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 š“š‘›š‘‹(š‘›)(š‘”)+š“š‘›āˆ’1š‘‹(š‘›āˆ’1)(š‘”)+ā‹Æ+š“1š‘‹ī…ž(š‘”)+š“0š‘‹(š‘”)=š•†,(1.1) where š“š‘–, š‘–=0,1,ā€¦, š‘›āˆˆā„³(š‘šĆ—š‘š;š”½), (i.e., the algebra of square matrices with elements in the field š”½) with š‘‹āˆˆš’žāˆž(š”½,ā„³(š‘šĆ—1;š”½)). For the sake of simplicity we set ā„³š‘š=ā„³(š‘šĆ—š‘š;š”½) and ā„³š‘›š‘š=ā„³(š‘›Ć—š‘š;š”½). In the sequel we adopt the following notations: š‘Œ1š‘Œ(š‘”)=š‘‹(š‘”),2(š‘”)=š‘‹ī…žā‹®š‘Œ(š‘”),š‘›āˆ’1(š‘”)=š‘‹(š‘›āˆ’1)š‘Œ(š‘”),š‘›(š‘”)=š‘‹(š‘›āˆ’1)š‘Œ(š‘”),ī…ž1(š‘”)=š‘‹ī…ž(š‘”)=š‘Œ2š‘Œ(š‘”),ī…ž2(š‘”)=š‘‹ī…žī…ž(š‘”)=š‘Œ3(ā‹®š‘Œš‘”),š‘›āˆ’1(š‘”)=š‘‹(š‘›āˆ’1)(š‘”)=š‘Œš‘›(š“š‘”),š‘›š‘Œī…žš‘›(š‘”)=š“š‘›š‘‹(š‘›)(š‘”)=āˆ’š“š‘›āˆ’1š‘Œš‘›(š‘”)āˆ’ā‹Æāˆ’š“1š‘Œ2(š‘”)āˆ’š“0š‘Œ1(š‘”).(1.2) Or in Matrix form š¹š‘Œī…ž(š‘”)=šŗš‘Œ(š‘”),(1.3) where š‘Œ(š‘”)=[š‘Œš‘‡1(š‘”)š‘Œš‘‡2(š‘”)ā‹Æš‘Œš‘‡š‘›(š‘”)]š‘‡ (where ()š‘‡ is the transpose tensor) and the coefficient matrices š¹, šŗ are given by āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£š¼š…=š‘šš•†ā‹Æš•†š•†š•†š¼š‘šā‹Æš•†š•†ā‹®ā‹®ā‹±ā‹®ā‹®š•†š•†ā‹Æš¼š‘šš•†š•†š•†ā‹Æš•†š“š‘›āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£,š†=š•†š¼š‘šš•†ā‹Æš•†š•†š•†š¼š‘šā‹Æš•†ā‹®ā‹®ā‹®ā‹±ā‹®š•†š•†š•†ā‹Æš¼š‘šāˆ’š“0āˆ’š“1āˆ’š“2ā‹Æāˆ’š“š‘›āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦(1.4) with corresponding dimension of š…, š†, and š˜(š‘”), š‘šš‘›Ć—š‘šš‘› and š‘šš‘›Ć—1, respectively. Matrix pencil theory has been extensively used for the study of linear differential equations (LDEs) with time invariant coefficients, see for instance [1ā€“5]. Systems of type (1.1) are more general, including the special case when š“š‘›=š¼š‘›, where š¼š‘› is the identity matrix of ā„³š‘›, since the well-known class of higher-order linear matrix differential equations of Apostol-Kolodner type is derived straightforwardly, see [6] for š‘›=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 š‘”=š‘”0. Hence, it is derived that (1.1) should adopt a generalized solution, in the sense of Dirac š›æ-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 š‘ š¹āˆ’šŗ.

Definition 2.1. Given š¹,šŗāˆˆā„³š‘›š‘š and an indeterminate š‘ āˆˆš”½, the matrix pencil š‘ š¹āˆ’šŗ is called regular when š‘š=š‘› and det(š‘ š¹āˆ’šŗ)ā‰ 0. In any other case, the pencil will be called singular.

