4.3 Systems on unbounded time intervals
4.3.3 Reformulation of non-advanced systems
Changing the variablex=V ¡d
dt
¢yand scaling (4.24) withU, we obtain the system 2
6 6 6 6 6 4
ΣP ∗ ∗ . . . ∗ 0 ∗ . . . ∗
0 ∗
. .. ...
0 3 7 7 7 7 7 5
y(t)= 2 6 6 6 6 6 4
Q11 ∗ ∗ . . . ∗ Σq ∗ . . . ∗ Σq−1 ∗ . .. ...
Σ1
3 7 7 7 7 7 5
y(t−τ)+Uf(t),
which is uniquely solvable, due to Lemma4.30. Therefore, the DDAE (4.25) is regular.
In particular, the restriction of Corollary 4.31to the DDAE (4.1a) gives us the fol-lowing corollary.
Corollary 4.32. Consider the IVP(4.1)for the DDAE (4.1a)and assume that an initial function φ is sufficiently smooth. Then the IVP (4.1) has a unique solution for every inhomogeneity f ∈C∞([0,∞),Cm)if and only if system(4.1a)is of square size and the associated polynomialq(λ,ω)=det(λE−A−ωB)is not identically zero.
Remark 4.33. From Corollary4.32, one sees that the regularity of a DDAE is indepen-dent of the (positive) delayτ. This property is important for control applications, for example where one applies a time-delayed feedback controlu(t)=K x(t−τ) to a de-scriptor system of the formEx(t)˙ = Ax(t)+Bu(t). There the delayτis used as a pa-rameter to achieve a desired behavior of the descriptor system, and in general it is necessary to verify the regularity of the closed-loop control system for each delay. Nev-ertheless, Corollary4.32shows that it suffices to verify the regularity of the system for an arbitrary delay.
4.3. Systems on unbounded time intervals 60 (4.1a)gives the resulting system
2 4
Eˆ1 0 0
3 5x(t)˙ =
2 4
Aˆ1 Aˆ2 0
3 5x(t)+
2 4
Bˆ1
Bˆ2 Bˆ3 3
5x(t−τ)+ 2 4
0 0 Bˆ4
3
5x(t˙ −τ)+ 2 6 4
fˆ1(t) fˆ2(t) fˆ3(t) 3 7 5,
dµ aµ vµ
(4.29)
where£Eˆ1T AˆT2⁄T
is of full row rank.
Proof. The proof is directly obtained by using Lemma2.21, where the functionx(t−τ) is considered as the function parameter.
From (4.29), one sees that the initial functionφmust satisfy the consistency condi-tion
0=Bˆ3φ(t−τ)+Bˆ4φ˙(t−τ)+fˆ3(t), (4.30) for allt∈(0,τ). In this case, the DDAE (4.1a) has the same solution set as the equation
2 4
Eˆ1 0
−Bˆ4
3 5x(t)˙ =
2 4
Aˆ1 Aˆ2 Bˆ3
3 5x(t)+
2 4
Bˆ1 Bˆ2 0
3
5x(t−τ)+ 2 6 4
fˆ1(t) fˆ2(t) fˆ3(t+τ)
3 7 5.
dµ aµ vµ
(4.31)
Furthermore, since the strangeness-free formulation preserves the associated polyno-mial det(λE−A−ωB), one sees that
det(λE−A−ωB) = det 0
@λ 2 4
Eˆ1 0 0
3 5−
2 4
Aˆ1
Aˆ2 0
3 5−ω
2 4
Bˆ1 Bˆ2 Bˆ3 3 5−λω
2 4
0 0 Bˆ4
3 5 1 A
= ωvµdet 0
@λ 2 4
Eˆ1 0
−Bˆ4 3 5−
2 4
Aˆ1 Aˆ2 Bˆ3
3 5−ω
2 4
Bˆ1 Bˆ2 0
3 5 1 A.
Thus, by transforming the DDAE (4.1a) to the new form (4.31), the degree ofωin the as-sociated polynomial decreases byvµ. Since the associated polynomial det(λE−A−ωB) is not identically zero, the degree ofωis finite. Thus, the following algorithm termi-nates after a finite number of iterations.
