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

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.

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 𝐴_𝑛𝑋^(𝑛)(𝑡)+𝐴_𝑛−1𝑋^(𝑛−1)(𝑡)+⋯+𝐴₁𝑋^(𝑡)+𝐴₀𝑋(𝑡)=𝕆,(1.1) where 𝐴_𝑖, 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 ℳ_𝑚=ℳ(𝑚×𝑚;𝔽) and ℳ_𝑛𝑚=ℳ(𝑛×𝑚;𝔽). In the sequel we adopt the following notations: 𝑌𝑌₁(𝑡)=𝑋(𝑡),₂𝑌(𝑡)=⋮𝑋^(𝑡),_𝑛−1𝑌(𝑡)=𝑋^(𝑛−1)(𝑡),_𝑛𝑌(𝑡)=𝑋^(𝑛−1)(𝑡),_1𝑌(𝑡)=𝑋^(𝑡)=𝑌₂(𝑡),_2𝑌(𝑡)=𝑋⋮^(𝑡)=𝑌₃(𝑡),_𝑛−1𝐴(𝑡)=𝑋^(𝑛−1)(𝑡)=𝑌_𝑛(𝑡),_𝑛𝑌^𝑛(𝑡)=𝐴_𝑛𝑋^(𝑛)(𝑡)=−𝐴_𝑛−1𝑌_𝑛(𝑡)−⋯−𝐴₁𝑌₂(𝑡)−𝐴₀𝑌₁(𝑡).(1.2) Or in Matrix form 𝐹𝑌^(𝑡)=𝐺𝑌(𝑡),(1.3) where 𝑌(𝑡)=[𝑌^𝑇1(𝑡)𝑌^𝑇2(𝑡)⋯𝑌^𝑇𝑛(𝑡)]^𝑇 (where ()^𝑇 is the transpose tensor) and the coefficient matrices 𝐹, 𝐺 are given by 𝐅=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝐼^𝑚𝕆𝐼𝕆_𝑚⋯𝕆𝕆⋮𝕆⋮𝕆⋱⋯⋯𝐼⋮𝕆_𝑚⋮𝕆𝕆𝕆⋯𝕆𝐴𝕆_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝐆=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝕆𝐼^𝑚𝕆𝕆𝐼𝕆_𝑚⋯𝕆⋮𝕆⋮𝕆⋮𝕆⋱⋯⋯𝐼⋮𝕆_𝑚−𝐴₀−𝐴₁−𝐴₂⋯−𝐴_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎦(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 [1–5]. Systems of type (1.1) are more general, including the special case when 𝐴_𝑛=𝐼_𝑛, where 𝐼_𝑛 is the identity matrix of ℳ_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 𝑠𝐹−𝐺.

Definition 2.1. Given 𝐹,𝐺∈ℳ_𝑛𝑚 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 pencil 𝑠𝐹−𝐺 is said to be strictly equivalent to the pencil 𝑠𝐹−𝐺 if and only if there exist nonsingular 𝑃∈ℳ_𝑛 and 𝑄∈ℳ_𝑚 such as 𝑃(𝑠𝐹−𝐺)𝑄=𝑠𝐹−𝐺.(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 ℳ_𝑛𝑔∶=(𝑃,𝑄)∶𝑃,𝑄∈ℳ_𝑛,𝑃,𝑄nonsingular(2.2) and a composition rule ∗ defined on 𝑔 as follows: ∗∶𝑔×𝑔suchthat𝑃₁,𝑄₁∗𝑃₂,𝑄₂∶=𝑃₁⋅𝑃₂,𝑄₂⋅𝑄₁.(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 ℒ^reg_𝑛 is defined as ∘∶𝑔×ℒ^reg_𝑛→ℒ^reg_𝑛 such that ((𝑃,𝑄),𝑠𝐹−𝐺)⟶(𝑃,𝑄)∘(𝑠𝐹−𝐺)∶=𝑃(𝑠𝐹−𝐺)𝑄.(2.4) This group has the following properties: (a)(𝑃₁,𝑄₁)∘[(𝑃₂,𝑄₂)∘(𝑠𝐹−𝐺)]=(𝑃₁,𝑄₁)∗(𝑃₂,𝑄₂)∘(𝑠𝐹−𝐺) for every nonsingular 𝑃₁,𝑃₂∈ℳ_𝑛 and 𝑄₁,𝑄₂∈ℳ_𝑛.(b)𝑒_𝑔∘(𝑠𝐹−𝐺)=𝑠𝐹−𝐺, 𝑠𝐹−𝐺∈ℒ^reg_𝑛 where 𝑒_𝑔=(𝐼_𝑛,𝐼_𝑛) is the identity element of the group (𝑔,∗) on the set of ℒ^reg_𝑛 defines a transformation group denoted by 𝒩, see [9].

