Table of Contents
ISRN Mathematical Analysis
Volume 2011 (2011), Article ID 183795, 14 pages
Research Article

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

Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece

Received 7 March 2011; Accepted 20 April 2011

Academic Editor: T. Yamazaki

Copyright © 2011 Ioannis K. Dassios. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.

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 𝐴𝑛𝑋(𝑛)(𝑡)+𝐴𝑛1𝑋(𝑛1)(𝑡)++𝐴1𝑋(𝑡)+𝐴0𝑋(𝑡)=𝕆,(1.1) where 𝐴𝑖, 𝑖=0,1,, 𝑛(𝑚×𝑚;𝔽), (i.e., the algebra of square matrices with elements in the field 𝔽) with 𝑋𝒞(𝔽,(𝑚×1;𝔽)). For the sake of simplicity we set 𝑚=(𝑚×𝑚;𝔽) and 𝑛𝑚=(𝑛×𝑚;𝔽). In the sequel we adopt the following notations: 𝑌1𝑌(𝑡)=𝑋(𝑡),2(𝑡)=𝑋𝑌(𝑡),𝑛1(𝑡)=𝑋(𝑛1)𝑌(𝑡),𝑛(𝑡)=𝑋(𝑛1)𝑌(𝑡),1(𝑡)=𝑋(𝑡)=𝑌2𝑌(𝑡),2(𝑡)=𝑋(𝑡)=𝑌3(𝑌𝑡),𝑛1(𝑡)=𝑋(𝑛1)(𝑡)=𝑌𝑛(𝐴𝑡),𝑛𝑌𝑛(𝑡)=𝐴𝑛𝑋(𝑛)(𝑡)=𝐴𝑛1𝑌𝑛(𝑡)𝐴1𝑌2(𝑡)𝐴0𝑌1(𝑡).(1.2) Or in Matrix form 𝐹𝑌(𝑡)=𝐺𝑌(𝑡),(1.3) where 𝑌(𝑡)=[𝑌𝑇1(𝑡)𝑌𝑇2(𝑡)𝑌𝑇𝑛(𝑡)]𝑇 (where ()𝑇 is the transpose tensor) and the coefficient matrices 𝐹, 𝐺 are given by 𝐼𝐅=𝑚𝕆𝕆𝕆𝕆𝐼𝑚𝕆𝕆𝕆𝕆𝐼𝑚𝕆𝕆𝕆𝕆𝐴𝑛,𝐆=𝕆𝐼𝑚𝕆𝕆𝕆𝕆𝐼𝑚𝕆𝕆𝕆𝕆𝐼𝑚𝐴0𝐴1𝐴2𝐴𝑛(1.4) 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 [15]. Systems of type (1.1) are more general, including the special case when 𝐴𝑛=𝐼𝑛, where 𝐼𝑛 is the identity matrix of 𝑛, 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 Section 2, 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 𝑠𝐹𝐺.

Definition 2.1. Given 𝐹,𝐺𝑛𝑚 and an indeterminate 𝑠𝔽, the matrix pencil 𝑠𝐹𝐺 is called regular when 𝑚=𝑛 and det(𝑠𝐹𝐺)0. In any other case, the pencil will be called singular.

Definition 2.2. The pencil 𝑠𝐹𝐺 is said to be strictly equivalent to the pencil 𝑠𝐺𝐹 if and only if there exist nonsingular 𝑃𝑛 and 𝑄𝑚 such as 𝑃(𝑠𝐹𝐺)𝑄=𝑠𝐹𝐺.(2.1)

In this paper, we consider the case that pencil is regular. 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 𝑛𝑔=(𝑃,𝑄)𝑃,𝑄𝑛,𝑃,𝑄nonsingular(2.2) and a composition rule defined on 𝑔 as follows: 𝑃𝑔×𝑔suchthat1,𝑄1𝑃2,𝑄2𝑃=1𝑃2,𝑄2𝑄1.(2.3) 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 reg𝑛 is defined as 𝑔×reg𝑛reg𝑛 such that ((𝑃,𝑄),𝑠𝐹𝐺)(𝑃,𝑄)(𝑠𝐹𝐺)=𝑃(𝑠𝐹𝐺)𝑄.(2.4) This group has the following properties: (a)(𝑃1,𝑄1)[(𝑃2,𝑄2)(𝑠𝐹𝐺)]=(𝑃1,𝑄1)(𝑃2,𝑄2)(𝑠𝐹𝐺) for every nonsingular 𝑃1,𝑃2𝑛 and 𝑄1,𝑄2𝑛.(b)𝑒𝑔(𝑠𝐹𝐺)=𝑠𝐹𝐺, 𝑠𝐹𝐺reg𝑛 where 𝑒𝑔=(𝐼𝑛,𝐼𝑛) is the identity element of the group (𝑔,) on the set of reg𝑛 defines a transformation group denoted by 𝒩, see [9].