Algorithm 4.1
1: Input:The DDAE (4.1a). Setj=0,Ee0=E,Ae0=A,Be0=B, ˜f0(t)=f(t).
2: Apply Algorithm2.1to the DDAE
Eejx(t)˙ =Aejx(t)+Bejx(t−τ)+f˜j(t),
to obtain system (4.29) and the characteristic quantitiesdµ,aµ,vµ.
3: ifvµ=0thenSTOP,
4: elseproceed to 6.
5: end if
6: Shift forward the last equation of (4.29) byτto obtain (4.31). Increase j by 1, set
Eej= 2 4
Eˆ1 0
−Bˆ4 3 5, Aej=
2 4
Aˆ1 Aˆ2 Bˆ3 3 5, Bej=
2 4
Bˆ1 Bˆ2 0
3
5, ˜fj(t)= 2 6 4
fˆ1(t) fˆ2(t) fˆ3(t+τ)
3 7 5, and go back to 2.
Algorithm4.1applied to the DDAE (4.1a) results in the following theorem.
Theorem 4.36. Consider the IVP(4.1)for the square DDAE(4.1a). Let Assumption4.34 hold and assume that the matrix triple(E,A,B)is regular. Furthermore, suppose that the initial functionφsatisfies the consistency condition(4.30)in every iterations of Al-gorithm4.1. Then, the DDAE(4.1a)has the same solution set as the system
•Id 0 0 0
‚ •y˙1(t)
˙ y2(t)
‚
=
•A11 0 0 Ia
‚ •y1(t) y2(t)
‚ +
•B11 B12 B21 B22
‚ •y1(t−τ) y2(t−τ)
‚ +
•γ1(t) γ2(t)
‚
, (4.32) where x=Q y, for some nonsingular matrix Q∈Cn,n. Consequently, the IVP(4.1)has a unique solution.
Proof. The second claim is straightforward from the first one, since the corresponding IVP for the DDAE (4.32) has a unique solution. Therefore, we only need to prove the first claim. Applying Algorithm4.1to the DDAE (4.1a), we obtain the system
•Eˆ1 0
‚
˙ x(t)=
•Aˆ1 Aˆ2
‚ x(t)+
•Bˆ1 Bˆ2
‚
x(t−τ)+
•fˆ1(t) fˆ2(t)
‚
, (4.33)
where£Eˆ1T AˆT2⁄T
is nonsingular. Thus, there exists nonsingular matricesP,Q∈Cn,n such that
P
•Eˆ1 0
‚ Q=
•Id 0 0 0
‚ , P
•Aˆ1 Aˆ2
‚ Q=
•A11 0 0 Ia
‚ .
Scaling system (4.33) withPand changing the variablex=Q y, one obtains the desired form (4.32).
The consistency condition for the initial functionφis given by the following corol-lary.
Corollary 4.37. Consider the IVP(4.1)for the square DDAE(4.1a). Let Assumption4.34 hold and assume that the matrix triple(E,A,B)is regular. Then, the initial functionφis consistent if and only if the following conditions are satisfied.
i) The identity(4.30)holds for every iteration of Algorithm4.1.
ii) The second block equation of (4.33)is satisfied at t=0, which means that Aˆ2(0)φ(0)+Bˆ2(0)φ(−τ)+fˆ2(0)=0.
Now let us illustrate the validity of Algorithm4.1and Theorem4.36by the following example.
4.3. Systems on unbounded time intervals 62 Example 4.38. Consider the DDAE
•1 0 0 0
‚
| {z }
E
˙ x(t)=
•0 0 1 0
‚
| {z }
A
x(t)+
•1 0 0 −1
‚
| {z }
B
x(t−τ)+
•f1(t) f2(t)
‚
, (4.34)
on the time interval[0,∞). The associated polynomialq(λ,ω)=(λ−ω)ωis not identi-cally zero, so the matrix triple(E,A,B)is regular. Algorithm4.1applied to(4.34)reads in detail.
• j=0.
