# Solution Properties of Linear Descriptor (Singular) Matrix Differential Systems of Higher Order with (Non-) Consistent Initial Conditions

Authors: Athanasios A. Pantelous; Athanasios D. Karageorgos; Grigoris I. Kalogeropoulos; Kostas G. Arvanitis
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

In some interesting applications in control and system theory, linear descriptor (singular) matrix differential equations of higher order with time-invariant coefficients and (non-) consistent initial conditions have been used. In this paper, we provide a study for the solution properties of a more general class of the Apostol-Kolodner-type equations with consistent and nonconsistent initial conditions.


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## Body

## 1. Introduction

Linear Time-Invariant  (LTI) (i.e., with constant matrix coefficients) descriptor matrix differential systems of type (1.1) with several kinds of inputs(1.1)FX(r)(t)=AX(t)+BU(t),
where F,A∈ℳ(n×m;𝔽), B∈ℳ(n×μ;𝔽), and U∈𝒞∞(𝔽,ℳ(μ×m;𝔽)), often appear in control and system theory. For instance, (1.1) identifies and models effectively many physical, engineering, mechanical, as well as financial phenomena. For instance, we can provide in economy, the well-known, famous input-output Leontief model and its several important extensions, advice [1, 2]. Moreover, in the beginning of this introductive section, we should point out that singular perturbations arise often in systems whose dynamics have sufficiently separate slow and fast parts. Now by considering the classical proportional feedback controller(1.2)U(t)=-F̃X(t),
we can obtain (1.3), where G=A-BF̃.Our long-term purpose is to study the solution of LTI descriptor matrix differential systems of higher order (1.1) into the mainstream of matrix pencil theory, that is,(1.3)FX(r)(t)=GX(t),
where, for (1.1), (1.2), and (1.3), rth is the order of the systems, F,G∈ℳ(n×m;𝔽) (where matrix F is singular), and X∈𝒞∞(𝔽,ℳ(m×l;𝔽) (note that 𝔽 can be either ℝ or ℂ). For the sake of simplicity we set in the sequel ℳn=ℳ(n×n;𝔽) and ℳn,m=ℳ(n×m;𝔽).Matrix pencil theory has been extensively used for the study of LTI descriptor differential equations of first order; see, for instance, [3–6]. Systems of type (1.3) are more general, including the special case when F=In, where In is the identity matrix of ℳn, since the well-known class of higher order linear matrix differential of Apostol-type equations is derived straightforwardly; see [7–10]. In the same way, system (1.1) might be considered as the more general class of higher order linear descriptor matrix differential equations of Apostol-Kolodner type, since Kolodner has also studied such systems in nondescriptor form; see also [8].Recently, in [5], the regular case of higher order linear descriptor matrix differential equations of Apostol-Kolodner type has been investigated. The regular case is simpler, since it considers square matrix coefficients and the Weierstrass canonical form has been applied. Actually, the recent work is a nonstraight generalization of [5]. Analytically, in this article, we study the linear descriptor matrix differential equations of higher order whose coefficients are rectangular constant matrices, that is, the singular case is examined. Adopting several different methods for computing the matrix powers and exponential, new formulas representing auxiliary results are obtained. This allows us to prove properties of a large class of linear matrix differential equations of higher order; in particular results of Apostol and Kolodner are recovered; see also [5, 8].Finally, it should be mentioned that in the classical theory of linear (descriptor) differential systems, see, for instance, [1, 2, 11–13], one of the important features is that not every initial condition X0 admits a functional solution. Thus, we shall call X0a consistent initial conditionfor (1.3) at to if there is a solution for (1.3), which is defined on some interval [to,to+γ], γ>0 such that X(to)=X0.On the other hand, it is not rare to appear in some practical significant applications that the assumption of theinitial conditions for (1.3) can be nonconsistent,that is, X(to)≠X0.


## 2. Mathematical Background and Notations

In this preliminary section, some well-known concepts and definitions for matrix pencils are introduced. This discussion is highly important, in order to understand better the results of Section3.Definition 2.1.
GivenF,G∈ℳnm and an indeterminate s∈𝔽, the matrix pencil sF-G is called regular when m=n and det(sF-G)≠0 (where 0 is the zero element of ℳ(1,𝔽)). In any other case, the pencil is called singular.In this paper, as we are going to see in the next paragraph, we consider the case that the pencil issingular. The next definition is very important, since the notion of strict equivalence between two pencils is presented.Definition 2.2.
The pencilsF-G is said to be strictly equivalent to the pencil sF̃-G̃ if and only if there exist nonsingular P∈ℳn and Q∈ℳm such that
(2.1)P(sF-G)Q=sF̃-G̃.The characterization of singular pencils requires the definition of additional sets of invariants known as the minimal indices.Let us assume thatr=rank𝔽(s)(sF-G), where 𝔽(s) denotes the field of rational functions in s having coefficients in the field 𝔽. The equations(2.2)(sF-G)x̲(s)=0̲,ψ̲T(s)(sF-G)=0̲T
have nonzero solutions x̲(s) and ψ̲(s) which are vectors in the rational vector spaces(2.3)𝒩right(s)≜𝒩right(sF-G),𝒩left(s)≜𝒩left(sF-G),
respectively, where(2.4)𝒩right(s)≜{x̲(s)∈𝔽(s)m:(sF-G)x̲(s)=0̲n},𝒩left(s)={ψ̲(s)∈𝔽(s)n:ψ̲T(s)(sF-G)=0̲mT}.