Definition 2.2. The pencil š‘ š¹āˆ’šŗ is said to be strictly equivalent to the pencil š‘ ī‚ī‚šŗš¹āˆ’ if and only if there exist nonsingular š‘ƒāˆˆā„³š‘› and š‘„āˆˆā„³š‘š such as ī‚ī‚š‘ƒ(š‘ š¹āˆ’šŗ)š‘„=š‘ š¹āˆ’šŗ.(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 ā„³š‘›ī€½š‘”āˆ¶=(š‘ƒ,š‘„)āˆ¶š‘ƒ,š‘„āˆˆā„³š‘›ī€¾,š‘ƒ,š‘„nonsingular(2.2) and a composition rule āˆ— defined on š‘” as follows: ī€·š‘ƒāˆ—āˆ¶š‘”Ć—š‘”suchthat1,š‘„1ī€øāˆ—ī€·š‘ƒ2,š‘„2ī€øī€·š‘ƒāˆ¶=1ā‹…š‘ƒ2,š‘„2ā‹…š‘„1ī€ø.(2.3) It can be easily verified that (š‘”,āˆ—) forms a nonabelian group. Furthermore, an action āˆ˜ of the group (š‘”,āˆ—) on the set of regular matrix pencils ā„’regš‘› is defined as āˆ˜āˆ¶š‘”Ć—ā„’regš‘›ā†’ā„’regš‘› such that ((š‘ƒ,š‘„),š‘ š¹āˆ’šŗ)āŸ¶(š‘ƒ,š‘„)āˆ˜(š‘ š¹āˆ’šŗ)āˆ¶=š‘ƒ(š‘ š¹āˆ’šŗ)š‘„.(2.4) This group has the following properties: (a)(š‘ƒ1,š‘„1)āˆ˜[(š‘ƒ2,š‘„2)āˆ˜(š‘ š¹āˆ’šŗ)]=(š‘ƒ1,š‘„1)āˆ—(š‘ƒ2,š‘„2)āˆ˜(š‘ š¹āˆ’šŗ) for every nonsingular š‘ƒ1,š‘ƒ2āˆˆā„³š‘› and š‘„1,š‘„2āˆˆā„³š‘›.(b)š‘’š‘”āˆ˜(š‘ š¹āˆ’šŗ)=š‘ š¹āˆ’šŗ, š‘ š¹āˆ’šŗāˆˆā„’regš‘› where š‘’š‘”=(š¼š‘›,š¼š‘›) is the identity element of the group (š‘”,āˆ—) on the set of ā„’regš‘› defines a transformation group denoted by š’©, see [9].

For š‘ š¹āˆ’šŗāˆˆā„’regš‘›, the subset š‘”āˆ˜(š‘ š¹āˆ’šŗ)āˆ¶={(š‘ƒ,š‘„)āˆ˜(š‘ š¹āˆ’šŗ)āˆ¶(š‘ƒ,š‘„)āˆˆš‘”}āŠ†ā„’regš‘›(2.5) will be called the orbit of š‘ š¹āˆ’šŗ at š‘”. Also š’© defines an equivalence relation on ā„’regš‘› which is called a strict-equivalence relation and is denoted by ā„°š‘ āˆ’š‘’.

So, (š‘ š¹āˆ’šŗ)ā„°š‘ āˆ’š‘’ī‚ī‚(š‘ š¹āˆ’šŗ) if and only if ī‚ī‚šŗš‘ƒ(š‘ š¹āˆ’šŗ)š‘„=š‘ š¹āˆ’, where š‘ƒ,š‘„āˆˆā„³š‘› are nonsingular elements of algebra ā„³š‘›.

The class of ā„°š‘ āˆ’š‘’(š‘ š¹āˆ’šŗ) is characterized by a uniquely defined element, known as a complex Weierstrass canonical form, š‘ š¹š‘¤āˆ’š‘„š‘¤, see [9], specified by the complete set of invariants of ā„°š‘ āˆ’š‘’(š‘ š¹āˆ’šŗ).

This is the set of elementary divisors (e.d.) obtained by factorizing the invariant polynomials š‘“š‘–(š‘ ,Ģ‚š‘ ) into powers of homogeneous polynomials irreducible over field š”½. In the case where š‘ š¹āˆ’šŗ is a regular, we have e.d. of the following type:(i) e.d. of the type š‘ š‘are called zero finite elementary divisors (z. f.e.d.), (ii) e.d. of the type (š‘ āˆ’š‘Ž)šœ‹, š‘Žā‰ 0are called nonzero finite elementary divisors (nz. f.e.d.), (iii) e.d. of the type Ģ‚š‘ š‘ž are called infinite elementary divisors (i.e.d.).

