4.3 Systems on unbounded time intervals
4.3.2 A matrix polynomial approach
4.3. Systems on unbounded time intervals 54 ii) Since the matrix triple (N22E,JA,B22) is commutative, so is the triple (N22E, (JA)−1,B22).
Furthermore, the matrix (JA)−1N22E is nilpotent of the same nilpotency index asN22E. By rewriting equation (4.21b) as
(JA)−1N22E x˙2(t)=x2(t)+(JA)−1B22x2(t−τ)+(JA)−1f2(t), the formula (4.22b) is straightforward due to the identity
I−(JA)−1N22E d dt
¶−1
=
n−1
X
i=0
¡(JA)−1N22E¢i d dt
¶i
, and the commutativity of the matrix triple (N22E , (JA)−1,B22).
iii) In order to obtain (4.22c), we first rewrite (4.21b) as I−(JA)−1N22E d
dt +(JA)−1B22∆τ
¶
x2(t)= −(JA)−1f2(t).
SinceB22 is nilpotent, due to the commutativity of the matrix triple (N22E, (JA)−1,B22) one can directly verify that the operator ˜L :=(JA)−1N22E dtd−(JA)−1B22∆τsatisfies ˜Ln=0.
Therefore, the inverse of the operatorI−L˜ exists and is given by
n−1
X
i=0
L˜i =
n−1
X
i=0
(JA)−i ˆ i
X
j=0
ˆi j
!
(N22E )j(−B22)i−j d dt
¶j
(∆τ)i−j
! . Thus, we have
x2(t)= −
n−1X
i=0
(JA)−(i+1)
i
X
j=0
ˆi j
!
(N22E )j(−B22)i−j d dt
¶j
(∆τ)i−jf2(t),
which is exactly (4.22c). Note that 06i−j6n, so this representation forx2only makes sense fort≥nτ.
Except for the two special cases presented above, using constant equivalent trans-formations to study the structure of matrix triple coefficients in DDAEs is still an open problem. The author believes that, to study the structure of more than two matrices, with more than one operator acting on them, the class of constant equivalent transfor-mations is too restricted. We will consider a richer class of transfortransfor-mations in the next subsection.
where the coefficients satisfyRi,Sj∈Cm,nfor alli=K+, . . . , 0, j=K−, . . . , 0.
Following the terminology in [55], by amatrix polynomial of degree k we understand a matrix-valued function of a complex variable of the formL(ξ)=Pk
i=0Liξi, where L0, . . . ,Lk are complex matrices of sizem byn. The space ofm byn complex matrix polynomials is denoted byCm,n[ξ]. Therefore, we can rewrite system (4.23) as
P d dt
¶
x(t)=Q d dt
¶
x(t−τ)+f(t). (4.24)
whereP(ξ) :=PK+
i=0Riξi, andQ(ξ) :=PK− j=0Sjξj.
In the case of square systems, i.e., if m =n, by the associated polynomialof system (4.23) we mean the two variable polynomial
q(λ,ω) :=det(P(λ)−ωQ(λ)).
Within this section, we will analyze the solvability of the DDAE (4.23) and also estab-lish the relation between the solution property of (4.23) and the associated polynomial q(λ,ω). Instead of using constant equivalent transformations as in Subsection4.3.1, we will use the richer class of unimodular transformations.
A matrix polynomialM∈Cn,n[ξ] is calledunimodularif det(M) is a nonzero constant.
For a decomposition for matrix polynomials, there exists the following Smith canonical form.
Theorem 4.26. (Smith canonical form) LetP ∈Cm,n[ξ]. Then there exist unimodular matrix polynomialsU ∈Cm,m[ξ]andV ∈Cn,n[ξ]such that
U P V =
•diag(p1,p2, . . . ,pr) 0r,n−r
0m−r,r 0m−r,n−r
‚
where p1,p2, . . . ,pr ∈C1,1[ξ]are monic, i.e., their leading coefficients are equal to 1, and pk+1is divisible by pkfor k=1, 2, . . . ,r .
Proof. For the proof, see Theorem S1.1, [55].
To deal with (4.24), we make use of unimodular transformations defined as follows.
Definition 4.27. Two pairs of matrix polynomial (P1,Q1) and (P2,Q2) in (Cm,n[ξ])2are calledequivalent if there exist unimodular matricesU ∈Cm,m[ξ],V ∈Cn,n[ξ] such that
(P2,Q2)=(U P1V,U Q1V).
If this is the case, we write (P1,Q1)∼(P2,Q2).