The sets of the minimal degrees {vi,1≤i≤m-r} and {uj,1≤j≤n-r} are known as column minimal indices (c.m.i.) and row minimal indices (r.m.i.) of sF-G, respectively. Furthermore, if r=rank𝔽(s)(sF-G), it is evident such that(2.5)r=∑i=g+1m-rvi+∑j=h+1n-ruj+rank𝔽(s)(sFw-Gw),
where sFw-Gw is the complex Weierstrass canonical form; see [3].LetB1,B2,...,Bn be elements of ℳn.The direct sum of them denoted byB1⊕B2⊕⋯⊕Bn is the blockdiag{B1,B2,…,Bn}.Thus, there existsP∈ℳm and Q∈ℳn such that the complex Kronecker formsFk-Gk of the singular pencil sF-G is defined as follows:(2.6)sFk-Gk≜𝕆h,g⊕sΛv-λv⊕sΛuT-λuT⊕sIp-Jp⊕sHq-Iq,
where v=∑i=g+1m-rvi, u=∑j=h+1n-ruj, p=∑j=1κpj, and q=∑j=1σqj (see below). In more details, the following are given.(S1) Matrix𝕆h,g is uniquely defined by the sets {0,0,…,0}︸g and {0,0,…,0}︸h of zero column and row minimal indices, respectively.(S2) The second normal blocksΛv-λv is uniquely defined by the set of nonzero column minimal indices (a new arrangement of the indices of v must be noted in order to simplify the notation) {vg+1≤⋯≤vm-r} of sF-Q and has the form(2.7)sΛv-λv≜sΛvg+1-λvg+1⊕⋯⊕sΛvi-λvi⊕⋯⊕sΛvm-r-λvm-r,
where Λvi=[Ivi⋮0̲]∈ℳvi,vi+1, λvi=[Hvi⋮ε̲vi]∈ℳvi,vi+1 for every i=g+1,g+2,...,m-r, and Ivi and Hvi denote the vi×vi identity and the nilpotent (with index of nilpotency vi) matrix, respectively. 0̲ and ε̲vi=[0⋯01]T are the zero column and the column with element 1 at the viplace, respectively.(S3) The third normal blocksΛuT-λuT is uniquely determined by the set of nonzero row minimal indices (a new arrangement of the indices of u must be noted in order to simplify the notation) {uh+1≤⋯≤un-r} of sF-G and has the form(2.8)sΛuT-λuT≜sΛuh+1T-λuh+1T⊕⋯⊕sΛujT-λujT⊕⋯⊕sΛun-rT-λun-rT,
where ΛujT=[e̲ujT⋯Huj]∈ℳuj+1,uj, λujT=[0̲T⋯Iuj]∈ℳuj+1,uj for every j=h+1,h+2,…,m-r, and Iuj and Huj denote the uj×uj identity and nilpotent (with index of nilpotency uj) matrix, respectively. 0̲ and e̲uj=[1⋯00]T are the zero column and the column with element 1 at the first place, respectively.(S4-S5) The forth and the fifth normal matrix block is the complex Weierstrass formsFw-Gw of the singular pencil sF-G which is defined by(2.9)sFw-Qw≜sIp-Jp⊕sHq-Iq,
where the first normal Jordan-type element is uniquely defined by the set of finite elementary divisors (f.e.d.)(2.10)(s-a1)p1,…,(s-aκ)pκ,∑j=1κpj=p
of sF-G and has the form(2.11)sIp-Jp≜sIp1-Jp1(a1)⊕⋯⊕sIpκ-Jpκ(aκ).
And also the q blocks of the second uniquely defined block sHq-Iq correspond to the infinite elementary divisors (i.e.d.)(2.12)ŝq1,…,ŝqσ,∑j=1σqj=q
of sF-G and have the form(2.13)sHq-Iq≜sHq1-Iq1⊕⋯⊕sHqσ-Iqσ.ThusHq is a nilpotent element of ℳn with index q̃=max{qj:j=1,2,…,σ}, where(2.14)Hqq̃=𝕆,
and Ipj,Jpj(aj),Hqj are the matrices(2.15)Ipj=[10⋯001⋯0⋮⋮⋱⋮00⋯1]∈ℳpj,Jpj(aj)=[aj10⋯00aj1⋯0⋮⋮⋱⋮⋮000aj10000aj]∈ℳpj,Hqj=[010⋯0001⋯0⋮⋮⋱⋮⋮0000100000]∈ℳqj.In the last part of this introductive section, some elements for the analytic computation ofeA(t-to), t∈[to,∞) are provided. To perform this computation, many theoretical and numerical methods have been developed.Thus, the interested reader might consult papers in [1, 2, 7–10, 14–16], and the references therein. In order to obtain more analytic formulas, the following known results should be mentioned.Lemma 2.3 (see [15]).
