# Solutions of Higher-Order Homogeneous Linear Matrix Differential Equations for Consistent and Non-Consistent Initial Conditions: Regular Case

Authors: Ioannis K. Dassios
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

We study a class of
linear matrix differential equations (regular
case) of higher order whose coefficients are
square constant matrices. By using matrix pencil
theory and the Weierstrass canonical form of
the pencil we obtain formulas for the solutions
and we show that the solution is unique for
consistent initial conditions and infinite for
nonconsistent initial conditions. Moreover we
provide some numerical examples. These kinds of
systems are inherent in many physical and
engineering phenomena.


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## Body

## 1. Introduction

Linear matrix differential equations (LMDEs) are inherent in many physical, engineering, mechanical, and financial/actuarial models. Having in mind such applications, for instance in finance, we provide the well-known input-output Leondief model and its several important extensions, see [1, 2]. In this paper, our long-term purpose is to study the solution of LMDEs of higher order (1.1) into the mainstream of matrix pencil theory. This effort is significant, since there are numerous applications. Thus, we consider (1.1)AnX(n)(t)+An-1X(n-1)(t)+⋯+A1X′(t)+A0X(t)=O,
where Ai, i=0,1,…, n∈ℳ(m×m;𝔽), (i.e., the algebra of square matrices with elements in the field 𝔽) with X∈𝒞∞(𝔽,ℳ(m×1;𝔽)). For the sake of simplicity we set ℳm=ℳ(m×m;𝔽) and ℳnm=ℳ(n×m;𝔽). In the sequel we adopt the following notations:
(1.2)Y1(t)=X(t),Y2(t)=X′(t),⋮Yn-1(t)=X(n-1)(t),Yn(t)=X(n-1)(t),Y1′(t)=X′(t)=Y2(t),Y2′(t)=X′′(t)=Y3(t),⋮Yn-1(t)=X(n-1)(t)=Yn(t),AnYn′(t)=AnX(n)(t)=-An-1Yn(t)-⋯-A1Y2(t)-A0Y1(t).
Or in Matrix form (1.3)FY′(t)=GY(t),
where Y(t)=[Y1T(t)Y2T(t)⋯YnT(t)]T (where ()T is the transpose tensor) and the coefficient matrices F, G are given by (1.4)F=[ImO⋯OOOIm⋯OO⋮⋮⋱⋮⋮OO⋯ImOOO⋯OAn],G=[OImO⋯OOOIm⋯O⋮⋮⋮⋱⋮OOO⋯Im-A0-A1-A2⋯-An]
with corresponding dimension of F, G, and Y(t), mn×mn and mn×1, respectively. Matrix pencil theory has been extensively used for the study of linear differential equations (LDEs) with time invariant coefficients, see for instance [1–5]. Systems of type (1.1) are more general, including the special case when An=In, where In is the identity matrix of ℳn, since the well-known class of higher-order linear matrix differential equations of Apostol-Kolodner type is derived straightforwardly, see [6] for n=2, [7, 8].The paper is organized as follows: in Section2, some notations and the necessary preliminary concepts from matrix pencil theory are presented. Section 3 contains the case that system (1.1) has consistent initial conditions. In Section 4, the nonconsistent initial condition case is fully discussed. In this case, the arbitrarily chosen initial conditions which have physical meaning for (regular) systems, in some sense, can be created or structurally changed at a fixed time t=t0. Hence, it is derived that (1.1) should adopt a generalized solution, in the sense of Dirac δ-solutions.