Let šµ1,šµ2,ā€¦,šµš‘› be elements of ā„³š‘›. The direct sum of them denoted by šµ1āŠ•šµ2āŠ•ā‹ÆāŠ•šµš‘› is the block diag{šµ1,šµ2,ā€¦,šµš‘›}.

Then, the complex Weierstrass form š‘ š¹š‘¤āˆ’š‘„š‘¤ of the regular pencil š‘ š¹āˆ’šŗ is defined by š‘ š¹š‘¤āˆ’š‘„š‘¤āˆ¶=š‘ š¼š‘āˆ’š½š‘āŠ•š‘ š»š‘žāˆ’š¼š‘ž, where the first normal Jordan type element is uniquely defined by the set of f.e.d. ī€·š‘ āˆ’š‘Ž1ī€øš‘1ī€·,ā€¦,š‘ āˆ’š‘Žšœˆī€øš‘šœˆ,šœˆī“š‘—=1š‘š‘—=š‘(2.6) of š‘ š¹āˆ’šŗ and has the form š‘ š¼š‘āˆ’š½š‘āˆ¶=š‘ š¼š‘1āˆ’š½š‘1ī€·š‘Ž1ī€øāŠ•ā‹ÆāŠ•š‘ š¼š‘šœˆāˆ’š½š‘šœˆī€·š‘Žšœˆī€ø.(2.7) And also the š‘ž blocks of the second uniquely defined block š‘ š»š‘žāˆ’š¼š‘ž correspond to the i.e.d. Ģ‚š‘ š‘ž1,ā€¦,Ģ‚š‘ š‘žšœŽ,šœŽī“š‘—=1š‘žš‘—=š‘ž(2.8) of š‘ š¹āˆ’šŗ and has the form š‘ š»š‘žāˆ’š¼š‘žāˆ¶=š‘ š»š‘ž1āˆ’š¼š‘ž1āŠ•ā‹ÆāŠ•š‘ š»š‘žšœŽāˆ’š¼š‘žšœŽ.(2.9) Thus, š»š‘ž is a nilpotent element of ā„³š‘› with index Ģƒš‘ž=max{š‘žš‘—āˆ¶š‘—=1,2,ā€¦,šœŽ}, where š»Ģƒš‘žš‘ž=š•†,(2.10) and š¼š‘š‘—, š½š‘š‘—(š‘Žš‘—), š»š‘žš‘— are defined as š¼š‘š‘—=āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦10ā‹Æ001ā‹Æ0ā‹®ā‹®ā‹±ā‹®00ā‹Æ1āˆˆā„³š‘š‘—,š½š‘š‘—ī€·š‘Žš‘—ī€ø=āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£š‘Žš‘—10ā‹Æ00š‘Žš‘—1ā‹Æ0ā‹®ā‹®ā‹±ā‹®ā‹®000š‘Žš‘—10000š‘Žš‘—āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦āˆˆā„³š‘š‘—,š»š‘žš‘—=āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦010ā‹Æ0001ā‹Æ0ā‹®ā‹®ā‹±ā‹®ā‹®0000100000āˆˆā„³š‘žš‘—.(2.11) In the last part of this section, some elements for the analytic computation of š‘’š“(š‘”āˆ’š‘”0), š‘”āˆˆ[š‘”0,āˆž) are provided. To perform this computation, many theoretical and numerical methods have been developed. Thus, the interesting readers might consult