(2.16)eJpj(aj)(t-to)=[dk1k2]pj,
where
(2.17)dk1k2={eaj(t-to)(t-to)k2-k1(k2-k1)!,1≤k1≤k2≤pj,0,otherwise.Another expression for the exponential matrix of Jordan block, see (2.18), is provided by the following lemma.Lemma 2.4 (see [15]).
(2.18)eJpj(aj)(t-to)=∑i=0pj-1fi(t-to)[Jpj(aj)]i,
where the fk(t-to),s satisfy the following system of pjequations:
(2.19)∑i=kpj-1(ik)aji-kfi(t-to)=(t-to)kk!eajt,k=1,2,...,pj,[Jpj(aj)]i=[βk1k2(i)]pj,for1≤k1,k2≤pj,
where βk1k2(i)=(ik2-k1)aji-(k2-k1).


## 3. Solution Space for Consistent Initial Conditions

In this section, the main results for consistent initial conditions are analytically presented for the singular case. The whole discussion extends the existing literature; see, for instance [8]. Now, in order to obtain a solution, we deal with consistent initial value problem. More analytically, we consider the system(3.1)FX(r)(t)=GX(t),
with known
(3.2)X(to),X′(to),…,X(r-1)(to),
where F,G∈ℳn,m (where matrix F is singular), and X∈𝒞∞(𝔽,ℳm,l).From the singularity ofsrF-G, there exist nonsingular matrices P∈ℳn and Q∈ℳm such that (see also Section 2)(3.3)PFQ=Fk=𝕆h,g⊕Λv⊕ΛuT⊕Ip⊕Hq,PGQ=Gk=𝕆h,g⊕λv⊕λuT⊕Jp⊕Iq,
where Λv,λv, ΛuT, λuT, Ip,Jp,Hq, and Iq are given by(3.4)Λv=Λvg+1⊕⋯⊕Λvi⊕⋯⊕Λvm-r,λv≜λvg+1⊕⋯⊕λvi⊕⋯⊕λvm-r,ΛuT≜Λuh+1T⊕⋯⊕ΛujT⊕⋯⊕Λun-rT,λuT≜λuh+1T⊕⋯⊕λujT⊕⋯⊕λun-rT,Ip=Ip1⊕⋯⊕Ipκ,Jp=Jp1(a1)⊕⋯⊕Jpκ(aκ),Hq≜Hq1⊕⋯⊕Hqσ,Iq=Iq1⊕⋯⊕Iqσ.By using the Kronecker canonical form, we might rewrite system (1.3), as the following lemma denotes.Lemma 3.1.
System (1.3) may be divided into five subsystems:
(3.5)𝕆h,gYg(r)(t)=𝕆h,gYg(t),(3.6)ΛvYv(r)(t)=λvYv(t),(3.7)ΛuTYu(r)(t)=λuTYu(t),
the so-called slow  subsystem
(3.8)Yp(r)(t)=JpYp(t),
and the relative fast  subsystem
(3.9)HqYq(r)(t)=Yq(t).Proof.
Consider the transformation(3.10)X(t)=QY(t),
where Q∈ℳm and Y∈𝒞∞(𝔽,ℳm,l). Substituting the previous expression into (1.3), we obtain
(3.11)FQY(r)(t)=GQY(t).
Whereby, multiplying by P, we arrive at
(3.12)FkY(r)(t)=GkY(t).
Moreover, we can write Y(t) as
(3.13)Y(t)=[YgT(t)YvT(t)YuT(t)YpT(t)YqT(t)]T∈ℳm,l,
where Yg(t)∈𝒞∞(𝔽,ℳg,l), Yv(t)∈𝒞∞(𝔽,ℳv,l), Yv(t)∈𝒞∞(𝔽,ℳu,l), Yp(t)∈𝒞∞(𝔽,ℳp,l), and Yq(t)∈𝒞∞(𝔽,ℳq,l). Note that g is the number of zero column entries, v=∑i=g+1m-rvi, u=∑j=h+1n-ruj, p=∑j=1κpj, and q=∑j=1σqj.
And taking into account the above expressions, we arrive easily at (3.5)–(3.9).Proposition 3.2.
For system (3.5), the elements of the matrix Yg(t)∈𝒞∞(𝔽,ℳ(g×l;𝔽)) can be chosen arbitrarily.Proof.
Since𝕆h,g, it is profound that any g-column vector can be chosen.Proposition 3.3.
The analytic solution of system(3.14)ΛviYvi(r)(t)=λviYvi(t)
is given by the expression
(3.15)Yvi(t)=[Y̲1(t)Y̲2(t)⋯Y̲l(t)]=[yλ,j(t)]λ=1,2,…,vij=1,2,…,l,
where
(3.16)yλ,j(t)=∫⋯∫yλ+1,j(t)dt⋯dt︸r-times+∑ξ=1rcλ,r-ξ+1tξ-1(ξ-1)!,
where yλ+1,j(t) is an arbitrary function, for every λ=1,2,…,vi, i=g+1,…,m-r, and j=1,2,…,l. (Note that cλ,r-ξ+1 should be uniquely determined via the given initial conditions.)Proof.