## 2. Mathematical Background and Notation

This brief section introduces some preliminary concepts and definitions from matrix pencil theory, which are being used throughout the paper. Linear systems of type (1.1) are closely related to matrix pencil theory, since the algebraic geometric and dynamic properties stem from the structure by the associated pencil sF-G.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. In any other case, the pencil will be called singular.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 as
(2.1)P(sF-G)Q=sF̃-G̃.In this paper, we consider the case that pencil isregular. Thus, the strict equivalence relation can be defined rigorously on the set of regular pencils as follows. Here, we regard (2.2) as the set of pair of nonsingular elements of ℳn(2.2)g∶={(P,Q):P,Q∈Mn,P,Qnonsingular}
and a composition rule * defined on g as follows: (2.3)*:g×gsuchthat(P1,Q1)*(P2,Q2)∶=(P1⋅P2,Q2⋅Q1).
It can be easily verified that (g,*) forms a nonabelian group. Furthermore, an action ∘ of the group (g,*) on the set of regular matrix pencils ℒnreg is defined as ∘:g×ℒnreg→ℒnreg such that (2.4)((P,Q),sF-G)⟶(P,Q)∘(sF-G)∶=P(sF-G)Q.
This group has the following properties:  (a)
(P1,Q1)∘[(P2,Q2)∘(sF-G)]=(P1,Q1)*(P2,Q2)∘(sF-G) for every nonsingular P1,P2∈ℳn and Q1,Q2∈ℳn.(b)
eg∘(sF-G)=sF-G, sF-G∈ℒnreg where eg=(In,In) is the identity element of the group (g,*) on the set of ℒnreg defines a transformation group denoted by 𝒩, see [9].ForsF-G∈ℒnreg, the subset (2.5)g∘(sF-G)∶={(P,Q)∘(sF-G):(P,Q)∈g}⊆Lnreg
will be called the orbit of sF-G at g. Also 𝒩 defines an equivalence relation on ℒnreg which is called a strict-equivalence relation and is denoted by ℰs-e.So,(sF-G)ℰs-e(sF̃-G̃) if and only if P(sF-G)Q=sF̃-G̃, where P,Q∈ℳn are nonsingular elements of algebra ℳn.The class ofℰs-e(sF-G) is characterized by a uniquely defined element, known as a complex Weierstrass canonical form, sFw-Qw, see [9], specified by the complete set of invariants of ℰs-e(sF-G).This is the set ofelementary divisors (e.d.) obtained by factorizing the invariant polynomials fi(s,ŝ) into powers of homogeneous polynomials irreducible over field 𝔽. In the case where sF-G is a regular, we have e.d. of the following type:(i)
e.d. of the typespare called zero finite elementary divisors (z. f.e.d.),(ii)
e.d. of the type(s-a)π, a≠0are called nonzero finite elementary divisors (nz. f.e.d.),(iii)
e.d. of the typeŝq are called infinite elementary divisors (i.e.d.).LetB1,B2,…,Bn be elements of ℳn. The direct sum of them denoted by B1⊕B2⊕⋯⊕Bn is the block diag{B1,B2,…,Bn}.Then, the complex Weierstrass formsFw-Qw of the regular pencil sF-G is defined by sFw-Qw∶=sIp-Jp⊕sHq-Iq, where the first normal Jordan type element is uniquely defined by the set of f.e.d. 						(2.6)(s-a1)p1,…,(s-aν)pν,∑j=1νpj=p
of sF-G and has the form (2.7)sIp-Jp∶=sIp1-Jp1(a1)⊕⋯⊕sIpν-Jpν(aν).
And also the q blocks of the second uniquely defined block sHq-Iq correspond to the i.e.d. (2.8)ŝq1,…,ŝqσ,∑j=1σqj=q
of sF-G and has the form (2.9)sHq-Iq∶=sHq1-Iq1⊕⋯⊕sHqσ-Iqσ.
Thus, Hq is a nilpotent element of ℳn with index q̃=max{qj:j=1,2,…,σ}, where (2.10)Hqq̃=O,
and Ipj, Jpj(aj), Hqj are defined as (2.11)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 section, some elements for the analytic computation of eA(t-t0), t∈[t0,∞) are provided. To perform this computation, many theoretical and numerical methods have been developed. Thus, the interesting readers might consult papers [7, 8, 10–12] and the references therein. In order to have computational formulas, see the following Sections 3 and 4, the following known results should firstly be mentioned.Lemma 2.3 (see [10]).
eJpj(aj)(t-t0)=(dk1k2)pj, where
(2.12)dk1k2={eaj(t-t0)(t-t0)k2-k1(k2-k1)!,1≤k1≤k2≤pj,0,otherwise.Another expression for the exponential matrix of Jordan block, see (2.11), is provided by the following lemma.Lemma 2.4 (see [12]).
(2.13)eJpj(aj)(t-t0)=∑i=0pj-1fi(t-t0)[Jpj(aj)]i,
where the fi(t-t0)'s are given analytically by the following pj equations:
(2.14)fpj-1-k(t-t0)=eaj(t-t0)∑i=0kbk,iajk-i(t-t0)pj-1-i(pj-1-i)!,k=0,1,2,…,pj-1,
where
(2.15)bk,i=∑l=0k-i(pjl)(k-li)(-1)l,[Jpj(aj)]i=(ck1k2(i))pj,for1≤k1,k2≤pj,
where
(2.16)ck1k2(i)=(ik2-k1)aji-(k2-k1).