• System(4.29)takes the form
•0 0 0 0
‚
˙ x(t)=
•0 0 1 0
‚ x(t)+
•1 0 0 −1
‚
x(t−τ)+
•0 −1
0 0
‚
x(t−τ)+˙
•f1(t)+f˙2(t) f2(t)
‚
, vµ=1 aµ=1. Since vµ=16=0, we proceed to line 6 of Algorithm4.1.
• System(4.31)takes the form
•0 1 0 0
‚
˙ x(t)=
•1 0 1 0
‚ x(t)+
•0 0 0 −1
‚
x(t−τ)+
•f1(t+τ)+f˙2(t+τ) f2(t)
‚
. dµ=1 aµ=1
• j= 1. Since vµ=0, STOP.
Consequently, the formulation(4.32)in Theorem4.36is obtained by performing Gaus-sian elimination
•0 1 0 0
‚
˙ x(t)=
•0 0 1 0
‚ x(t)+
•0 1 0 −1
‚
x(t−τ)+
•f1(t+τ)+f˙2(t+τ)−f2(t) f2(t)
‚ ,
followed by a column permutation x(t)=
•0 1 1 0
‚ y(t).
Solvability Analysis of General Linear Time Varying DDAEs
“Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things.”
Isaac Newton Cited inRules for methodizing the Apocalypse, Rule 9, from a manuscript published in The Religion of Isaac Newton(1974) by Frank E. Manuel, p. 120.
Having studied the theoretical properties of linear time invariant DDAEs in Chap-ter4, the main goal of this chapter is to address the solvability of DDAEs in such a way that it can be used later for the numerical integration of the IVP (1.2). Our object now is general systems with linear time varying coefficients of the form (1.2a). As seen in Sec-tion3.4, systems with multiple commensurate delays can be transformed into single delay systems, and hence all the results in this chapter can also be extended without any difficulty to the multiple delay case. However, for notational simplicity we only discuss the single delay case.
With the goal of the numerical solution to the IVP (1.2), we need to handle two problems. First, we must generalize the strangeness-free formulation (2.13) of non-delayed DAEs (Section 2.2) to DDAEs. Second, we must determine the smoothness requirements and the consistency conditions for an initial functionφand an inhomo-geneityf. Recall that for the IVP (1.2), we use the piecewise continuously differentiable solution concept. Furthermore, we assume the existence and uniqueness of a solution as follows.
Assumption 5.1. Assume that for any consistent inhomogeneity function f and any consistent initial functionφwith respect to f , the IVP(1.2)has a unique solution.
Recall that by introducing new functions as in the method of steps, the DDAE (1.2a) can be rewritten as a sequence of DAEs
Ej(t) ˙xj(t)=Aj(t)xj(t)+Bj(t)xj−1(t)+fj(t), j =1, . . . ,`, (5.1) for allt∈(0,τ), and due to the continuity ofxat the pointsiτ, 16i6`, we obtain
xi(0)=xi−1(τ), (5.2)
63
5.1. Generalization of the method of steps 64 fori=1, . . . ,`, with the note thatx0(τ)=φ(0).
The failure of the method of steps for dealing with noncausal DDAEs, as have seen in Section3.3, motivates two new approaches:
The first approach aims to generalize the method of steps, so that we can still compute the sequence of functions {xi|i=1, . . . ,`} step by step, starting fromx1. This approach will be our main concern in Sections5.1-5.3.
The second approach is to rewrite the set of DAEs (5.1) as one DAE on the time inter-val (0,τ) of the combined variableX =£
x1T . . . x`T⁄T
. This DAE is coupled with the boundary conditions (5.2) to form a differential-algebraic BVP. Thus, by computingX from this BVP, we are able to determine all the functions xi, i =1, . . . ,`not consecu-tively as the method of steps, but at the same time. This approach will be examined in Section5.4. This chapter is completed with the comparison of these two approaches and some illustrative examples.