For 𝑠𝐹𝐺reg𝑛, the subset 𝑔(𝑠𝐹𝐺)={(𝑃,𝑄)(𝑠𝐹𝐺)(𝑃,𝑄)𝑔}reg𝑛(2.5) will be called the orbit of 𝑠𝐹𝐺 at 𝑔. Also 𝒩 defines an equivalence relation on reg𝑛 which is called a strict-equivalence relation and is denoted by 𝑠𝑒.

So, (𝑠𝐹𝐺)𝑠𝑒(𝑠𝐹𝐺) if and only if 𝐺𝑃(𝑠𝐹𝐺)𝑄=𝑠𝐹, where 𝑃,𝑄𝑛 are nonsingular elements of algebra 𝑛.

The class of 𝑠𝑒(𝑠𝐹𝐺) is characterized by a uniquely defined element, known as a complex Weierstrass canonical form, 𝑠𝐹𝑤𝑄𝑤, see [9], specified by the complete set of invariants of 𝑠𝑒(𝑠𝐹𝐺).

This is the set of elementary divisors (e.d.) obtained by factorizing the invariant polynomials 𝑓𝑖(𝑠,̂𝑠) into powers of homogeneous polynomials irreducible over field 𝔽. In the case where 𝑠𝐹𝐺 is a regular, we have e.d. of the following type:(i) e.d. of the type 𝑠𝑝are called zero finite elementary divisors (z. f.e.d.), (ii) e.d. of the type (𝑠𝑎)𝜋, 𝑎0are called nonzero finite elementary divisors (nz. f.e.d.), (iii) e.d. of the type ̂𝑠𝑞 are called infinite elementary divisors (i.e.d.).

Let 𝐵1,𝐵2,,𝐵𝑛 be elements of 𝑛. The direct sum of them denoted by 𝐵1𝐵2𝐵𝑛 is the block diag{𝐵1,𝐵2,,𝐵𝑛}.

Then, the complex Weierstrass form 𝑠𝐹𝑤𝑄𝑤 of the regular pencil 𝑠𝐹𝐺 is defined by 𝑠𝐹𝑤𝑄𝑤=𝑠𝐼𝑝𝐽𝑝𝑠𝐻𝑞𝐼𝑞, where the first normal Jordan type element is uniquely defined by the set of f.e.d. 𝑠𝑎1𝑝1,,𝑠𝑎𝜈𝑝𝜈,𝜈𝑗=1𝑝𝑗=𝑝(2.6) of 𝑠𝐹𝐺 and has the form 𝑠𝐼𝑝𝐽𝑝=𝑠𝐼𝑝1𝐽𝑝1𝑎1𝑠𝐼𝑝𝜈𝐽𝑝𝜈𝑎𝜈.(2.7) And also the 𝑞 blocks of the second uniquely defined block 𝑠𝐻𝑞𝐼𝑞 correspond to the i.e.d. ̂𝑠𝑞1,,̂𝑠𝑞𝜎,𝜎𝑗=1𝑞𝑗=𝑞(2.8) of 𝑠𝐹𝐺 and has the form 𝑠𝐻𝑞𝐼𝑞=𝑠𝐻𝑞1𝐼𝑞1𝑠𝐻𝑞𝜎𝐼𝑞𝜎.(2.9) Thus, 𝐻𝑞 is a nilpotent element of 𝑛 with index ̃𝑞=max{𝑞𝑗𝑗=1,2,,𝜎}, where 𝐻̃𝑞𝑞=𝕆,(2.10) and 𝐼𝑝𝑗, 𝐽𝑝𝑗(𝑎𝑗), 𝐻𝑞𝑗 are defined as 𝐼𝑝𝑗=100010001𝑝𝑗,𝐽𝑝𝑗𝑎𝑗=𝑎𝑗1000𝑎𝑗10000𝑎𝑗10000𝑎𝑗𝑝𝑗,𝐻𝑞𝑗=010000100000100000𝑞𝑗.(2.11) In the last part of this section, some elements for the analytic computation of 𝑒𝐴(𝑡𝑡0), 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, 1012] 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)(𝑡𝑡0)=(𝑑𝑘1𝑘2)𝑝𝑗, where 𝑑𝑘1𝑘2=𝑒𝑎𝑗(𝑡𝑡0)𝑡𝑡0𝑘2𝑘1𝑘2𝑘1!,1𝑘1𝑘2𝑝𝑗,0,otherwise.(2.12)