## 3. Solution Space Form of Consistent Initial Conditions

In this section, the main results for consistent initial conditions are analytically presented for the regular case. Moreover, it should be stressed out that these results offer the necessary mathematical framework for interesting applications; see also Introduction. Now, in order to obtain a unique solution, we deal with consistent initial value problem. More analytically, we consider the system(3.1)AnX(n)(t)+An-1X(n-1)(t)+⋯+A1X′(t)+A0X(t)=O
with known initial conditions (3.2)X(t0),X′(t0),…,X(n-1)(t0).
Analytically, we consider the system (3.3)FY′(t)=GY(t),Y(t0).
From the regularity of sF-G, there exist nonsingular ℳ(mn×mn,𝔽) matrices P and Q such that (see also Section 2), such as (3.4)PFQ=Fw=Ip⊕Hq,PGQ=Gw=Jp⊕Iq,
where Ip, Jp, Hq, and Iq are given by (2.11) where (3.5)Ip=Ip1⊕⋯⊕Ipν,Jp=Jp1(a1)⊕⋯⊕Jpν(aν),Hq=Hq1⊕⋯⊕Hqσ,Iq=Iq1⊕⋯⊕Iqσ.
Note that ∑j=1νpj=p and ∑j=1σqj=q, where p+q=n.Lemma 3.1.
System (3.1) is divided into two subsystems: the so-called slow subsystem
(3.6)Zp′(t)=JpZp(t),
and the relative fast subsystem
(3.7)HqZp′(t)=Zq(t).Proof.
Consider the transformation(3.8)Y(t)=QZ(t).
Substituting the previous expression into (3.1) we obtain
(3.9)FQZ′(t)=GQZ(t).
Whereby, multiplying by P, we arrive at
(3.10)FwZ(t)=GwZ(t).
Moreover, we can write Z(t) as Z(t)=[Zp(t)Zq(t)]. Taking into account the above expressions, we arrive easily at (3.6) and (3.7).Proposition 3.2.
The sub-system (3.6) has the unique solution
(3.11)Zp(t)=eJp(t-t0)Zp(t0),t≥t0,
where ∑j=1vpj=p.Proof.
See [5, 12].Proposition 3.3.
The fast subsystem (3.7) has only the zero solution.Proof.
Letq* be the index of the nilpotent matrix Hq, that is, Hqq*=𝕆, we obtain the following equations:
(3.12)HqZq′(t)=Zq(t),HqLZq′(t)=LZq(t),Hq[sY(s)-Zq(t0)]=Y(s),(sHq-Iq)Y(s)=HqZq,
where Y(s)=LZq(t)=∫t0∞Zq(t)e-s(t-t0)dt is by definition the Laplace transform of Zq. It is easy to show that det(sHq-Iq)≠0 and that (sHq-Iq)-1=-∑n=0qmax-1(sHq)n, while Hqn=0 for n≥q*(3.13)Y(s)=(sHq-Iq)-1HqZq,Y(s)=-∑n=0q*-1(sHq)nHqZq=-∑n=0q*-1snHqn+1Zq,Y(s)=-∑n=1q*-2(s)n-1HqnZq,L-1Y(s)=-∑n=1q*-2L-1sn-1HqnZq(t0),Zq=-∑n=1q*-2δn(t-t0)HqnZq(t0),Zq=0,
where δ(t-t0) is by definition the Dirac function
(3.14)∫-∞∞δ(t-t0)dt=1,t=t0,δ(t-t0)=0,t≠t0.
The conclusion, that is, Yq(t)=𝕆, is obtained by repetitive substitution of each equation in the next one, and using the fact that Hqq*=𝕆.Theorem 3.4.
Consider the system (3.1)-(3.2).Then the solution is unique if and only if the initial conditions are consistent. Moreover the analytic solution of (3.1)-(3.2) is given by
(3.15)X(t)=Qp1eJp(t-t0)Zp(t0).Proof.
LetQ=[QpQq] where Qp∈ℳ(mn)p and Qq∈ℳ(mn)q; combining (3.8) and (3.17), we obtain
(3.16)Y(t)=QZ(t)=[QpQq][Zp(t)O]=QpeJp(t-t0)Zp(t0),
solution that exists if and only if Y(t0)=QpZp(t0) or Y(t0)∈colspanQp. The columns of Qp are the p eigenvectors of the finite elementary divisors (eigenvalues) of the pencil sA-B. In that case the system has the unique solution
(3.17)X(t)=Qp1eJp(t-t0)Zp(t0),
where Qp1 is defined as
(3.18)Qp=[Qp1Qp2]
and Qp1∈ℳpp.