5.1 Generalization of the method of steps
Similar to the method of steps, the core of the generalization is the problem of deter-mining the solutionxof the IVP (1.2) on the time interval [(i−1)τ,iτ], provided that xis known on the interval [−τ, (i−1)τ]. This problem can be restated as determining functionxi, provided that functionsxi−1, . . . ,x0are already known. Clearly, the subset of (5.1) that contains the firsti−1 DAEs
Ej(t) ˙xj(t)=Aj(t)xj(t)+Bj(t)xj−1(t)+fj(t), j=1, . . . ,i−1,
becomes a set of redundant equations and as a result, the solvability ofxi is governed by the set of DAEs
Ej(t) ˙xj(t)=Aj(t)xj(t)+Bj(t)xj−1(t)+fj(t), j =i, . . . ,`. (5.3) The failure of the method of steps for general DDAEs is due to the reason that it uses only the first equation of system (5.3) (i.e.,j=i) to determinexi. Therefore, the natural idea of a new method is computingxi(t) from (5.3). Note that depending on the time interval, the set of DAEs (5.3) may have finite equations (`< ∞) or infinitely many equations (`= ∞). In the second case, one certainly cannot use the whole set (5.3) to determinexi, and this fact motivates theshift indexconcept in the next definition.
Definition 5.2. For a fixedi6`, consider the set of DAEs (5.3). The minimum number k∈N0such that the so-calledshift-inflated system
Ei+j(t) ˙xi+j(t)=Ai+j(t)xi+j(t)+Bi+j(t)xi+j−1(t)+fi+j(t), j =0, . . . ,k, (5.4) has a unique solution xi, provided a function xi−1 and a consistent initial vector xi(0)=xi−1(τ), is called theshift indexof the set of DAEs (5.1) with respect toi, and be denoted byκ(i).
The idea for a generalization of the method of steps is carried out in the next algo-rithm.
Algorithm 5.1Generalization of the method of steps
1: Input:The IVP (1.2) for the linear DDAE (1.2a).
2: Return:The solutionxon consecutive time intervals [(i−1)τ,iτ], 16i6`.
3: Rewrite the IVP (1.2) as a sequence of DAEs (5.1) by the method of steps.
4: fori=1 to`do
5: Setk=0.
6: Construct the shift-inflated DAE system (5.4).
7: Determine whether one can computexi uniquely from (5.4).
8: if YES thenκ(i)=k, solvexi from (5.4),
9: else increasekby 1 and go back to 6.
10: end if
11: end for
We demonstrate the applicability of Algorithm5.1in the next example.
Example 5.3. Consider the DDAE
•1 0 0 0
‚
˙ x(t)=
•0 0 1 0
‚ x(t)+
•0 1 0 0
‚
x(t−τ)+
•f(t) g(t)
‚ , for all t∈(0,∞). The DAE sequence(5.1)reads
•1 0 0 0
‚
˙ xi(t)=
•0 0 1 0
‚
xi(t)+
•0 1 0 0
‚
xi−1(t)+
•fi(t) gi(t)
‚
, (5.5)
for i=1,2, . . .. Let us fix some i∈Nand see how Algorithm5.1proceeds to compute xi. 1.) With k=0, the set of DAEs(5.4)is nothing else than the DAE(5.5). However, one can easily see that the function xi is not uniquely determined from(5.5), so this fact means that: first, the method of steps fails to compute xi, and second, the shift indexκ(i)must be bigger than zero. Now we increase k by1and go back to line 6 of Algorithm5.1.
2.) With k=1, the set(5.4)reads
•1 0 0 0
‚
x˙i(t)=
•0 0 1 0
‚
xi(t)+
•0 1 0 0
‚
xi−1(t)+
•fi(t) gi(t)
‚
, (5.6a)
•1 0 0 0
‚
˙
xi+1(t)=
•0 0 1 0
‚
xi+1(t)+
•0 1 0 0
‚ xi(t)+
•fi+1(t) gi+1(t)
‚
. (5.6b)
Adding the derivative of the second equation of (5.6b)to its first equation, it follows that 0=£
0 1⁄
xi(t)+fi+1(t)+g˙i+1(t),
which gives an explicit formula for the second component of xi. On the other hand, the first component of xiis clearly fixed by the second equation of (5.6a)and therefore system (5.6)uniquely determines xi. Thus, the shift indexκ(i)is1for all i . More important, we observe that to compute xi numerically, one must work with the system(5.6), which is under-determined for the variable xi+1. We, however, do not care about this since our function of interest xi is still uniquely computable.