System (3.14) is rewritten as
(3.17)[Ivi⋮0̲]Yvι(r)(t)=[Hvi⋮ε̲vi]Yvι(t),
for every i=g+1,g+2,…,m-r. Now, we denote
(3.18)Yvi(t)=[Ψvi(t)y̲1(t)],
where Ψvi(t)∈ℳvil, Ψvi(t)=[Y̲1(t)Y̲2(t)⋯Y̲l(t)] with Y̲j(t)=[y1,j(t)y2,j(t)⋯yvi,j(t)]T, and y̲1(t)∈ℳ1l (vector, 1×l).
Thus,
(3.19)[Ivi⋮0̲][Ψvi(r)(t)y̲1(r)(t)]=[Hvi⋮ε̲vi][Ψvi(t)y̲1(t)],
or, equivalently, we obtain
(3.20)Ψvi(r)(t)=HviΨvi(t)+ε̲viy̲1(t).
Note that ε̲viy̲1(t) is a matrix with vi×vi-elements as follows
(3.21)ε̲viy̲1(t)=[Ỹ̲1(t)Ỹ̲2(t)⋯Ỹ̲l(t)]≜[𝕆vi-1,lyvi+1,1(t)yvi+1,2(t)⋯yvi,+1l(t)]
where Ỹ̲j(t)=[00⋯yvi+1,j(t)]T, for j=1,2,…,l.
Consequently, (3.20) is rewritten as follows:
(3.22)[Y̲1(r)(t)Y̲2(r)(t)⋯Y̲l(r)(t)]=[HviY̲1(t)HviY̲2(t)⋯HviY̲l(t)]+[Ỹ̲1(t)Ỹ̲2(t)⋯Ỹ̲l(t)],
or, equivalently,
(3.23)Y̲j(r)(t)=HviY̲j(t)+Ỹ̲j(t),
and eventually, as a scalar system, we obtain
(3.24)y1,j(r)(t)=y2,j(t),y2,j(r)(t)=y3,j(t),…,yvi-1,j(r)(t)=yvi,j(t),yvi,j(r)(t)=yvi+1,j(t).
Denote that elementyvi+1,j(t) is an arbitrary function; then the solution is given iteratively, as follows.
Firstly, we take the equationyvi,j(r)(t)=yvi+1,j(t) for every j=1,2,…,l,
(3.25)yvi,j(r-1)(t)=∫yvi+1,j(t)dt+cvi,1,…,yvi,j(t)=∫⋯∫yvi+1,j(t)dt⋯dt︸r-times+∑ξ=1rcvi,r-ξ+1tξ-1(ξ-1)!.
We continue the procedure, for yvi-1,j(r)(t)=yvi,j(t), and so forth. Thus, we finally obtain (3.15).With the following remark, we obtain the solution of subsystem (3.6).Remark 3.4.
The solution of subsystem (3.6) is given by
(3.26)Yv(t)=Yvg+1(t)⊕⋯⊕Yvi(t)⊕⋯⊕Yvm-r(t),
where the results of Proposition 3.3 are also considered.Remark 3.5.
Considering the solution (3.14), and therefore the system (3.6), it should be pointed out that the solution is not unique, since the last component of the solution vector is chosen arbitrary. Moreover, it is worth to be emphasized here that the solution of the singular system (1.3) is not unique.Proposition 3.6.
The system(3.27)ΛujTYuj(r)(t)=λujTYuj(t)
has only the zero solution.Proof.
Consider that system (3.27) can be rewritten as follows:
(3.28)[e̲ujT⋯Huj]Yuj(r)(t)=[0̲T⋯Iuj]Yuj(t)
for every j=h+1,h+2,…,m-r.
Afterwards, we obtain straightforwardly the following system:
(3.29)[e̲ujTYuj(r)(t)HujYuj(r)(t)]=[0̲TYuj(t)]⇔{e̲ujTYuj(r)(t)=0̲T,HujYuj(r)(t)=Yuj(t).
Now, by successively taking rth derivatives with respect to t on both sides of
(3.30)HujYuj(r)(t)=Yuj(t),
and left multiplying by the matrix Huj, uj-1 times (where uj is the index of the nilpotent matrix Huj, i.e., Hujuj=𝕆), we obtain the following equations:
(3.31)Huj2Yuj(2r)(t)=HujYuj(r)(t),…,HujujYuj(ujr)(t)=Hujuj-1Yuj((uj-1)r)(t).
Thus, we conclude to the following expression:
(3.32)Yuj(t)=HujYuj(r)(t)=Huj2Yuj(2r)(t)=⋯=HujujYuj(ujr)(t)=𝕆.Remark 3.7.
Consequently, the subsystem (3.7) has also the zero solution.Proposition 3.8 (see [5]).
(a) The analytic solution of the so-called slow  subsystem (3.8) is given by
(3.33)Yp(t)=LR⊕j=1κ⊕k=0r-1eJjk(λjk)(t-to)R-1Z(to),
where L=[Ip𝕆⋯𝕆]∈ℳp,pr; R∈ℳprsuch that J=R-1AR.