## 4. Form on Nonconsistent Initial Condition

In this short section, we describe the impulse behavior of the original system (1.1), at time t0. In that case, we reformulate Theorem 3.4, so the impulse solution is finally obtained.Proposition 4.1.
Consider the system (3.3). Then for nonconsistent initial conditions (Y(t0)∉colspanQp) the system has infinite solutions.Proof.
LetQp,Qq be the matrices defined in Theorem 3.4. If the initial conditions are nonconsistent then Y(t0)∉colspanQp and Zq(t0)≠ 0. Moreover Y(t0)=QpZp(t0)+QqZq(t0).
This means (3.3) is defined for t≠t0 because if t=t0 then FY′(t0)=GY(t0) and Zq(t0)=𝕆 which is a contradiction. Let H(t-t0) be the Heaviside function and
(4.1)f(t)=H(t-t0)-H(t0-t)={1,t>t0,0,t=t0,g(t)=H(t0-t)={1,t=t0,0,t≠t0.
Then the system can be written as
(4.2)f(t)FY′(t)=GY(t)-g(t)GY(t0),t≥t0.
This is a linear matrix differential equation of first order with Yh(t)=f(t)QpeJp(t-t0)C being the solution of the homogeneous and Yp(t)=g(t)Y(t0) a partial solution. And we obtain the general solution
(4.3)Y(t)=f(t)QpeJp(t-t0)C+g(t)Y(t0),t≥t0,
where C=[C1C2⋯Cp]T is constant vector and the dimension of the solution vector space is p.Theorem 4.2.
Consider the system (3.1)-(3.2) with nonconsistent initial conditions. Then the system has infinite solutions.Proof.
We rewrite the system (3.1) in the following form:
(4.4)AnX(n)(t)+An-1X(n-1)(t)+⋯+A1X′(t)+A0X(t)=O,t>t0,∑i=0nAiX(i)(t)=O,t>t0,A0X(t)+f(t)∑i=1nAiX(i)(t)=g(t)A0∑i=0n-1(t-t0)ii!X(i)(t0),t≥t0,
where f(t) and g(t) are the functions defined in Theorem 4.2. Combining the results of Theorem 4.2 and the above discussion the solution of the system is
(4.5)X(t)=f(t)QpeJp(t-t0)C+g(t)∑i=0n-1(t-t0)ii!X(i)(t0),t≥t0.
The dimension of the solution vector space is p.Remark 4.3.
Fort>t0, it is obvious that (4.5) is satisfied. Thus, we should stress out that the system (3.1)-(3.2) has the above impulse behaviour at time instant where a non-consistent initial value is assumed, while it returns to smooth behaviour at any subsequent time instant.