Now we discuss the critical part of Algorithm5.1: How to verify the unique
solvabil-5.1. Generalization of the method of steps 66 ity of a functionxi and to computexi from the set of DAEs (5.4). Observing that (5.4) can be decoupled into two parts. The first part is the equation
Ei(t) ˙xi(t)=Ai(t)xi(t)+Bi(t)xi−1(t)+fi(t), (5.7a) that involves onlyxi, and the second part contains the remaining equations that in-volve bothxi and the other unknown functionsxi+1, . . . ,xi+k,
Ei+α(t) ˙xi+α(t)=Ai+α(t)xi+α(t)+Bi+α(t)xi+α−1(t)+fi+α(t), α=1, . . . ,k, (5.7b) which are represented as follows
2 6 6 6 4
Ei+1 Ei+2
. ..
Ei+k 3 7 7 7 5
2 6 6 6 4
˙ xi+1
˙ xi+2
...
˙ xi+k
3 7 7 7 5
= 2 6 6 6 4
Ai+1 Bi+2 Ai+2
. .. . ..
Bi+k Ai+k 3 7 7 7 5
2 6 6 6 4
xi+1 xi+2
... xi+k
3 7 7 7 5
+ 2 6 6 6 4
Bi+1
0 ... 0
3 7 7 7 5
xi+ 2 6 6 6 4
fi+1
fi+2 ... fi+k
3 7 7 7 5 .
Introducing
Ei := 2 6 6 6 4
Ei+1 Ei+2
. ..
Ei+k 3 7 7 7 5
, Ai:= 2 6 6 6 4
Ai+1 Bi+2 Ai+2
. .. . ..
Bi+k Ai+k 3 7 7 7 5 ,
Bi :=
2 6 6 6 4
Bi+1 0
... 0
3 7 7 7 5
, y:=
2 6 6 6 4
xi+1 xi+2
... xi+k
3 7 7 7 5
, gi:=
2 6 6 6 4
fi+1 fi+2
... fi+k
3 7 7 7 5 ,
we consider system (5.7b) as a DAE in the variabley, with a function parameterxi Eiy˙=Aiy+Bixi+gi. (5.8) Assuming that the strangeness indexµ=µ(Ei,Ai) is well-defined for the function pair (Ei,Ai), we apply Algorithm2.1to (5.8) to obtain the system
2 4
Ei,1
0 0
3 5y˙=
2 4
Ai,1
Ai,2 0
3 5y+
µ
X
j=0
2 4
Bi,j
Ci,j Di,j
3 5x(ij)+
2 4
gi,1
gi,2 gi,3 3
5, (5.9)
where
•Ei,1 Ai,2
‚
has pointwise full row rank.
The following lemma completely extracts all the constraints ofxi hidden in the set
of DAEs (5.4). More important, these extracted constraints give us the sufficient and necessary condition for the unique solvability ofxi.
Lemma 5.4. Consider a fixed number i ∈N, 16i 6`, and assume that the function xi−1is known. Furthermore, assume that the strangeness indexµis well-defined for the DAE(5.8). Then all the constraints of xi hidden in the set of DAEs(5.4)are given by the following high-order DAE in the variable xi
• 0 Di,µ
‚
x(iµ)+ · · · +
• 0 Di,2
‚ xi(2)+
• Ei Di,1
‚
˙ xi+
•−Ai Di,0
‚ xi=
•Bi 0
‚ xi−1+
• fi
−gi,3
‚
. (5.10) Here the functions Di,j, j =0, . . . ,µand gi,3 come out from the system (5.9). Conse-quently, xi is uniquely determined from the set of DAEs(5.4)if and only if it is uniquely determined from the DAE(5.10).
Proof. First note that the strangeness-free formulation (5.9) does not alter the solution set of the DAE (5.8) and consequently,xi is uniquely determined from the set of DAEs (5.4) if and only if it is also uniquely determined from the system
Eix˙i = Aixi+Bixi−1+fi, (5.11a) Ei,1y˙ = Ai,1y+
µ
X
j=0
Bi,jxi(j)+gi,1, (5.11b) 0 = Ai,2y+
µ
X
j=0
Ci,jxi(j)+gi,2, (5.11c) 0 =
µ
X
j=0
Di,jx(ij)+gi,3. (5.11d)
Since
•Ei,1 Ai,2
‚
has pointwise full row rank, we see that in the system (5.11b)-(5.11c) xi
only plays the role of a function parameter. Therefore, all the constraints ofxiare given by the two equations (5.11a) and (5.11d), which form the DAE (5.10).