Another expression for the exponential matrix of Jordan block, see (2.11), is provided by the following lemma.

Lemma 2.4 (see [12]). 𝑒𝐽𝑝𝑗(𝑎𝑗)(𝑡𝑡0)=𝑝𝑗1𝑖=0𝑓𝑖𝑡𝑡0𝐽𝑝𝑗𝑎𝑗𝑖,(2.13) where the 𝑓𝑖(𝑡𝑡0)'s are given analytically by the following 𝑝𝑗 equations: 𝑓𝑝𝑗1𝑘𝑡𝑡0=𝑒𝑎𝑗(𝑡𝑡0)𝑘𝑖=0𝑏𝑘,𝑖𝑎𝑗𝑘𝑖𝑡𝑡0𝑝𝑗1𝑖𝑝𝑗!1𝑖,𝑘=0,1,2,,𝑝𝑗1,(2.14) where  𝑏𝑘,𝑖=𝑘𝑖𝑙=0𝑝𝑗𝑙𝑖𝑘𝑙(1)𝑙,𝐽𝑝𝑗𝑎𝑗𝑖=𝑐𝑘(𝑖)1𝑘2𝑝𝑗,for1𝑘1,𝑘2𝑝𝑗,(2.15) where 𝑐𝑘(𝑖)1𝑘2=𝑖𝑘2𝑘1𝑎𝑖(𝑘2𝑘1)𝑗.(2.16)

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 𝐴𝑛𝑋(𝑛)(𝑡)+𝐴𝑛1𝑋(𝑛1)(𝑡)++𝐴1𝑋(𝑡)+𝐴0𝑋(𝑡)=𝕆(3.1) with known initial conditions 𝑋𝑡0,𝑋𝑡0,,𝑋(𝑛1)𝑡0.(3.2) Analytically, we consider the system 𝐹𝑌𝑡(𝑡)=𝐺𝑌(𝑡),𝑌0.(3.3) From the regularity of 𝑠𝐹𝐺, there exist nonsingular (𝑚𝑛×𝑚𝑛,𝔽) matrices 𝑃 and 𝑄 such that (see also Section 2), such as 𝑃𝐹𝑄=𝐹𝑤=𝐼𝑝𝐻𝑞,𝑃𝐺𝑄=𝐺𝑤=𝐽𝑝𝐼𝑞,(3.4) where 𝐼𝑝, 𝐽𝑝, 𝐻𝑞, and 𝐼𝑞 are given by (2.11) where 𝐼𝑝=𝐼𝑝1𝐼𝑝𝜈,𝐽𝑝=𝐽𝑝1𝑎1𝐽𝑝𝜈𝑎𝜈,𝐻𝑞=𝐻𝑞1𝐻𝑞𝜎,𝐼𝑞=𝐼𝑞1𝐼𝑞𝜎.(3.5) Note that 𝜈𝑗=1𝑝𝑗=𝑝 and 𝜎𝑗=1𝑞𝑗=𝑞, where 𝑝+𝑞=𝑛.