## 5. Numerical Example

Let(5.1)[1100]X′′′(t)+[2100]X′′(t)+[-2311]X′(t)+[4-2-1-1]X(t)=O,
where X(t)=[X1(t)TX2(t)T]T. We adopt the following notations: (5.2)Y1(t)=X(t),Y2(t)=X′(t),Y3(t)=X′′(t),Y1′(t)=X′(t)=Y2(t),Y2′(t)=X′′(t)=Y3(t),A3Y3′(t)=A3X′′′(t)=-A2Y3(t)-A1Y2(t)-A0Y1(t).
Or in Matrix form (5.3)FY′(t)=GY(t),
where Y(t)=[Y1T(t)Y2T(t)Y3T(t)]T (where ()T is the transpose tensor) and the coefficient matrices F, G are given by (5.4)F=[I2OOOI2OOOA3],G=[OI2OOOI2-A0-A1-A2],sF-G=s[100000010000001000000100000011000000]-[001000000100000010000001-422-3-2-111-1-100]s-1, s-2, s-3 finite elementary divisors and s̅3 the infinte elementary divisor of degree 3 of the pencil sF-G. There exist matrices nonsingular P, Q such that PAQ=Fw and PGQ=Gw, where (5.5)Fw=[I3OOH3],Gw=[J3OOI3].
Let Y(t)=QZ(t) then (5.6)FY′(t)=GY(t),PFQZ′(t)=PGQZ(t),FwZ′(t)=GwZ(t).
For consistent initial conditions the solution is (5.7)Y(t)=Q31eJ3(t-t0)Z3(t0)X(t)=Q31eJ3(t-t0)Z3(t0)
and for nonconsistent initial conditions the solution is (5.8)X(t)=f(t)Q3eJ3(t-t0)C+g(t)∑i=02(t-t0)ii!X(i)(t0),
where (5.9)J3=[100020003],eJ3t=[et000e2t000e3t].
The columns of Qp are the eigenvectors of the eigenvalues 1,2,3 (5.10)QpT=[3-53-53-51-12-24-41-13-39-9]T.
Let the initial values of the system be (5.11)X(0)=[1-3],X′(0)=[-20],X′′(0)=[-108],Y(0)T=[1-3-20-108]T.
Then Y(0)∈colspanQp (consistent initial conditions) and the solution of the system is (5.12)Y(t)=Q3eJ3tZ3(0)
and by calculating Zp(0) we get (5.13)Y(0)=Q3Z3(0),Z3(0)T=[1-1-1]T
and the solution of the system is (5.14)Y(t)=Q3eJ3tZ3(0),Y(t)=[3et-e2t-e3t-5et+e2t+e3t3et-2e2t-3e3t-5et+2e2t+3e3t3et-4e2t-9e2t-5et+4e2t+9e3t],X(t)=[3et-e2t-e3t-5et+e2t+e3t].
Next assume the initial conditions (5.15)X(0)=[00],X′(0)=[00],X′′(0)=[11],Y(0)T=[000011]T.
Then Y(0)∉colspanQp nonconsistent initial conditions and the solution is (5.16)X(t)=f(t)Q31eJ3tC+g(t)∑i=02tii!X(i)(0),X(t)=f(t)[3etc1+e2tc2+e3tc3-5etc1-e2tc2-e3tc3]+g(t)t22[11],t≥0.
The dimension of the domain that describes the solutions of the system is 3.


## 6. Conclusions

In this paper we investigate systems of the form suggested in [1–3, 5, 7], but from another point of view. By taking into consideration that the relevant pencil is regular, we use the Weierstrass canonical form in order to decompose the differential system into two sub-systems. Afterwards, we give necessary and sufficient conditions for existence and uniqness of solutions for that general class of linear matrix differential equations of higher order and we provide analytical formulas when we have consistent and non-consistent initial conditions. Moreover, as a further extension of the present paper, we can discuss the case where the pencil is singular. Thus, the Kronecker canonical form is required. The non-homogeneous case has also a special interest, since it appears often in applications. For all these, there is some research in progress.



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