Remark 5.5. i) Even though system (5.8) looks like a control system (a descriptor sys-tem), one should not confuse the strangeness-free formulation for a DAE [75] with the one for a descriptor system [76]. Herexi is specified as a function parameter, and it will not be reinterpreted as a part of a new behavior variable as in [76].
ii) We further note that sinceµ=µ(Ei,Ai), so in generalµdepends oni. However, for notational convenience, we will writeµinstead ofµ(i).
From Lemma5.4, we can deduce the existence and uniqueness of the shift index, even if`= ∞, in the next theorem.
Theorem 5.6. Consider the IVP(1.2)and the sequence of functions{xi|16i 6`} aris-ing from applyaris-ing the generalized method of steps to(1.2a). Let Assumption5.1hold.
Furthermore, suppose that for each16i6`the strangeness indices are well-defined for the two DAEs(5.8)and(5.10). Then for each16i6`, there exists a unique shift index κ(i)for the set of DAEs(5.1)with respect to i .
5.1. Generalization of the method of steps 68 Proof. Consider a fixedi. From Lemma5.4, we see that for eachk≥0, the set of DAEs (5.4) has the same solutionxias the DAE (5.10). Letukbe the number of undetermined variables contained in the solution x of the DAE (5.10), we then obtain a sequence {uk}k≥0. Introducing
Mk := {xi: [0,τ]→Cn| there exist functionsxi+1, . . . ,xi+kthat satisfy (5.4)}, Nk := {xi: [0,τ]→Cn|xi solves the DAE (5.10)},
we see thatMk=Nkfor everyk≥0 due to Lemma5.4.
Since the sequence {Mk}k≥0is decreasing in the sense that for everykwe haveMk+1⊆ Mk, so is the sequence {Nk}k≥0 and hence the sequence {uk}k≥0 is decreasing. The boundedness from below of the sequence {uk}k≥0implies that this sequence becomes stationary.
Moreover, due to Assumption5.1, the set of DAEs (5.4) has a unique solution for some k6`−i, no matter whether`is finite or not. Therefore lim
k↑(`−i)uk=0 and hence, there exists a finite numberk such thatuk=0. Letκ(i) :=min{k≥0|uk=0}, we obtain the existence and uniqueness of the shift indexκ(i) for the set of DAEs (5.1).
Remark 5.7. It would be ideal if in (5.10) we haveDi,j=0 for all j ≥2. Then, with the initial vectorxi(0)=xi−1(τ), one can computexi by solving the corresponding IVP for the first order DAE
• Ei Di,1
‚
˙ xi+
•−Ai Di,0
‚ xi =
•Bi 0
‚ xi−1+
• fi
−gi,3
‚ .
However, in general, as pointed out in [65], a high-order constraint forxmay be hidden in (1.2a), which results in a high-order DAE (5.10). This situation is demonstrated in Example5.8below.
Example 5.8. Consider the IVP consisting of the DDAE 2
4
0 0 1 0 0 0 0 1 0 3 5
2 4
˙ x(t)
˙ y(t)
˙ z(t)
3 5=
2 4
0 1 0 0 0 1 0 0 0 3 5 2 4
x(t) y(t) z(t) 3 5+
2 4
0 0 0 1 0 0 0 0 0 3 5
2 4
x(t−1) y(t−1) z(t−1) 3 5+
2 4
−t
−1−et−1 1
3
5, (5.12) on the time intervalI=[0,∞), together with an initial function
£x(t) y(t) z(t)⁄T
=£
φ1(t) φ2(t) φ3(t)⁄T
, t∈[−1, 0].