papers [7, 8, 10ā€“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]). š‘’š½š‘š‘—(š‘Žš‘—)(š‘”āˆ’š‘”0)=(š‘‘š‘˜1š‘˜2)š‘š‘—, where š‘‘š‘˜1š‘˜2=āŽ§āŽŖāŽØāŽŖāŽ©š‘’š‘Žš‘—(š‘”āˆ’š‘”0)ī€·š‘”āˆ’š‘”0ī€øš‘˜2āˆ’š‘˜1ī€·š‘˜2āˆ’š‘˜1ī€ø!,1ā‰¤š‘˜1ā‰¤š‘˜2ā‰¤š‘š‘—,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]). š‘’š½š‘š‘—(š‘Žš‘—)(š‘”āˆ’š‘”0)=š‘š‘—āˆ’1ī“š‘–=0š‘“š‘–ī€·š‘”āˆ’š‘”0ī€øī‚ƒš½š‘š‘—ī€·š‘Žš‘—ī€øī‚„š‘–,(2.13) where the š‘“š‘–(š‘”āˆ’š‘”0)'s are given analytically by the following š‘š‘— equations: š‘“š‘š‘—āˆ’1āˆ’š‘˜ī€·š‘”āˆ’š‘”0ī€ø=š‘’š‘Žš‘—(š‘”āˆ’š‘”0)š‘˜ī“š‘–=0š‘š‘˜,š‘–š‘Žš‘—š‘˜āˆ’š‘–ī€·š‘”āˆ’š‘”0ī€øš‘š‘—āˆ’1āˆ’š‘–ī€·š‘š‘—ī€ø!āˆ’1āˆ’š‘–,š‘˜=0,1,2,ā€¦,š‘š‘—āˆ’1,(2.14) whereā€‰ š‘š‘˜,š‘–=š‘˜āˆ’š‘–ī“š‘™=0īƒ©š‘š‘—š‘™š‘–īƒŖīƒŖīƒ©š‘˜āˆ’š‘™(āˆ’1)š‘™,ī‚ƒš½š‘š‘—ī€·š‘Žš‘—ī€øī‚„š‘–=ī‚€š‘š‘˜(š‘–)1š‘˜2ī‚š‘š‘—,for1ā‰¤š‘˜1,š‘˜2ā‰¤š‘š‘—,(2.15) where š‘š‘˜(š‘–)1š‘˜2=īƒ©š‘–š‘˜2āˆ’š‘˜1īƒŖš‘Žš‘–āˆ’(š‘˜2āˆ’š‘˜1)š‘—.(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 š“š‘›š‘‹(š‘›)(š‘”)+š“š‘›āˆ’1š‘‹(š‘›āˆ’1)(š‘”)+ā‹Æ+š“1š‘‹ī…ž(š‘”)+š“0š‘‹(š‘”)=š•†(3.1) with known initial conditions š‘‹ī€·š‘”0ī€ø,š‘‹ī…žī€·š‘”0ī€ø,ā€¦,š‘‹(š‘›āˆ’1)ī€·š‘”0ī€ø.(3.2) Analytically, we consider the system š¹š‘Œī…žī€·š‘”(š‘”)=šŗš‘Œ(š‘”),š‘Œ0ī€ø.(3.3) From the regularity of š‘ š¹āˆ’šŗ, there exist nonsingular ā„³(š‘šš‘›Ć—š‘šš‘›,š”½) matrices š‘ƒ and š‘„ such that (see also Section 2), such as š‘ƒš¹š‘„=š¹š‘¤=š¼š‘āŠ•š»š‘ž,š‘ƒšŗš‘„=šŗš‘¤=š½š‘āŠ•š¼š‘ž,(3.4) where š¼š‘, š½š‘, š»š‘ž, and š¼š‘ž are given by (2.11) where š¼š‘=š¼š‘1āŠ•ā‹ÆāŠ•š¼š‘šœˆ,š½š‘=š½š‘1ī€·š‘Ž1ī€øāŠ•ā‹ÆāŠ•š½š‘šœˆī€·š‘Žšœˆī€ø,š»š‘ž=š»š‘ž1āŠ•ā‹ÆāŠ•š»š‘žšœŽ,š¼š‘ž=š¼š‘ž1āŠ•ā‹ÆāŠ•š¼š‘žšœŽ.(3.5) Note that āˆ‘šœˆš‘—=1š‘š‘—=š‘ and āˆ‘šœŽš‘—=1š‘žš‘—=š‘ž, where š‘+š‘ž=š‘›.