Note thatJ∈ℳpr is the Jordan Canonical form of matrix
(3.34)A=[𝕆Ip𝕆⋯𝕆𝕆𝕆Ip⋯𝕆⋮⋮⋮⋱⋮𝕆𝕆𝕆⋯IpJp𝕆𝕆⋯𝕆],
and Z(to)=[YpT(to)Y'pT(to)⋯Yp(r-1)T(to)]T∈ℳpr,l.
The eigenvalues of the matrixA are given by
(3.35)λjk=|aj|r(cos2kπ+φjr+zsin2kπ+φjr),
where aj=|aj|(cosφj+zsinφj) (aj finite elementary divisors) and z2=-1 for every j=1,2,…,κ and k=0,1,2,…,r-1.
(b) However, the relative fast subsystem (3.9) has only the zero solution.
It is worth to say that the results of Ben Taher and Rachidi [14] can be compared with the results of Proposition 3.8, which has been discussed extensively in [5].Remark 3.9.
The characteristic polynomial ofA is φ(λ)=∏j=1κ(λr-aj)pj, with ai≠aj for i≠j and ∑j=1κpj=p. Without loss of generality, we define that
(3.36)d1=τ1,d2=τ2,…,dl=τl,anddl+1<τl+1,…,dκ<τκ,
where dj,τj,j=1,2,…,κ, are the geometric and algebraic multiplicities of the given eigenvalues aj, respectively.
(i) Consequently, whendj=τj, then
(3.37)Jjk(λjκ)=[λjkλjk⋱λjk]∈ℳτj
is also a diagonal matrix with diagonal elements of the eigenvalue λjk, for j=1,…,l.
(ii) Whendj<τj, then
(3.38)Jjk,zj=[λjk1λjk1λjk⋱⋱1λjk]∈ℳzj
for j=l+1,l+2,…,κ, and zj=1,2,…,dj.
Hence, the set of consistent initial conditions for system
(3.39)FkY(r)(t)=GkY(t)
has the following form:
(3.40)Y(k)(to)={[Yg(k)T(to)Yv(k)T(to)𝕆uTYp(k)T(to)𝕆qT]T;k=0,…,r-1}.
In more details, since we have considered (3.14) and we can denote
(3.41)Q=[Qn,gQn,vQn,uQn,pQn,q],
then we can derive the following expression:
(3.42)X(to)=[Qn,gQn,vQn,uQn,pQn,q][YgT(to)YvT(to)𝕆uTYpT(to)𝕆qT]T=Qn,gYg(to)+Qn,vYv(to)+Qn,pYp(to).
Then, the set of consistent initial conditions for (1.3) is given by
(3.43){Qn,gYg(to)+Qn,vYv(to)+Qn,pYp(to)Qn,gYg′(to)+Qn,vYv′(to)+Qn,pYp′(to)⋯Qn,gYg(r-1)(to)+Qn,vYv(r-1)(to)+Qn,pYp(r-1)(to)}.
Now, taking into consideration (3.2) and (3.43), we conclude to
(3.44)X(to)=Qn,gYg(to)+Qn,vYv(to)+Qn,pYp(to),X'(to)=Qn,gYg′(to)+Qn,vYv′(to)+Qn,pYp′(to),⋮X(r-1)(to)=Qn,gYg(r-1)(to)+Qn,vYv(r-1)(to)+Qn,pYp(r-1)(to).Theorem 3.10.
The analytic solution of (3.2) is given by
(3.45)X(t)=Qn,gYg(t)+Qn,vYv(t)+Qn,pLR⨁j=1κ⨁k=0r-1eJjk(λjk)(t-to)R-1Z(to)
for Yv(t)=Yvg+1(t)⊕⋯⊕Yvi(t)⊕⋯⊕Yvm-r(t), where Yvi(t)=[yλ,j]λ=1,2,…,vi,j=1,2,…,l, for i=g+1,…,m-r and
(3.46)yλ,j(t)=∫⋯∫yλ+1,j(t)dt⋯dt︸r-times+∑ξ=1rcλ,r-ξ+1tξ-1(ξ-1)!.
(Note that cλ,r-ξ+1 should be uniquely determined via the given initial conditions.)
The matrixYg(t) is arbitrarily chosen. Moreover
(3.47)L=[Ip𝕆⋯𝕆]∈ℳp,pr,R∈ℳpr
such that J=R-1AR, where J∈ℳpr is the Jordan Canonical form  of matrix A, and
(3.48)Z(to)=[YpT(to)Y'pT(to)⋯Yp(r-1)T(to)]T∈ℳpr,l.Proof.
Using the results of Lemma3.1, Propositions 3.2–3.8, Remarks 3.4 and 3.7 and (3.10) then we obtain
(3.49)X(t)=QY(t)=[Qn,gQn,vQn,uQn,pQn,q]·[YgT(t)YvT(t)𝕆uTYpT(t)𝕆qT]T=Qn,gYg(t)+Qn,vYv(t)+Qn,pYp(t).
Finally, (3.45) is derived.The next remark connects the solution with the set of initial condition for the system (1.3).Remark 3.11.