For 𝑠𝐹−𝐺∈ℒ^reg_𝑛, the subset 𝑔∘(𝑠𝐹−𝐺)∶={(𝑃,𝑄)∘(𝑠𝐹−𝐺)∶(𝑃,𝑄)∈𝑔}⊆ℒ^reg_𝑛(2.5) will be called the orbit of 𝑠𝐹−𝐺 at 𝑔. Also 𝒩 defines an equivalence relation on ℒ^reg_𝑛 which is called a strict-equivalence relation and is denoted by ℰ_𝑠−𝑒.

So, (𝑠𝐹−𝐺)ℰ_𝑠−𝑒(𝑠𝐹−𝐺) if and only if 𝑃(𝑠𝐹−𝐺)𝑄=𝑠𝐹−𝐺, where 𝑃,𝑄∈ℳ_𝑛 are nonsingular elements of algebra ℳ_𝑛.

The class of ℰ_𝑠−𝑒(𝑠𝐹−𝐺) is characterized by a uniquely defined element, known as a complex Weierstrass canonical form, 𝑠𝐹_𝑤−𝑄_𝑤, see [9], specified by the complete set of invariants of ℰ_𝑠−𝑒(𝑠𝐹−𝐺).

This is the set of elementary divisors (e.d.) obtained by factorizing the invariant polynomials 𝑓_𝑖(𝑠,𝑠)̂ 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 type 𝑠^𝑝are 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 type 𝑠^𝑞̂ are called infinite elementary divisors (i.e.d.).

Let 𝐵₁,𝐵₂,…,𝐵_𝑛 be elements of ℳ_𝑛. The direct sum of them denoted by 𝐵₁⊕𝐵₂⊕⋯⊕𝐵_𝑛 is the block diag{𝐵₁,𝐵₂,…,𝐵_𝑛}.