Lemma 3.1. System (3.1) is divided into two subsystems: the so-called slow subsystem š‘ī…žš‘(š‘”)=š½š‘š‘š‘(š‘”),(3.6) and the relative fast subsystem š»š‘žš‘ī…žš‘(š‘”)=š‘š‘ž(š‘”).(3.7)

Proof. Consider the transformation š‘Œ(š‘”)=š‘„š‘(š‘”).(3.8) Substituting the previous expression into (3.1) we obtain š¹š‘„š‘ī…ž(š‘”)=šŗš‘„š‘(š‘”).(3.9) Whereby, multiplying by š‘ƒ, we arrive at š¹š‘¤š‘(š‘”)=šŗš‘¤š‘(š‘”).(3.10) Moreover, we can write š‘(š‘”) as ī‚ƒš‘(š‘”)=š‘š‘š‘(š‘”)š‘ž(š‘”)ī‚„. 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 š‘š‘(š‘”)=š‘’š½š‘(š‘”āˆ’š‘”0)š‘š‘ī€·š‘”0ī€ø,š‘”ā‰„š‘”0,(3.11) where āˆ‘š‘£š‘—=1š‘š‘—=š‘.

Proof. See [5, 12].

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

Proof. Let š‘žāˆ— be the index of the nilpotent matrix š»š‘ž, that is, š»š‘žāˆ—š‘ž=š•†, we obtain the following equations: š»š‘žš‘ī…žš‘ž(š‘”)=š‘š‘žš»(š‘”),š‘žšæš‘ī…žš‘ž(š‘”)=šæš‘š‘žš»(š‘”),š‘žī€ŗš‘ š‘Œ(š‘ )āˆ’š‘š‘žī€·š‘”0ī€·ī€øī€»=š‘Œ(š‘ ),š‘ š»š‘žāˆ’š¼š‘žī€øš‘Œ(š‘ )=š»š‘žš‘š‘ž,(3.12) where š‘Œ(š‘ )=šæš‘š‘žāˆ«(š‘”)=āˆžš‘”0š‘š‘ž(š‘”)š‘’āˆ’š‘ (š‘”āˆ’š‘”0)š‘‘š‘” is by definition the Laplace transform of š‘š‘ž. It is easy to show that det(š‘ š»š‘žāˆ’š¼š‘ž)ā‰ 0 and that (š‘ š»š‘žāˆ’š¼š‘ž)āˆ’1āˆ‘=āˆ’š‘žmaxāˆ’1š‘›=0(š‘ š»š‘ž)š‘›, while š»š‘›š‘ž=0 for š‘›ā‰„š‘žāˆ—ī€·š‘Œ(š‘ )=š‘ š»š‘žāˆ’š¼š‘žī€øāˆ’1š»š‘žš‘š‘ž,š‘Œ(š‘ )=āˆ’š‘žāˆ—āˆ’1ī“š‘›=0ī€·š‘ š»š‘žī€øš‘›š»š‘žš‘š‘ž=āˆ’š‘žāˆ—āˆ’1ī“š‘›=0š‘ š‘›š»š‘žš‘›+1š‘š‘ž,š‘Œ(š‘ )=āˆ’š‘žāˆ—āˆ’2ī“š‘›=1(š‘ )š‘›āˆ’1š»š‘›š‘žš‘š‘ž,šæāˆ’1š‘Œ(š‘ )=āˆ’š‘žāˆ—āˆ’2ī“š‘›=1šæāˆ’1š‘ š‘›āˆ’1š»š‘›š‘žš‘š‘žī€·š‘”0ī€ø,š‘š‘ž=āˆ’š‘žāˆ—āˆ’2ī“š‘›=1š›æš‘›ī€·š‘”āˆ’š‘”0ī€øš»š‘›š‘žš‘š‘žī€·š‘”0ī€ø,š‘š‘ž=0,(3.13) where š›æ(š‘”āˆ’š‘”0) is by definition the Dirac function ī€œāˆžāˆ’āˆžš›æī€·š‘”āˆ’š‘”0ī€øš‘‘š‘”=1,š‘”=š‘”0,š›æī€·š‘”āˆ’š‘”0ī€ø=0,š‘”ā‰ š‘”0.(3.14) The conclusion, that is, š‘Œš‘ž(š‘”)=š•†, is obtained by repetitive substitution of each equation in the next one, and using the fact that š»š‘žāˆ—š‘ž=š•†.