In order to study the structure of pairs of matrix polynomials, we construct a con-densed form in the next lemma.
Lemma 4.28. For any pair of matrix polynomials(P,Q)∈(Cm,n[ξ])2, the following as-sertions hold.
4.3. Systems on unbounded time intervals 56 i) There exist unimodular matrix polynomialsU ∈Cm,m[ξ],V ∈Cn,n[ξ]such that
U P V = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
ΣP 0 ∗ ∗ . . . ∗ 0 ∗ . . . ∗ 0 . . . ∗ . .. ...
0 0 ∗ . . . ∗ 0 . . . ∗ . .. ...
0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
, U QV = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
Q11 Q12 ∗ ∗ . . . ∗ Σq ∗ . . . ∗ Σq−1 . . . ∗ . .. ...
Σ1
0 ∗ . . . ∗
0 . . . ∗ . .. ...
0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ,
(4.25) whereΣj=diag(pj,1,pj,2, . . . ,pj,rj)with monic pj,1, pj,2, . . . ,pj,rj ∈C1,1[ξ].
ii) Consequently, the associated polynomialq(λ,ω)is not identically zero if and only if (4.25)becomes
U P V = 2 6 6 6 6 6 4
ΣP ∗ ∗ . . . ∗ 0 ∗ . . . ∗
0 ∗
. .. ...
0 3 7 7 7 7 7 5
, U QV = 2 6 6 6 6 6 4
Q11 ∗ ∗ . . . ∗ Σq ∗ . . . ∗ Σq−1 . . . ∗ . .. ...
Σ1
3 7 7 7 7 7 5
. (4.26)
Proof. The transformation (4.25) is derived in a constructive way, which is based on the condensed form approach proposed in [23] and [24]. Consider the recursive algorithm:
Initial:Let (P1,Q1)=(P,Q)∈(Cm,n[ξ])2and seti =1.
Step 1. LetU1,V1be unimodular matrix polynomials that produce the Smith form of P1and partitioningU1Q1V1conformably, we get
P2:=U1P1V1=
• Σe 0 0 0
‚
, Q2:=U1Q1V1=
•Q1,1 Q1,2
Q2,1 Q2,2
‚
. di
m−di If£
Q2,1 Q2,2
⁄=0 thenStop,otherwiseproceed to Step 2.
Step 2. Let ˜U2,V2be unimodular matrix polynomials that produce the Smith form of
£Q2,1 Q2,2
⁄:
U˜2
£Q2,1 Q2,2
⁄V2=
•0 Σi
0 0
‚
. ai vi Set
U2:=
•Idi 0 0 U˜2
‚
,P3:=U2P2V2= 2 4
Pˆ11 Pˆ12
0 0
0 0
3
5,Q3:=U2Q3V2= 2 4
Qˆ11 Qˆ12
0 Σi
0 0
3 5,
di ai vi increasei by 1 and repeat the process from Step 1 by applying transformation for the pair ( ˆP11, ˆQ11) with appropriate embedding into the complete matrix polynomial.
End.
Clearly, this algorithm terminates after a finite number of iterations whenai=0, which
yields the matrix polynomial pair ( ˜P, ˜Q) as follows
P˜ := 2 6 6 6 6 6 6 6 6 6 6 6 6 4
Σp 0 ∗ ∗ . . . ∗ 0 ∗ . . . ∗ 0 ∗ . . . ∗ 0 . . . ∗ 0 . . . ∗ . .. ...
0 0
3 7 7 7 7 7 7 7 7 7 7 7 7 5
, ˜Q= 2 6 6 6 6 6 6 6 6 6 6 6 6 4
Q11 Q12 ∗ ∗ . . . ∗ Σq ∗ . . . ∗
0 ∗ . . . ∗
Σq−1 . . . ∗ 0 . . . ∗ . .. ...
Σ1
0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 .
Permuting the 3r d, 5t h, 7t h, . . . block rows to the end, we then have (4.25).
The second claim is obtained by direct calculation of det(P(ξ)−ωQ(ξ)) using the trans-formation (4.25).
One well-known disadvantage of the matrix polynomial approach is that the or-der of dtd inV is not restricted by the orders ofP andQ, and consequently, one has to assume that an initial functionφ is sufficiently smooth. Under this condition, by applying Lemma4.28to the DDAE (4.24), we obtain the following theorem.