Lemma 3.1. System (3.1) is divided into two subsystems: the so-called slow subsystem 𝑍𝑝(𝑡)=𝐽𝑝𝑍𝑝(𝑡),(3.6) and the relative fast subsystem 𝐻𝑞𝑍𝑝(𝑡)=𝑍𝑞(𝑡).(3.7)

Proof. Consider the transformation 𝑌(𝑡)=𝑄𝑍(𝑡).(3.8) Substituting the previous expression into (3.1) we obtain 𝐹𝑄𝑍(𝑡)=𝐺𝑄𝑍(𝑡).(3.9) Whereby, multiplying by 𝑃, we arrive at 𝐹𝑤𝑍(𝑡)=𝐺𝑤𝑍(𝑡).(3.10) Moreover, we can write 𝑍(𝑡) as 𝑍(𝑡)=𝑍𝑝𝑍(𝑡)𝑞(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 𝑍𝑝(𝑡)=𝑒𝐽𝑝(𝑡𝑡0)𝑍𝑝𝑡0,𝑡𝑡0,(3.11) where 𝑣𝑗=1𝑝𝑗=𝑝.

Proof. See [5, 12].

Proposition 3.3. The fast subsystem (3.7) has only the zero solution.

Proof. Let 𝑞 be the index of the nilpotent matrix 𝐻𝑞, that is, 𝐻𝑞𝑞=𝕆, we obtain the following equations: 𝐻𝑞𝑍𝑞(𝑡)=𝑍𝑞𝐻(𝑡),𝑞𝐿𝑍𝑞(𝑡)=𝐿𝑍𝑞𝐻(𝑡),𝑞𝑠𝑌(𝑠)𝑍𝑞𝑡0=𝑌(𝑠),𝑠𝐻𝑞𝐼𝑞𝑌(𝑠)=𝐻𝑞𝑍𝑞,(3.12) where 𝑌(𝑠)=𝐿𝑍𝑞(𝑡)=𝑡0𝑍𝑞(𝑡)𝑒𝑠(𝑡𝑡0)𝑑𝑡 is by definition the Laplace transform of 𝑍𝑞. It is easy to show that det(𝑠𝐻𝑞𝐼𝑞)0 and that (𝑠𝐻𝑞𝐼𝑞)1=𝑞max1𝑛=0(𝑠𝐻𝑞)𝑛, while 𝐻𝑛𝑞=0 for 𝑛𝑞𝑌(𝑠)=𝑠𝐻𝑞𝐼𝑞1𝐻𝑞𝑍𝑞,𝑌(𝑠)=𝑞1𝑛=0𝑠𝐻𝑞𝑛𝐻𝑞𝑍𝑞=𝑞1𝑛=0𝑠𝑛𝐻𝑞𝑛+1𝑍𝑞,𝑌(𝑠)=𝑞2𝑛=1(𝑠)𝑛1𝐻𝑛𝑞𝑍𝑞,𝐿1𝑌(𝑠)=𝑞2𝑛=1𝐿1𝑠𝑛1𝐻𝑛𝑞𝑍𝑞𝑡0,𝑍𝑞=𝑞2𝑛=1𝛿𝑛𝑡𝑡0𝐻𝑛𝑞𝑍𝑞𝑡0,𝑍𝑞=0,(3.13) where 𝛿(𝑡𝑡0) is by definition the Dirac function 𝛿𝑡𝑡0𝑑𝑡=1,𝑡=𝑡0,𝛿𝑡𝑡0=0,𝑡𝑡0.(3.14) 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 𝑋(𝑡)=𝑄1𝑝𝑒𝐽𝑝(𝑡𝑡0)𝑍𝑝𝑡0.(3.15)

Proof. Let 𝑄=[𝑄𝑝𝑄𝑞] where 𝐐𝐩(𝑚𝑛)𝑝 and 𝐐𝐪(𝑚𝑛)𝑞; combining (3.8) and (3.17), we obtain 𝑄𝑌(𝑡)=𝑄𝑍(𝑡)=𝑝𝑄𝑞𝑍𝑝(𝕆𝑡)=𝑄𝑝𝑒𝐽𝑝(𝑡𝑡0)𝑍𝑝𝑡0,(3.16) 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 𝑠𝐴𝐵. In that case the system has the unique solution 𝑋(𝑡)=𝑄1𝑝𝑒𝐽𝑝(𝑡𝑡0)𝑍𝑝𝑡0,(3.17) where 𝑄1𝑝 is defined as 𝑄𝑝=𝑄1𝑝𝑄2𝑝(3.18) and 𝐐𝟏𝐩𝑝𝑝.