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 š‘‹(š‘”)=š‘„1š‘š‘’š½š‘(š‘”āˆ’š‘”0)š‘š‘ī€·š‘”0ī€ø.(3.15)

Proof. Let š‘„=[š‘„š‘š‘„š‘ž] where šš©āˆˆā„³(š‘šš‘›)š‘ and ššŖāˆˆā„³(š‘šš‘›)š‘ž; combining (3.8) and (3.17), we obtain ī€ŗš‘„š‘Œ(š‘”)=š‘„š‘(š‘”)=š‘š‘„š‘žī€»īƒ¬š‘š‘(š•†īƒ­š‘”)=š‘„š‘š‘’š½š‘(š‘”āˆ’š‘”0)š‘š‘ī€·š‘”0ī€ø,(3.16) solution that exists if and only if š‘Œ(š‘”0)=š‘„š‘š‘š‘(š‘”0) or š‘Œ(š‘”0)āˆˆcolspanš‘„š‘. The columns of š‘„š‘ are the š‘ eigenvectors of the finite elementary divisors (eigenvalues) of the pencil š‘ š“āˆ’šµ. In that case the system has the unique solution š‘‹(š‘”)=š‘„1š‘š‘’š½š‘(š‘”āˆ’š‘”0)š‘š‘ī€·š‘”0ī€ø,(3.17) where š‘„1š‘ is defined as š‘„š‘=īƒ¬š‘„1š‘š‘„2š‘īƒ­(3.18) and ššŸš©āˆˆā„³š‘š‘.

4. Form on Nonconsistent Initial Condition

In this short section, we describe the impulse behavior of the original system (1.1), at time š‘”0. 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 (š‘Œ(š‘”0)āˆ‰colspanš‘„š‘) the system has infinite solutions.

Proof. Let š‘„š‘,š‘„š‘ž be the matrices defined in Theorem 3.4. If the initial conditions are nonconsistent then š‘Œ(š‘”0)āˆ‰colspanš‘„š‘ and š‘š‘ž(š‘”0)ā‰  0. Moreover š‘Œ(š‘”0)=š‘„š‘š‘š‘(š‘”0)+š‘„š‘žš‘š‘ž(š‘”0).
This means (3.3) is defined for š‘”ā‰ š‘”0 because if š‘”=š‘”0 then š¹š‘Œī…ž(š‘”0)=šŗš‘Œ(š‘”0) and š‘š‘ž(š‘”0)=š•† which is a contradiction. Let š»(š‘”āˆ’š‘”0) be the Heaviside function and ī€·š‘“(š‘”)=š»š‘”āˆ’š‘”0ī€øī€·š‘”āˆ’š»0ī€ø=īƒÆāˆ’š‘”1,š‘”>š‘”0,0,š‘”=š‘”0,š‘”ī€·š‘”(š‘”)=š»0ī€ø=īƒÆāˆ’š‘”1,š‘”=š‘”0,0,š‘”ā‰ š‘”0.(4.1) Then the system can be written as š‘“(š‘”)š¹š‘Œī…žī€·š‘”(š‘”)=šŗš‘Œ(š‘”)āˆ’š‘”(š‘”)šŗš‘Œ0ī€ø,š‘”ā‰„š‘”0.(4.2) This is a linear matrix differential equation of first order with š‘Œā„Ž(š‘”)=š‘“(š‘”)š‘„š‘š‘’š½š‘(š‘”āˆ’š‘”0)š¶ being the solution of the homogeneous and š‘Œš‘(š‘”)=š‘”(š‘”)š‘Œ(š‘”0) a partial solution. And we obtain the general solution š‘Œ(š‘”)=š‘“(š‘”)š‘„š‘š‘’š½š‘(š‘”āˆ’š‘”0)ī€·š‘”š¶+š‘”(š‘”)š‘Œ0ī€ø,š‘”ā‰„š‘”0,(4.3) where š¶=[š¶1š¶2ā‹Æš¶š‘]š‘‡ is constant vector and the dimension of the solution vector space is š‘.

