantees the existence of the so-calledstrangeness indexof the DAE (2.3a) defined byµ=min{i∈N0, ri=ri+1}. We also callµthe strangeness index of the pair (E,A).
Theorem 2.16. Consider the DAE(2.3a)and assume that its strangeness indexµis well defined. Then, the DAE(2.3a)has the same solution set as the resulting DAE, which we denote by
2 4
Mˆ11dtd −Mˆ12
−Mˆ22
0
3 5x=
2 6 4
fˆ1 fˆ2 fˆ3 3 7 5,
dµ aµ vµ
(2.13)
where
•Mˆ11 Mˆ22
‚
has pointwise full row rank. The functions fˆ2, fˆ3depend on f , f˙, . . . ,f(µ), while the function fˆ1depends only on f .
The quantitiesdµ,aµ,vµ anduµ:=n−dµ−aµare called thecharacteristic quan-titiesof the DAE (2.3a). In particular,uµisthe number of undetermined variables con-tained in the state vector-valued functionx. Following the notation in [75], we also call (2.13) thestrangeness-free formulationof the DAE (2.3a).
As a direct result, the solvability of the DAE (2.3a) is analyzed in the following corollary.
Corollary 2.17. Consider the DAE(2.3a)and assume that its strangeness indexµis well defined. Then, the following assertions hold.
i) The DAE(2.3a)is solvable if and only if in(2.13), one has either fˆ3=0or vµ=0.
ii) The initial condition x0is consistent if and only if in addition,−Mˆ22(0)x0=fˆ2(0). In this case,(2.3a)has the same solution set as theunderlying linear system
• Mˆ11
−Mˆ22
‚
˙ x=
"
Mˆ12
M˙ˆ22
# x+
"
fˆ1
f˙ˆ2
#
. (2.14)
iii) Furthermore, (2.3a)is regular if and only if in addition uµ=0. If this is the case, (2.14)becomes an ODE, which is often calledan underlying ODEin the literature, see [6,22,82].
Remark 2.18. Whereas other index concepts, such as differentiation index [22], per-turbation index [66], or tractability index [82], aiming at the resulting system is ei-ther an underlying ODE or an inherent ODE, the goal of the strangeness index is the strangeness-free formulation (2.13) and the underlying linear system (2.14). Conse-quently, the strangeness index is suitable for general DAEs, which can be underdeter-mined or overdeterunderdeter-mined.
The validity of Algorithm2.1is illustrated by the next example.
Example 2.19. We apply Algorithm2.1to the following DAE
•−t t2
−1 t
‚
˙ x+
•1 0 0 1
‚ x=
•0 0
‚
. (2.15)
First scaling the system with P1=
• 0 −1
−1 t
‚
yields system(2.12)
• d
dt −tdtd −1
−1 t
‚ x=
•0 0
‚
. (2.16)
Then, applying the operator
P2=
•1 dtd 0 1
‚
to(2.16)using dtdt=1+tdtd, we obtain the strangeness-free formulation(2.13)
• 0 0
−1 t
‚ x=
•0 0
‚ .
The strangeness-index is µ=1, and the characteristic invariants are dµ =0, aµ =1, vµ=1, uµ=1.
In order to understand the effect of the reformulation procedure performed in Al-gorithm2.1to DDAEs, we apply it to a modification of the DAE (1.3) including a
func-2.2. Time Varying Differential-Algebraic Equations 20 tion parameter that reads
E(t) ˙x(t)=A(t)x(t)+T(t)λ(t)+f(t), (2.17a) together with an initial vector
x(0)=x0. (2.17b)
Hereλ:I→CnandT :I→Cm,n. We further assume thatE,A,T andf are sufficiently smooth.
The smoothness comparison between the function parameterλand the state variable xgives rise to the system classification as follows.
Definition 2.20. The parameter dependent DAE (2.17a) is called:
i) retarded if for any continuous function λ, there exists a solution x to the IVP (2.17). Since the solutionxis continuously differentiable, formally, we will sayx is smoother thanλ.
ii) neutralif for any continuously differentiable functionλ, there exists a solutionx to the IVP (2.17). Formally, we will sayxis at least as smooth asλ.
iii) The remaining case, whereλmust be at least two times continuously differen-tiable to guarantee the existence of a solutionxto (2.17), is calledadvanced.
This classification leads to different forms of the resulting DAE (2.13) as in the fol-lowing lemma.
Lemma 2.21. Suppose that the parameter dependent DAE(2.17a)is not advanced and the strangeness indexµis well-defined for the function pair(E,A)of (2.17a). Consid-ering T(t)λ(t)+f(t) as a new inhomogeneity, then Algorithm2.1 applied to(2.17a) results in a system
2 4
Eˆ1(t) 0 0
3 5x(t˙ )=
2 4
Aˆ1(t) Aˆ2(t)
0 3 5x(t)+
2 4
Tˆ10(t) Tˆ20(t) Tˆ30(t) 3 5λ(t)+
2 4
0 0 Tˆ31(t)
3 5λ(t˙ )+
2 6 4
fˆ1(t) fˆ2(t) fˆ3(t) 3 7
5, (2.18)
where the matrix function
•Eˆ1 Aˆ2
‚
is of pointwise full row rank. In addition, if (2.17a)is of retarded type thenTˆ20=0andTˆ31=0.
Proof. When applying the strangeness-free formulation to the DAE (2.17a), the as-sumption that the system is not advanced ensures that all algebraic constraints of (2.17a) have the form
0=Ae2(t)x(t)+Te2(t)λ(t)+f˜2(t),
for some matrix functions Ae2, Te2, ˜f2. On the other hand, all differential equations of (2.17a) have the form
Ee1(t) ˙x(t)=Ae1(t)x(t)+Te1(t)λ(t)+f˜1(t),
for some matrix functionsEe1, Ae1, Te1, ˜f1. Moreover, consistency conditions forλand for the inhomogeneity of the DAE (2.17a) can only arise from one of the following three sources:
i) Adding an algebraic equation to another algebraic equation;
ii) Adding a differential equation to another differential equation;
iii) Adding the derivative of some algebraic equation to a differential equation.
Therefore, the consistency condition for the inhomogeneity of (2.17a) does not contain derivatives of λof order bigger than one. This means that Algorithm2.1 applied to (2.17a) results in the DAE of the form (2.18).