IfQ̃n,p is the existing left inverse of Qn,p, then considering also (3.10)
(3.50)Z(to)=[Yp(to)Y'p(to)⋮Yp(r-1)(to)]=[Q̃p,nXp(to)Q̃p,nX'p(to)⋮Q̃p,nXp(r-1)(to)]=[Q̃p,nQ̃p,n⋱Q̃p,n][Xp(to)X'p(to)⋮Xp(r-1)(to)]=Q̃Ψ(to).
Finally, the solution (3.45) is given by
(3.51)X(t)=Qn,gYg(t)+Qn,vYv(t)+Qn,pLR⨁j=1κ⨁k=0r-1eJjk(λjk)(t-to)R-1Q̃Ψ(to),
where Ψ(to)=[XpT(to)X′pT(to)⋯Xp(r-1)T(to)]T∈ℳpr,l and Q̃n,p is the existing left inverse of Qn,p.The following two expressions, that is, (3.52) and (3.54) are based on Lemmas 2.3 and 2.4. Thus, two new analytical formulas are derived which are practically very useful. Their proofs are straightforward exercise of Lemmas 2.3, 2.4, and (3.51).Proposition 3.12.
Considering the results of Lemma2.3, one obtains the expression
(3.52)X(t)=Qn,gYg(t)+Qn,vYv(t)+Qn,pLR[(⨁j=0l⨁k=0r-1eλjk(t-to)Iτjk)⨁(⨁j=l+1κ⨁k=0r-1⨁zj=1dj(dk1k2)zj)]R-1Q̃Ψ(to),
where
(3.53)dk1k2={eλjk(t-to)(t-to)k2-k1(k2-k1)!,1≤k1≤k2≤zj,0,otherwise
for j=l+1,l+2,…,κ, and zj=1,2,…,dj.Another expression for the exponential matrix of Jordan block, see (2.18), is provided by the following lemma.Proposition 3.13.
Considering the results of Lemma2.4, one obtains the expression
(3.54)X(t)=Qn,gYg(t)+Qn,vYv(t)+Qn,pLR[(⨁j=0l⨁k=0r-1eλjk(t-to)Iτjk)⨁(⨁j=l+1κ⨁k=0r-1⨁zj=1dj∑i=0zj-1fi(t-to)[Jzj(λjk)]i)]×R-1Q̃Ψ(to),
where the fk(t-to),s satisfy the following system of zj (for zj=1,2,…,dj) equations:
(3.55)∑i=kzj-1(ik)aji-kfi(t-to)=(t-to)kk!eλjkt,k=1,2,…,zj
and [Jzj(λjk)]i=[βk1k2(i)]zj, for 1≤k1,k2≤zj where βk1k2(i)=(ik2-k1)aji-(k2-k1).Analyzing more the results of this section, see Theorem3.10 and Lemma 2.3 and 2.4, we can present briefly a symbolical algorithm for the solution of system (1.3).Symbolical Algorithm
Step 1.
Determine the pencilsF-G.Step 2.
Calculate the expressions (3.3). Thus, we have to find the f.e.d, i.e.d., r.m.i, c.m.i, and so forth, (i.e., the complex Kronecker formsFk-Gk of the singular pencil sF-G here; it should be noticed that this step is not an easy task; some parts are still under research).Step 3.
Using the results of Step2, determine the matrices Λv,λv, ΛuT, λuT, Ip,Jp,Hq and Iq.Step 4.
DetermineQn,g,Qn,v, Qn,p, L, R (using the Jordan canonical form of matrix A), Q̃ and Ψ(to) (see Remark 3.9).Step 5.
Considering the transformation (3.10), that is, X(t)=QY(t), we obtain (3.54). Then the following.Sub step 5.1.
Choose an arbitrary matrixYg(t).Sub step 5.2.
Determine the matrixYv(t), that is,
(3.56)Yv(t)=Yvg+1(t)⊕…⊕Yvi(t)⊕…⊕Yvm-r(t),
where Yvi(t)=[yλ,j]λ=1,2,…,vi,j=1,2,…,l, for i=g+1,…,m-r and
(3.57)yλ,j(t)=∫⋯∫yλ+1,j(t)dt⋯dt︸r-times+∑ξ=1rcλ,r-ξ+1tξ-1(ξ-1)!.Step fta
Following the results ofLemma4, determine
(3.58)⨁j=1κ⨁k=0r-1eJjk(λjk)(t-to)=(⨁j=0l⨁k=0r-1eλjk(t-to)Iτjk)⨁(⨁j=l+1κ⨁k=0r-1⨁zj=1dj(dk1k2)zj),
where
(3.59)dk1k2={eλjk(t-to)(t-to)k2-k1(k2-k1)!,1≤k1≤k2≤zj,0,otherwise
for j=l+1,l+2,…,κ, and zj=1,2,…,dj.Step ftb
Following the results ofLemma5, determine
(3.60)⨁j=1κ⨁k=0r-1eJjk(λjk)(t-to)=(⨁j=0l⨁k=0r-1eλjk(t-to)Iτjk)⊕(⨁j=l+1κ⨁k=0r-1⨁zj=1dj∑i=0zj-1fi(t-to)[Jzj(λjk)]i),
where fk(t-to),s satisfy the following system of zj (for zj=1,2,…,dj) equations:
(3.61)∑i=kzj-1(ik)aji-kfi(t-to)=(t-to)kk!eλjkt,k=1,2,…,zj,
and [Jzj(λjk)]i=[βk1k2(i)]zj, for 1≤k1,k2≤zj, and βk1k2(i)=(ik2-k1)aji-(k2-k1).Example 3.14 (withconsistent initial condition).