Then, the complex Weierstrass form 𝑠𝐹_𝑤−𝑄_𝑤 of the regular pencil 𝑠𝐹−𝐺 is defined by 𝑠𝐹_𝑤−𝑄_𝑤∶=𝑠𝐼_𝑝−𝐽_𝑝⊕𝑠𝐻_𝑞−𝐼_𝑞, where the first normal Jordan type element is uniquely defined by the set of f.e.d. 𝑠−𝑎₁^𝑝1,…,𝑠−𝑎_𝜈^𝑝𝜈,^𝜈_𝑗=1𝑝_𝑗=𝑝(2.6) of 𝑠𝐹−𝐺 and has the form 𝑠𝐼_𝑝−𝐽_𝑝∶=𝑠𝐼_𝑝1−𝐽_𝑝1𝑎₁⊕⋯⊕𝑠𝐼_𝑝𝜈−𝐽_𝑝𝜈𝑎_𝜈.(2.7) And also the 𝑞 blocks of the second uniquely defined block 𝑠𝐻_𝑞−𝐼_𝑞 correspond to the i.e.d. 𝑠̂^𝑞1,…,𝑠̂^𝑞𝜎,^𝜎_𝑗=1𝑞_𝑗=𝑞(2.8) of 𝑠𝐹−𝐺 and has the form 𝑠𝐻_𝑞−𝐼_𝑞∶=𝑠𝐻_𝑞1−𝐼_𝑞1⊕⋯⊕𝑠𝐻_𝑞𝜎−𝐼_𝑞𝜎.(2.9) Thus, 𝐻_𝑞 is a nilpotent element of ℳ_𝑛 with index 𝑞̃=max{𝑞_𝑗∶𝑗=1,2,…,𝜎}, where 𝐻𝑞_𝑞̃=𝕆,(2.10) and 𝐼_𝑝𝑗, 𝐽_𝑝𝑗(𝑎_𝑗), 𝐻_𝑞𝑗 are defined as 𝐼_𝑝𝑗=⎡⎢⎢⎢⎢⎢⎣⋮100⋮010⋱⋯⋯⋯⋮001⎤⎥⎥⎥⎥⎥⎦∈ℳ_𝑝𝑗,𝐽_𝑝𝑗𝑎_𝑗=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝑎^𝑗0𝑎1_𝑗0⋯0⋮0⋮0⋱01⋯𝑎⋮_𝑗⋮00000𝑎1_𝑗⎤⎥⎥⎥⎥⎥⎥⎥⎦∈ℳ_𝑝𝑗𝐻_𝑞𝑗,=⎡⎢⎢⎢⎢⎢⎢⎢⎣⋮0000⋮1000⋱0100⋯⋯⋮00⋮0010⎤⎥⎥⎥⎥⎥⎥⎥⎦∈ℳ_𝑞𝑗.(2.11) In the last part of this section, some elements for the analytic computation of 𝑒^{𝐴(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]). 𝑒^{𝐽𝑝𝑗(𝑎𝑗)(𝑡−𝑡0)}=(𝑑_𝑘1𝑘2)_𝑝𝑗, where 𝑑_𝑘1𝑘2=⎧⎪⎨⎪⎩𝑒^{𝑎𝑗(𝑡−𝑡0)}𝑡−𝑡₀^{𝑘2−𝑘1}𝑘₂−𝑘₁!,1≤𝑘¹0,otherwi≤se.𝑘₂≤𝑝_𝑗,(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]). 𝑒^{𝐽𝑝𝑗(𝑎𝑗)(𝑡−𝑡0)}=^𝑝𝑗−1_𝑖=0𝑓_𝑖𝑡−𝑡₀𝐽_𝑝𝑗𝑎_𝑗^𝑖,(2.13) where the 𝑓_𝑖(𝑡−𝑡₀)'s are given analytically by the following 𝑝_𝑗 equations: 𝑓_{𝑝𝑗−1−𝑘}𝑡−𝑡₀=𝑒^{𝑎𝑗(𝑡−𝑡0)𝑘}_𝑖=0𝑏_𝑘,𝑖𝑎^{𝑗𝑘−𝑖}𝑡−𝑡₀^{𝑝𝑗−1−𝑖}𝑝_𝑗−1−𝑖!,𝑘=0,1,2,…,𝑝^𝑗−1,(2.14) where  𝑏_𝑘,𝑖=^𝑘−𝑖_𝑙=0𝑝𝑙^𝑗𝑘−𝑖𝑙(−1)^𝑙𝐽_𝑝𝑗,𝑎_𝑗^𝑖=𝑐^𝑘(𝑖)_1𝑘2_𝑝𝑗,for1≤𝑘₁,𝑘₂≤𝑝_𝑗,(2.15) where 𝑐^𝑘(𝑖)_1𝑘2=𝑘₂𝑖−𝑘₁𝑎^{𝑖−(𝑘2−𝑘1)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 𝐴_𝑛𝑋^(𝑛)(𝑡)+𝐴_𝑛−1𝑋^(𝑛−1)(𝑡)+⋯+𝐴₁𝑋^(𝑡)+𝐴₀𝑋(𝑡)=𝕆(3.1) with known initial conditions 𝑋𝑡₀,𝑋^𝑡₀,…,𝑋^(𝑛−1)𝑡₀.(3.2) Analytically, we consider the system 𝐹𝑌^(𝑡)=𝐺𝑌(𝑡),𝑌𝑡₀.(3.3) 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 𝑃𝑃𝐹𝐺𝑄𝑄==𝐹𝐺_𝑤𝑤=𝐼_𝑝⊕𝐻_𝑞,=𝐽_𝑝⊕𝐼_𝑞,(3.4) where 𝐼_𝑝, 𝐽_𝑝, 𝐻_𝑞, and 𝐼_𝑞 are given by (2.11) where 𝐽_𝑝𝐼_𝑝=𝐼_𝑝1⊕⋯⊕𝐼_𝑝𝜈,=𝐻𝐽_𝑝1𝑞𝑎₁⊕⋯⊕𝐽_𝑝𝜈𝑎_𝜈,𝐼_𝑞=𝐻_𝑞1⊕⋯⊕𝐻_𝑞𝜎,=𝐼_𝑞1⊕⋯⊕𝐼_𝑞𝜎.(3.5) Note that ∑^𝜈𝑗=1𝑝_𝑗=𝑝 and ∑^𝜎𝑗=1𝑞_𝑗=𝑞, where 𝑝+𝑞=𝑛.