Theorem 4.29. Consider the DDAE(4.24)and assume that an initial functionφis suf-ficiently smooth. Then system(4.24)is equivalent (in the sense that there is a bijective mapping between the solution spaces via a unimodular matrix polynomial) to the fol-lowing system
2 4
ΣP 0 P13
0 0 N1
0 0 N2
3 5
2 4
y1(t) y2(t) y3(t) 3 5=
2 4
Q11 Q12 Q13
0 0 ΣQ
0 0 N3
3 5 2 4
y1(t−τ) y2(t−τ) y3(t−τ) 3 5+
2 4
f1 f2
f3 3
5, (4.27)
where the partitioned blocks are as in(4.25), ΣP (resp. ΣQ) is diagonal (resp. upper triangular) with monic polynomials on the main diagonal andN1,N2,N3are strictly upper triangular.
Proof. For the pair of matrix polynomials (P,Q) associated with the DDAE (4.24), we apply Lemma4.28to obtain two unimodular matrix polynomials U,V such that (U P V,U QV) takes the form (4.25). Changing the variable x=V ¡d
dt
¢y, and scaling the system (4.24) withU, it immediately leads to (4.27).
Since the initial function of (4.24) isφ, the initial function of (4.27) will be ˜φ :=
V−1¡d
dt
¢φ. Note that sinceV is unimodular, its inverseV−1exists and it is also a ma-trix polynomial. Partitioning ˜φcorresponding to (4.27), then the solvability of (4.27) is characterized in the following lemma.
Lemma 4.30. Consider the IVP consisting of the DDAE(4.27)together with an initial functionφ˜. Then the following assertions hold:
i) The component y3is fixed by the equation N1
d dt
¶
y3(t)=ΣQ
d dt
¶
y3(t−τ)+f2(t). (4.28)
4.3. Systems on unbounded time intervals 58 ii) The third equation of (4.27)provides a consistency condition for f3.
iii) The second component y2can be freely chosen, so it can be reinterpreted as an input function.
Proof. The claims ii) and iii) are direct consequences of the claim i), so we only need to prove i). Rewriting equation (4.28) explicitly by utilizing the structure ofN1andΣQ, we have
2 6 6 6 4
0 ∗ . . . ∗ 0 . . . ∗ . .. ...
0 3 7 7 7 5
y3(t)= 2 6 6 6 4
Σq ∗ . . . ∗ Σq−1 . . . ∗ . .. ...
Σ1
3 7 7 7 5
y3(t−τ)+f2(t),
whereΣj=diag(pj,1,pj,2, . . . ,pj,rj) with monicpj,1, pj,2, . . . , pj,rj ∈C1,1[ξ].
Partitioningy3conformably asy3=h
zTq zq−1T . . . z1TiT
, we can recursively solve for the componentsz1,z2, . . . ,zq, and hence we obtainy3.
By utilizing Theorem4.29and Lemma4.30, we obtain the solvability properties of the DDAE (4.24) (and in particular, the DDAE (4.1a)). Besides that, the following corol-lary to Theorem4.29states that the DDAE (4.24) is regular if and only if the associated polynomialq(λ,ω) is not identically zero.
Corollary 4.31. Assume that an initial function x|[−τ,0]associated with the DDAE(4.24) is sufficiently smooth. Then the corresponding IVP for(4.24)has a unique solution for every inhomogeneity f ∈C∞([0,∞),Cm)if and only if system(4.24)is of square size and the associated polynomial q(λ,ω) is not identically zero.
Proof. [Necessity.]Due to Lemma4.30, in the condensed form (4.27), the second block column and the last block row do not occur. Therefore, the DDAE (4.24) is of square size and
det(U) det(P(ξ)−ωQ(ξ)) det(V)=det
•Σp−ωQ11 P13−ωQ13
0 N1−ωΣQ
‚¶
. Denote by ˆqthe size ofΣQ. SinceU,V are unimodular, we have
det(P(ξ)−ωQ(ξ))
=(det(U))−1(det(V))−1det(ΣP−ωQ11) det 0 B B B
@ 2 6 6 6 4
−ωΣq ∗ . . . ∗
−ωΣq−1 . . . ∗ . .. ...
−ωΣ1
3 7 7 7 5 1 C C C A
=(det(U))−1(det(V))−1det(ΣP−ωQ11) (−ω)qˆ
q
Y
i=1
det(Σi), which clearly is a non-zero polynomial.
[Sufficiency.] Assume that the associated polynomial q(λ,ω) is not identically zero.
Thus, Lemma 4.28implies the existence of two unimodular matrix polynomials U, V such thatU P V andU QV take the form (4.26).
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.