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. Let 𝑄𝑝,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 𝑡𝑡0 because if 𝑡=𝑡0 then 𝐹𝑌(𝑡0)=𝐺𝑌(𝑡0) and 𝑍𝑞(𝑡0)=𝕆 which is a contradiction. Let 𝐻(𝑡𝑡0) be the Heaviside function and 𝑓(𝑡)=𝐻𝑡𝑡0𝑡𝐻0=𝑡1,𝑡>𝑡0,0,𝑡=𝑡0,𝑔𝑡(𝑡)=𝐻0=𝑡1,𝑡=𝑡0,0,𝑡𝑡0.(4.1) Then the system can be written as 𝑓(𝑡)𝐹𝑌𝑡(𝑡)=𝐺𝑌(𝑡)𝑔(𝑡)𝐺𝑌0,𝑡𝑡0.(4.2) This is a linear matrix differential equation of first order with 𝑌(𝑡)=𝑓(𝑡)𝑄𝑝𝑒𝐽𝑝(𝑡𝑡0)𝐶 being the solution of the homogeneous and 𝑌𝑝(𝑡)=𝑔(𝑡)𝑌(𝑡0) a partial solution. And we obtain the general solution 𝑌(𝑡)=𝑓(𝑡)𝑄𝑝𝑒𝐽𝑝(𝑡𝑡0)𝑡𝐶+𝑔(𝑡)𝑌0,𝑡𝑡0,(4.3) where 𝐶=[𝐶1𝐶2𝐶𝑝]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: 𝐴𝑛𝑋(𝑛)(𝑡)+𝐴𝑛1𝑋(𝑛1)(𝑡)++𝐴1𝑋(𝑡)+𝐴0𝑋(𝑡)=𝕆,𝑡>𝑡0,𝑛𝑖=0𝐴𝑖𝑋(𝑖)(𝑡)=𝕆,𝑡>𝑡0,𝐴0𝑋(𝑡)+𝑓(𝑡)𝑛𝑖=1𝐴𝑖𝑋(𝑖)(𝑡)=𝑔(𝑡)𝐴0𝑛1𝑖=0𝑡𝑡0𝑖𝑋𝑖!(𝑖)𝑡0,𝑡𝑡0,(4.4) 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 𝑋(𝑡)=𝑓(𝑡)𝑄𝑝𝑒𝐽𝑝(𝑡𝑡0)𝐶+𝑔(𝑡)𝑛1𝑖=0𝑡𝑡0𝑖𝑋𝑖!(𝑖)𝑡0,𝑡𝑡0.(4.5) The dimension of the solution vector space is 𝑝.