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: š“š‘›š‘‹(š‘›)(š‘”)+š“š‘›āˆ’1š‘‹(š‘›āˆ’1)(š‘”)+ā‹Æ+š“1š‘‹ī…ž(š‘”)+š“0š‘‹(š‘”)=š•†,š‘”>š‘”0,š‘›ī“š‘–=0š“š‘–š‘‹(š‘–)(š‘”)=š•†,š‘”>š‘”0,š“0š‘‹(š‘”)+š‘“(š‘”)š‘›ī“š‘–=1š“š‘–š‘‹(š‘–)(š‘”)=š‘”(š‘”)š“0š‘›āˆ’1ī“š‘–=0ī€·š‘”āˆ’š‘”0ī€øš‘–š‘‹š‘–!(š‘–)ī€·š‘”0ī€ø,š‘”ā‰„š‘”0,(4.4) where š‘“(š‘”) and š‘”(š‘”) 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 š‘‹(š‘”)=š‘“(š‘”)š‘„š‘š‘’š½š‘(š‘”āˆ’š‘”0)š¶+š‘”(š‘”)š‘›āˆ’1ī“š‘–=0ī€·š‘”āˆ’š‘”0ī€øš‘–š‘‹š‘–!(š‘–)ī€·š‘”0ī€ø,š‘”ā‰„š‘”0.(4.5) The dimension of the solution vector space is š‘.