Consider the 2nd order system(3.62)FẌ(t)=GX(t),
where F,G∈ℳ(11×12;ℝ), and X∈𝒞∞(ℝ,ℳ(12×5;ℝ)), with
(3.63)F=[000000000000011-10-10-100000001000000000-1001100000000-11-101100000-100010100000000000000000100-1-10010000-100010001000-10011-10000101000-1000-100]∈ℳ11,12,G=[0000000000000-100110000000000000000000000010000000100-1-20000000000001000000-100010100000000-100011000-1000100010000000100001-101000-1000-101]∈ℳ11,12,with known consistent initial conditions X(to),X'(to).
From the singularity ofsF-G, there exist nonsingular matrices
(3.64)P=[1000000000001100000000001000000000001000000001011000000000001000000000001000000000001000000000001000100100001000000000101]∈ℳ11,Q=[110000001000010001000000001000010000000100000000010010000000000001000000000000100000010000010000000010001000010000000100000000000011000000000001]∈ℳ12
Then, using (3.3), we obtain
(3.65)PFQ=Fk=𝕆1,2⊕Λ3⊕Λ2T⊕I2⊕H2,PGQ=Gk=𝕆1,2⊕λ3⊕λ2T⊕J2⊕I2,
where Λ2,λ2, Λ2T, λ2T, I2,J2, and H2 are given by
(3.66)Λ3=[I3⋮0̲]=[100001000010]∈ℳ3,4,λ3=[H3⋮ε̲3]=[001000000001]∈ℳ2,3,Λ2T=[e̲2T⋯H2]=[100100]∈ℳ3,2,λ2T=[0̲T⋯I2]=[001001]∈ℳ3,2,I2=[1001],J2(1)=[1101],H2=[0100].
Considering the transformation (3.10), that is, X(t)=QY(t), we obtain the results of Lemma 2.3 (see also J, below):
(3.67)X(t)=Q12,g=2Yg=2(t)+Q12,v=2Yv=2(t)+Q12,p=2LR[⨁j=12⨁zj=12(dk1k2)zj]R-1Q̃Ψ(to),
where
(3.68)Q12,g=2=[110000001000010001000000],Q12,v=2=[001000010000000100000000],Q12,p=2=[000010001000010000000100].
(i)Yg=2(t)∈ℳ2,5 is an arbitrarily chosen matrix.
(ii)Yv=2(t)∈ℳ2,5 with Y2(t)=[yλ,j]λ=1,2,j=1,2,…,5, and
(3.69)yλ,j(t)=∬yλ+1,j(t)dtdt+∑ξ=12cλ,3-ξtξ-1(ξ-1)!.
(iii)
(3.70)L=[I2𝕆2]=[10000100]∈ℳ2,4.
(iv)
(3.71)R=[141414-140-12012-140140012012],R-1=[21-200-10121200101],J=R-1AR=[-11000-10000110001].
(We have only two eigenvalues, λ1=-1 and λ2=1.)
(v)
(3.72)Q̃Ψ(to)=[Q̃p=2,12Q̃p=2,12][Xp(to)X'p(to)],
where X(to),X'(to) are known, and
(3.73)Q̃p=2,12=[0100-1-10010000-10001000100].
(vi) With
(3.74)
Combining the above arithmetic results, the analytic solution of system (3.62) is given by considering (3.58).


## 4. Solution Space Form of Nonconsistent Initial Conditions

In this short section, we would like to describe briefly the impulse behaviour of the solution of the original system (1.3), at time to, see also [11–13]. In that case, we reformulate Proposition 3.6, so that the impulse solution is finally obtained. Note that in this part of the paper the condition that X∈𝒞∞(𝔽,ℳ(m×l;𝔽)) does not hold anymore, since we are interesting for solutions with impulsive behaviour (again 𝔽 can be either ℝ or ℂ).Moreover, we assume that the space of nonconsistent initial conditions is denoted by𝒞0*, which is called also redundancy space. Then, considering also Lemma 3.1, and especially (3.7) and (3.9), we have the nonconsistent initial condition that is(4.1)Yuj(k)(to)≠Yuj,o,forj=h+1,h+2,…,m-r,Yqz(k)(to)≠Yqz,o,forz=1,2,…,σ,
for k=0,1,…,r-1.In order to be able to find a solution, we use the classical method of Laplace transformation. This method has been applied several times in descriptor system theory; see for instance, [4, 5, 11].Proposition 4.1.
The analytic solution of the system (3.27) is given by
(4.2)Yuj(t)=-∑k=0r-1∑ζ=0uj-2δ(rζ+r-1-k)(t)Hujζ+1Yuj,o(k),
where δ and δ(k) are the delta function of Dirac and its derivatives, respectively.Proof.