Lemma 3.1. System (3.1) is divided into two subsystems: the so-called slow subsystem 𝑍^𝑝(𝑡)=𝐽_𝑝𝑍_𝑝(𝑡),(3.6) and the relative fast subsystem 𝐻_𝑞𝑍^𝑝(𝑡)=𝑍_𝑞(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 𝐹_𝑤𝑍(𝑡)=𝐺_𝑤𝑍(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 𝑍_𝑝(𝑡)=𝑒^{𝐽𝑝(𝑡−𝑡0)}𝑍_𝑝𝑡₀,𝑡≥𝑡₀,(3.11) where ∑^𝑣𝑗=1𝑝_𝑗=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 𝐻_𝑞, that is, 𝐻^𝑞∗_𝑞=𝕆, we obtain the following equations: 𝐻𝐻_𝑞𝑍^𝑞(𝑡)=𝑍_𝑞(𝑡),_𝑞𝐻𝐿𝑍^𝑞(𝑡)=𝐿𝑍_𝑞(𝑡),_𝑞𝑠𝐻𝑠𝑌_𝑞(𝑠)−𝑍_𝑞𝑡₀=𝑌(𝑠),−𝐼_𝑞𝑌(𝑠)=𝐻_𝑞𝑍_𝑞,(3.12) where 𝑌(𝑠)=𝐿𝑍_𝑞(𝑡)=∫^∞𝑡₀𝑍_𝑞(𝑡)𝑒^{−𝑠(𝑡−𝑡0)}𝑑𝑡 is by definition the Laplace transform of 𝑍_𝑞. It is easy to show that det(𝑠𝐻_𝑞−𝐼_𝑞)≠0 and that (𝑠𝐻_𝑞−𝐼_𝑞)⁻¹=−∑^{𝑞max−1𝑛=0}(𝑠𝐻_𝑞)^𝑛, while 𝐻^𝑛𝑞=0 for 𝑛≥𝑞_∗𝑌(𝑠)=−𝑌(𝑠_𝑞∗−1)=𝑠𝐻_𝑞−𝐼_𝑞⁻¹𝐻_𝑞𝑍_𝑞,_𝑛=0𝑠𝐻_𝑞^𝑛𝐻_𝑞𝑍_𝑞=−^𝑞∗−1_𝑛=0𝑌(𝑠)=−_𝑞∗−2𝑠^𝑛𝐻^𝑞𝑛+1𝑍_𝑞,_𝑛=1𝐿₋₁(𝑠)^𝑛−1𝐻^𝑛𝑞𝑍_𝑞,𝑌(𝑠)=−^𝑞∗−2_𝑛=1𝑍_𝑞𝐿⁻¹𝑠^𝑛−1𝐻^𝑛𝑞𝑍_𝑞𝑡₀,=−^𝑞∗−2_𝑛=1𝛿^𝑛𝑍_𝑞𝑡−𝑡₀𝐻^𝑛𝑞𝑍_𝑞𝑡₀,=0,(3.13) where 𝛿(𝑡−𝑡₀) is by definition the Dirac function ^∞−∞𝛿𝛿𝑡𝑡−−𝑡𝑡₀₀𝑑𝑡=1,𝑡=𝑡₀,=0,𝑡≠𝑡₀.(3.14) The conclusion, that is, 𝑌_𝑞(𝑡)=𝕆, is obtained by repetitive substitution of each equation in the next one, and using the fact that 𝐻^𝑞∗_𝑞=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)=𝑄^1𝑝𝑒^{𝐽𝑝(𝑡−𝑡0)}𝑍_𝑝𝑡₀.(3.15)