Applying Algorithm5.1to(5.12), we have the set of DAEs where the it h-DAE, i ∈N, reads 2
4
0 0 1 0 0 0 0 1 0 3 5
2 4
˙ xi(t)
˙ yi(t)
˙ zi(t)
3 5=
2 4
0 1 0 0 0 1 0 0 0 3 5
2 4
xi(t) yi(t) zi(t) 3 5+
2 4
0 0 0 1 0 0 0 0 0 3 5 2 4
xi−1(t) yi−1(t) zi−1(t) 3 5+
2 4
−t−i+1
−1−et+i−2 1
3
5, (5.13) for t ∈(0, 1). One sees that for every i , the function xi is not uniquely determined from (5.13)and hence,κ(i)>0. Thus, we proceed to the line 9 of Algorithm5.1. Constructing the set of DAEs(5.4)with k=1and applying Lemma5.4, we obtain the DAE(5.10)which
reads 2 4
0 0 1 0 0 0 0 1 0 3 5
2 4
˙ xi(t)
˙ yi(t)
˙ zi(t)
3 5 =
2 4
0 1 0 0 0 1 0 0 0 3 5
2 4
xi(t) yi(t) zi(t) 3 5+
2 4
0 0 0 1 0 0 0 0 0 3 5
2 4
xi−1(t) yi−1(t) zi−1(t) 3 5+
2 4
−t−i+1
−1−et+i−2 1
3 5, 0 = x¨i(t)−et+i−1,
which clearly implies the existence and uniqueness of£
xi yi zi⁄T
. As a result, the shift index isκ(i)=1and the DAE (5.10)is a second order DAE.
This high-order situation, however, does not lead to any theoretical difficulty, since the analysis of high-order DAEs has been studied in Section2.3. By utilizing Theorem 2.23and Corollaries2.25,2.26for the DAE (5.10), we can deduce the strangeness-free DDAE of the variablexi in the following theorem.
Theorem 5.9. Consider the IVP(1.2)and the sequence of functions{xi|16i 6`} aris-ing from applyaris-ing the generalized method of steps to(1.2a). Let Assumption5.1hold.
Furthermore, suppose that for each16i6`the strangeness indices are well-defined for the two DAEs(5.8)and(5.10). Then, for each i , the function xi is also the solution of the strangeness-free DDAE
2 6 6 6 6 6 6 4
Aˆiµ,1 Aˆiµ−1,1 . . . Aˆi0,1 Aˆiµ−1,2 . . . Aˆi0,2 . .. ...
Aˆi0,µ+1
0 0 . . . 0
3 7 7 7 7 7 7 5
2 6 6 6 4
x(µ)(t) x(µ−1)(t)
... x(t)
3 7 7 7 5
= 2 6 6 6 6 6 6 6 4
Bˆη,1i Bˆη−1,1i . . . Bˆi0,1 Bˆηi,2 Bˆη−i 1,2 . . . Bˆi0,2
... ... ...
Bˆη,µ+1i Bˆη−1,µ+1i . . . Bˆ0,µ+1i Bˆηi,µ+2 Bˆη−i 1,µ+2 . . . Bˆ0,iµ+2 3 7 7 7 7 7 7 7 5
2 6 6 6 4
x(η)(t−τ) x(η−1)(t−τ)
... x(t−τ)
3 7 7 7 5
+ 2 6 6 6 6 6 6 4
fˆ1i fˆ2i ... fˆµ+i 1 fˆµ+i 2 3 7 7 7 7 7 7 5
, (5.14)
with someµ,η∈N0, where the matrix-valued function Aˆµi :=h‡
Aˆiµ,1·T ‡
Aˆiµ−1,2·T
. . . ‡
Aˆi0,µ+1·TiT
is pointwise nonsingular. Furthermore, some of the block rows may not be present.
Proof. By applying Theorem 2.23to the DAE (5.10), we obtain the strangeness-free DDAE (5.14), where Aˆµi has pointwise full row rank. Furthermore, due to the unique-ness of the functionxi, it turns out that the matrix functionAˆiµis pointwise nonsingu-lar.
Analogous to Corollary2.25, the restriction of an underlying DDE of (1.2a) on the time interval [(i−1)τ,iτ] is derived from the strangeness-free DDAE (5.14) as in the next corollary.
5.2. Solvability analysis via system classification 70