Remark 4.3. For š‘”>š‘”0, 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 īƒ¬īƒ­š‘‹1100ī…žī…žī…žīƒ¬īƒ­š‘‹(š‘”)+2100ī…žī…žīƒ¬īƒ­š‘‹(š‘”)+āˆ’2311ī…žīƒ¬īƒ­(š‘”)+4āˆ’2āˆ’1āˆ’1š‘‹(š‘”)=š•†,(5.1) where š‘‹(š‘”)=[š‘‹1(š‘”)š‘‡š‘‹2(š‘”)š‘‡]š‘‡. We adopt the following notations: š‘Œ1š‘Œ(š‘”)=š‘‹(š‘”),2(š‘”)=š‘‹ī…žš‘Œ(š‘”),3(š‘”)=š‘‹ī…žī…žš‘Œ(š‘”),ī…ž1(š‘”)=š‘‹ā€²(š‘”)=š‘Œ2š‘Œ(š‘”),ī…ž2(š‘”)=š‘‹ī…žī…ž(š‘”)=š‘Œ3š“(š‘”),3š‘Œī…ž3(š‘”)=š“3š‘‹ī…žī…žī…ž(š‘”)=āˆ’š“2š‘Œ3(š‘”)āˆ’š“1š‘Œ2(š‘”)āˆ’š“0š‘Œ1(š‘”).(5.2) Or in Matrix form š¹š‘Œī…ž(š‘”)=šŗš‘Œ(š‘”),(5.3) where š‘Œ(š‘”)=[š‘Œš‘‡1(š‘”)š‘Œš‘‡2(š‘”)š‘Œš‘‡3(š‘”)]š‘‡ (where ()š‘‡ is the transpose tensor) and the coefficient matrices š¹, šŗ are given by āŽ”āŽ¢āŽ¢āŽ¢āŽ£š¼š…=2š•†š•†š•†š¼2š•†š•†š•†š“3āŽ¤āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ¢āŽ£,š†=š•†š¼2š•†š•†š•†š¼2āˆ’š“0āˆ’š“1āˆ’š“2āŽ¤āŽ„āŽ„āŽ„āŽ¦,āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦āˆ’āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦š‘ š¹āˆ’šŗ=š‘ 100000010000001000000100000011000000001000000100000010000001āˆ’422āˆ’3āˆ’2āˆ’111āˆ’1āˆ’100(5.4)š‘ āˆ’1, š‘ āˆ’2, š‘ āˆ’3 finite elementary divisors and š‘ 3 the infinte elementary divisor of degree 3 of the pencil š‘ š¹āˆ’šŗ. There exist matrices nonsingular š‘ƒ, š‘„ such that š‘ƒš“š‘„=š¹š‘¤ and š‘ƒšŗš‘„=šŗš‘¤, where š…š°=īƒ¬š¼3š•†š•†š»3īƒ­,š†š°=īƒ¬š½3š•†š•†š¼3īƒ­.(5.5) Let š‘Œ(š‘”)=š‘„š‘(š‘”) then š¹š‘Œī…ž(š‘”)=šŗš‘Œ(š‘”),š‘ƒš¹š‘„š‘ī…žš¹(š‘”)=š‘ƒšŗš‘„š‘(š‘”),š‘¤š‘ī…ž(š‘”)=šŗš‘¤š‘(š‘”).(5.6) For consistent initial conditions the solution is š‘Œ(š‘”)=š‘„13š‘’š½3(š‘”āˆ’š‘”0)š‘3ī€·š‘”0ī€øš‘‹(š‘”)=š‘„13š‘’š½3(š‘”āˆ’š‘”0)š‘3ī€·š‘”0ī€ø(5.7) and for nonconsistent initial conditions the solution is š‘‹(š‘”)=š‘“(š‘”)š‘„3š‘’š½3(š‘”āˆ’š‘”0)š¶+š‘”(š‘”)2ī“š‘–=0ī€·š‘”āˆ’š‘”0ī€øš‘–š‘‹š‘–!(š‘–)ī€·š‘”0ī€ø,(5.8) where š½3=āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦,š‘’100020003š½3š‘”=āŽ”āŽ¢āŽ¢āŽ¢āŽ£š‘’š‘”000š‘’2š‘”000š‘’3š‘”āŽ¤āŽ„āŽ„āŽ„āŽ¦.(5.9) The columns of š‘„š‘ are the eigenvectors of the eigenvalues 1,2,3 š‘„š‘‡š‘=āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦3āˆ’53āˆ’53āˆ’51āˆ’12āˆ’24āˆ’41āˆ’13āˆ’39āˆ’9š‘‡.(5.10) Let the initial values of the system be īƒ¬1īƒ­š‘‹(0)=āˆ’3,š‘‹ī…žīƒ¬0īƒ­(0)=āˆ’2,š‘‹ī…žī…žīƒ¬8īƒ­,(0)=āˆ’10š‘Œ(0)š‘‡=ī€ŗī€»1āˆ’3āˆ’20āˆ’108š‘‡.(5.11) Then š‘Œ(0)āˆˆcolspanš‘„š‘ (consistent initial conditions) and the solution of the system is š‘Œ(š‘”)=š‘„3š‘’š½3š‘”š‘3(0)(5.12) and by calculating š‘š‘(0) we get š‘Œ(0)=š‘„3š‘3š‘(0),3(0)š‘‡=ī€ŗī€»1āˆ’1āˆ’1š‘‡(5.13) and the solution of the system is š‘Œ(š‘”)=š‘„3š‘’š½3š‘”š‘3āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£(0),š‘Œ(š‘”)=3š‘’š‘”āˆ’š‘’2š‘”āˆ’š‘’3š‘”āˆ’5š‘’š‘”+š‘’2š‘”+š‘’3š‘”3š‘’š‘”āˆ’2š‘’2š‘”āˆ’3š‘’3š‘”āˆ’5š‘’š‘”+2š‘’2š‘”+3š‘’3š‘”3š‘’š‘”āˆ’4š‘’2š‘”āˆ’9š‘’2š‘”āˆ’5š‘’š‘”+4š‘’2š‘”+9š‘’3š‘”āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦,īƒ¬š‘‹(š‘”)=3š‘’š‘”āˆ’š‘’2š‘”āˆ’š‘’3š‘”āˆ’5š‘’š‘”+š‘’2š‘”+š‘’3š‘”īƒ­.(5.14) Next assume the initial conditions īƒ¬00īƒ­š‘‹(0)=,š‘‹ī…žīƒ¬00īƒ­(0)=,š‘‹ī…žī…žīƒ¬11īƒ­,(0)=š‘Œ(0)š‘‡=ī€ŗī€»000011š‘‡.(5.15) Then š‘Œ(0)āˆ‰colspanš‘„š‘ nonconsistent initial conditions and the solution is š‘‹(š‘”)=š‘“(š‘”)š‘„13š‘’š½3š‘”š¶+š‘”(š‘”)2ī“š‘–=0š‘”š‘–š‘‹š‘–!(š‘–)īƒ¬(0),š‘‹(š‘”)=š‘“(š‘”)3š‘’š‘”š‘1+š‘’2š‘”š‘2+š‘’3š‘”š‘3āˆ’5š‘’š‘”š‘1āˆ’š‘’2š‘”š‘2āˆ’š‘’3š‘”š‘3īƒ­š‘”+š‘”(š‘”)22īƒ¬11īƒ­,š‘”ā‰„0.(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 [1ā€“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.


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.