Proof. Let 𝑄=[𝑄_𝑝𝑄_𝑞] where 𝐐_𝐩∈ℳ_{(𝑚𝑛)𝑝} and 𝐐_𝐪∈ℳ_{(𝑚𝑛)𝑞}; combining (3.8) and (3.17), we obtain 𝑌(𝑡)=𝑄𝑍(𝑡)=𝑄_𝑝𝑄_𝑞𝑍^𝑝𝕆(𝑡)=𝑄_𝑝𝑒^{𝐽𝑝(𝑡−𝑡0)}𝑍_𝑝𝑡₀,(3.16) solution that exists if and only if 𝑌(𝑡₀)=𝑄_𝑝𝑍_𝑝(𝑡₀) or 𝑌(𝑡₀)∈colspan𝑄_𝑝. The columns of 𝑄_𝑝 are the 𝑝 eigenvectors of the finite elementary divisors (eigenvalues) of the pencil 𝑠𝐴−𝐵. In that case the system has the unique solution 𝑋(𝑡)=𝑄^1𝑝𝑒^{𝐽𝑝(𝑡−𝑡0)}𝑍_𝑝𝑡₀,(3.17) where 𝑄^1𝑝 is defined as 𝑄_𝑝=𝑄^1𝑝𝑄^2𝑝(3.18) and 𝐐^𝟏𝐩∈ℳ_𝑝𝑝.

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)∉colspan𝑄_𝑝) the system has infinite solutions.