Remark 4.3. For 𝑡>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 𝑋1100𝑋(𝑡)+2100𝑋(𝑡)+2311(𝑡)+4211𝑋(𝑡)=𝕆,(5.1) where 𝑋(𝑡)=[𝑋1(𝑡)𝑇𝑋2(𝑡)𝑇]𝑇. We adopt the following notations: 𝑌1𝑌(𝑡)=𝑋(𝑡),2(𝑡)=𝑋𝑌(𝑡),3(𝑡)=𝑋𝑌(𝑡),1(𝑡)=𝑋(𝑡)=𝑌2𝑌(𝑡),2(𝑡)=𝑋(𝑡)=𝑌3𝐴(𝑡),3𝑌3(𝑡)=𝐴3𝑋(𝑡)=𝐴2𝑌3(𝑡)𝐴1𝑌2(𝑡)𝐴0𝑌1(𝑡).(5.2) Or in Matrix form 𝐹𝑌(𝑡)=𝐺𝑌(𝑡),(5.3) where 𝑌(𝑡)=[𝑌𝑇1(𝑡)𝑌𝑇2(𝑡)𝑌𝑇3(𝑡)]𝑇 (where ()𝑇 is the transpose tensor) and the coefficient matrices 𝐹, 𝐺 are given by 𝐼𝐅=2𝕆𝕆𝕆𝐼2𝕆𝕆𝕆𝐴3,𝐆=𝕆𝐼2𝕆𝕆𝕆𝐼2𝐴0𝐴1𝐴2,𝑠𝐹𝐺=𝑠100000010000001000000100000011000000001000000100000010000001422321111100(5.4)𝑠1, 𝑠2, 𝑠3 finite elementary divisors and 𝑠3 the infinte elementary divisor of degree 3 of the pencil 𝑠𝐹𝐺. There exist matrices nonsingular 𝑃, 𝑄 such that 𝑃𝐴𝑄=𝐹𝑤 and 𝑃𝐺𝑄=𝐺𝑤, where 𝐅𝐰=𝐼3𝕆𝕆𝐻3,𝐆𝐰=𝐽3𝕆𝕆𝐼3.(5.5) Let 𝑌(𝑡)=𝑄𝑍(𝑡) then 𝐹𝑌(𝑡)=𝐺𝑌(𝑡),𝑃𝐹𝑄𝑍𝐹(𝑡)=𝑃𝐺𝑄𝑍(𝑡),𝑤𝑍(𝑡)=𝐺𝑤𝑍(𝑡).(5.6) For consistent initial conditions the solution is 𝑌(𝑡)=𝑄13𝑒𝐽3(𝑡𝑡0)𝑍3𝑡0𝑋(𝑡)=𝑄13𝑒𝐽3(𝑡𝑡0)𝑍3𝑡0(5.7) and for nonconsistent initial conditions the solution is 𝑋(𝑡)=𝑓(𝑡)𝑄3𝑒𝐽3(𝑡𝑡0)𝐶+𝑔(𝑡)2𝑖=0𝑡𝑡0𝑖𝑋𝑖!(𝑖)𝑡0,(5.8) where 𝐽3=,𝑒100020003𝐽3𝑡=𝑒𝑡000𝑒2𝑡000𝑒3𝑡.(5.9) The columns of 𝑄𝑝 are the eigenvectors of the eigenvalues 1,2,3 𝑄𝑇𝑝=353535112244113399𝑇.(5.10) Let the initial values of the system be 1𝑋(0)=3,𝑋0(0)=2,𝑋8,(0)=10𝑌(0)𝑇=1320108𝑇.(5.11) Then 𝑌(0)colspan𝑄𝑝 (consistent initial conditions) and the solution of the system is 𝑌(𝑡)=𝑄3𝑒𝐽3𝑡𝑍3(0)(5.12) and by calculating 𝑍𝑝(0) we get 𝑌(0)=𝑄3𝑍3𝑍(0),3(0)𝑇=111𝑇(5.13) and the solution of the system is 𝑌(𝑡)=𝑄3𝑒𝐽3𝑡𝑍3(0),𝑌(𝑡)=3𝑒𝑡𝑒2𝑡𝑒3𝑡5𝑒𝑡+𝑒2𝑡+𝑒3𝑡3𝑒𝑡2𝑒2𝑡3𝑒3𝑡5𝑒𝑡+2𝑒2𝑡+3𝑒3𝑡3𝑒𝑡4𝑒2𝑡9𝑒2𝑡5𝑒𝑡+4𝑒2𝑡+9𝑒3𝑡,𝑋(𝑡)=3𝑒𝑡𝑒2𝑡𝑒3𝑡5𝑒𝑡+𝑒2𝑡+𝑒3𝑡.(5.14) Next assume the initial conditions 00𝑋(0)=,𝑋00(0)=,𝑋11,(0)=𝑌(0)𝑇=000011𝑇.(5.15) Then 𝑌(0)colspan𝑄𝑝 nonconsistent initial conditions and the solution is 𝑋(𝑡)=𝑓(𝑡)𝑄13𝑒𝐽3𝑡𝐶+𝑔(𝑡)2𝑖=0𝑡𝑖𝑋𝑖!(𝑖)(0),𝑋(𝑡)=𝑓(𝑡)3𝑒𝑡𝑐1+𝑒2𝑡𝑐2+𝑒3𝑡𝑐35𝑒𝑡𝑐1𝑒2𝑡𝑐2𝑒3𝑡𝑐3𝑡+𝑔(𝑡)2211,𝑡0.(5.16) 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 [13, 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.


The author would like to express his sincere gratitude to my supervisor Professor G. I. Kalogeropoulos for making for him available personal notes and notations that he used in Sections 1 and 2 and for his fruitful discusion that improved the paper.


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