Let us start by observing that—as it is well known—there exists auj∈ℕ such that Hujuj=𝕆, that is, the index of nilpotency equals uj of Huj.
Moreover, system (3.27) can be rewritten as follows; see also proof of Proposition 3.6,
(4.3)[e̲ujTYuj(r)(t)HujYuj(r)(t)]=[0̲TYuj(t)]⇔{e̲ujTYuj(r)(t)=0̲t,HujYuj(r)(t)=Yuj(t).
Where by taking the Laplace transformation of HujYuj(r)(t)=Yuj(t), the following expression derives:
(4.4)Hujℑ{Yuj(r)(t)}=ℑ{Yuj(t)},
and by defining ℑ{Yuj(t)}=𝒳uj(s), we obtain
(4.5)(srHuj-Iuj)𝒳uj(s)=Huj∑k=0r-1sr-1-kYuj,o(k).
Since uj is the index nilpotency of Huj, it is known that
(4.6)(srHuj-Iuj)-1=-∑ζ=0uj-1(srHuj)ζ,
where Huj0=Iuj; see, for instance [4, 10]. Thus, substituting the above expression into (4.5), the following equation is being taken:
(4.7)𝒳uj(s)=-∑k=0r-1∑ζ=0uj-2srζ+r-1-kHujζ+1Yuj,o(k).
Since ℑ{δ(k)(t)}=sk, (4.7) is transformed into (4.8):
(4.8)𝒳uj(s)=-∑k=0r-1∑ζ=0uj-2ℑ{δ(rζ+r-1-k)}(t)Hujζ+1Yuj,o(k).
Now, by applying the inverseLaplace transformation into (4.8), the equation (4.2) is derived.Remark 4.2.
The analytic solution of the subsystem (3.7) is given by
(4.9)Yu(t)=Yuh+1(t)⊕⋯⊕Yuj(t)⊕⋯⊕Yun-r(t),
where the results of Proposition 4.1 are also considered.Similarly to Proposition4.1, we can prove the following proposition.Proposition 4.3.
The analytic solution of the system (3.9) is given by
(4.10)Yq(s)=-∑k=0r-1∑z=0q*-2δ(rz+r-1-k)(t)Hqz+1Yq,o(k).Theorem 4.4.
The analytic solution of (1.3) is given by
(4.11)X(t)=Qn,gYg(t)+Qn,vYv(t)+Qn,uYu(t)+Qn,pLR⊕j=0κ⊕k=0r-1eJjk(λjk)(t-to)R-1Z(to)-Qn,q∑k=0r-1∑z=0q*-2δ(rz+r-1-k)(t)Hqz+1Yq,o(k).
The matrix Yg(t) is arbitrarily chosen. For Yv(t)=Yvg+1(t)⊕⋯⊕Yvi(t)⊕⋯⊕Yvm-r(t)where Yvi(t)=[yλ,j]λ=1,2,…,vi,j=1,2,…,l, for i=g+1,…,m-r, and
(4.12)yλ,j(t)=∫⋯∫yλ+1,j(t)dt⋯dt︸r-times+∑ξ=1rcλ,r-ξ+1tξ-1(ξ-1)!.
For
(4.13)Yu(t)=Yuh+1(t)⊕⋯⊕Yuj(t)⊕⋯⊕Yun-r(t),
where Yuj(t)=-∑k=0r-1∑ζ=0uj-2δ(rζ+r-1-k)(t)Hujζ+1Yuj,o(k).
MoreoverL=[Ip𝕆⋯𝕆]∈ℳp,pr; R∈ℳpr such that J=R-1AR, where J∈ℳpr is the Jordan Canonical form  of matrix A, and
(4.14)Z(to)=[Yqt(to)Y'qt(to)⋯Yp(r-1)t(to)]t∈ℳpr,l.Proof.
Combining the results of Theorem3.10 and the above discussion, the solution is provided by (4.11).Remark 4.5.
Fort>to, it is obvious that (3.45) is satisfied. Thus, we should stress that the system (1.3) has the above impulse behaviour at time instant where a nonconsistent initial value is assumed, while it returns to smooth behaviour at time instant to.


## 5. Conclusions

In this paper, we study the class of LTI descriptor (singular) matrix descriptor differential equations of higher order whose coefficients are rectangular constant matrices. By taking into consideration that the relevant pencil is singular, we get affected by the Kronecker canonical form in order to decompose the differential system into five subsystems. Afterwards, we provide analytical formulas for this general class of Apostol-Kolodner type of equations when we have consistent and nonconsistent initial conditions.As a future research, we would like to compare more our results with those of the preceding papers of Kolodner [9], Ben Taher and Rachidi [8, 14], and Geerts [12, 13]. Analytically, following the work in [9], we want to investigate whether it is possible to apply our results to find the dynamical solution of LTI matrix descriptor (singular) differential systems. Moreover, it also our wish to apply some combinatorial method for computing the matrix powers and exponential, as it is appeared in [8]. Finally, it is worth to extend the results of Geerts [13] to the general class of LTI matrix descriptor (regular and singular) differential systems.



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