Proof. Let 𝑄_𝑝,𝑄_𝑞 be the matrices defined in Theorem 3.4. If the initial conditions are nonconsistent then 𝑌(𝑡₀)∉colspan𝑄_𝑝 and 𝑍_𝑞(𝑡₀)≠ 0. Moreover 𝑌(𝑡₀)=𝑄_𝑝𝑍_𝑝(𝑡₀)+𝑄_𝑞𝑍_𝑞(𝑡₀).
This means (3.3) is defined for 𝑡≠𝑡₀ because if 𝑡=𝑡₀ then 𝐹𝑌^(𝑡₀)=𝐺𝑌(𝑡₀) and 𝑍_𝑞(𝑡₀)=𝕆 which is a contradiction. Let 𝐻(𝑡−𝑡₀) be the Heaviside function and 𝑓(𝑡)=𝐻𝑡−𝑡₀−𝐻𝑡₀−𝑡=1,𝑡>𝑡⁰0,𝑡=𝑡₀,𝑔(𝑡)=𝐻𝑡₀,−𝑡=1,𝑡=𝑡⁰0,𝑡≠𝑡₀,.(4.1) Then the system can be written as 𝑓(𝑡)𝐹𝑌^(𝑡)=𝐺𝑌(𝑡)−𝑔(𝑡)𝐺𝑌𝑡₀,𝑡≥𝑡₀.(4.2) This is a linear matrix differential equation of first order with 𝑌_ℎ(𝑡)=𝑓(𝑡)𝑄_𝑝𝑒^{𝐽𝑝(𝑡−𝑡0)}𝐶 being the solution of the homogeneous and 𝑌_𝑝(𝑡)=𝑔(𝑡)𝑌(𝑡₀) a partial solution. And we obtain the general solution 𝑌(𝑡)=𝑓(𝑡)𝑄_𝑝𝑒^{𝐽𝑝(𝑡−𝑡0)}𝐶+𝑔(𝑡)𝑌𝑡₀,𝑡≥𝑡₀,(4.3) where 𝐶=[𝐶₁𝐶₂⋯𝐶_𝑝]^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: 𝐴_𝑛𝑋^(𝑛)(𝑡)+𝐴_𝑛−1𝑋^(𝑛−1)(𝑡)+⋯+𝐴₁𝑋^(𝑡)+𝐴₀𝑋(𝑡)=𝕆,𝑡>𝑡₀,_𝑛_𝑖=0𝐴₀𝐴_𝑖𝑋^(𝑖)(𝑡)=𝕆,𝑡>𝑡₀,𝑋(𝑡)+𝑓(𝑡)^𝑛_𝑖=1𝐴_𝑖𝑋^(𝑖)(𝑡)=𝑔(𝑡)𝐴_0𝑛−1_𝑖=0𝑡−𝑡₀^𝑖𝑖!𝑋^(𝑖)𝑡₀,𝑡≥𝑡₀,(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)𝑄_𝑝𝑒^{𝐽𝑝(𝑡−𝑡0)}𝐶+𝑔(𝑡)^𝑛−1_𝑖=0𝑡−𝑡₀^𝑖𝑖!𝑋^(𝑖)𝑡₀,𝑡≥𝑡₀.(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 1010𝑋^(𝑡)+2010𝑋^(𝑡)+−1231𝑋^(𝑡)+−41−−21𝑋(𝑡)=𝕆,(5.1) where 𝑋(𝑡)=[𝑋₁(𝑡)^𝑇𝑋₂(𝑡)^𝑇]^𝑇. We adopt the following notations: 𝑌𝑌₁(𝑡)=𝑋(𝑡),₂𝑌(𝑡)=𝑋^(𝑡),₃𝑌(𝑡)=𝑋^(𝑡),_1𝑌(𝑡)=𝑋′(𝑡)=𝑌₂(𝑡),_2𝐴(𝑡)=𝑋^(𝑡)=𝑌₃(𝑡),₃𝑌^3(𝑡)=𝐴₃𝑋^(𝑡)=−𝐴₂𝑌₃(𝑡)−𝐴₁𝑌₂(𝑡)−𝐴₀𝑌₁(𝑡).(5.2) Or in Matrix form 𝐹𝑌^(𝑡)=𝐺𝑌(𝑡),(5.3) where 𝑌(𝑡)=[𝑌^𝑇1(𝑡)𝑌^𝑇2(𝑡)𝑌^𝑇3(𝑡)]^𝑇 (where ()^𝑇 is the transpose tensor) and the coefficient matrices 𝐹, 𝐺 are given by 𝐅=⎡⎢⎢⎢⎣𝐼²𝕆𝐼𝕆₂𝕆𝕆𝕆𝐴𝕆₃⎤⎥⎥⎥⎦,𝐆=⎡⎢⎢⎢⎣𝕆𝐼²𝕆𝕆𝐼𝕆₂−𝐴₀−𝐴₁−𝐴₂𝑠𝐹−𝐺=𝑠⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣100000010000001000000100000010000010⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦−⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−000014200001−100021−−010031−00100⎤⎥⎥⎥⎦2,−000101⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(5.4)𝑠−1, 𝑠−2, 𝑠−3 finite elementary divisors and 𝑠³ the infinte elementary divisor of degree 3 of the pencil 𝑠𝐹−𝐺. There exist matrices nonsingular 𝑃, 𝑄 such that 𝑃𝐴𝑄=𝐹_𝑤 and 𝑃𝐺𝑄=𝐺_𝑤, where 𝐅_𝐰=𝐼³𝕆𝐻𝕆₃,𝐆^𝐰=𝐽³𝕆𝐼𝕆₃.(5.5) Let 𝑌(𝑡)=𝑄𝑍(𝑡) then 𝑃𝐹𝑄𝐹𝑍𝑌^_(𝑡)=𝐺𝑌(𝑡),𝐹(𝑡)=𝑃𝐺𝑄𝑍(𝑡),_𝑤𝑍^(𝑡)=𝐺_𝑤𝑍(𝑡).(5.6) For consistent initial conditions the solution is 𝑌(𝑡)=𝑄¹³𝑒^{𝐽3(𝑡−𝑡0)}𝑍₃𝑡₀𝑋(𝑡)=𝑄¹³𝑒^{𝐽3(𝑡−𝑡0)}𝑍₃𝑡₀(5.7) and for nonconsistent initial conditions the solution is 𝑋(𝑡)=𝑓(𝑡)𝑄₃𝑒^{𝐽3(𝑡−𝑡0)}𝐶+𝑔(𝑡)²_𝑖=0𝑡−𝑡₀^𝑖𝑖!𝑋^(𝑖)𝑡₀,(5.8) where 𝑒𝐽₃=⎡⎢⎢⎢⎣100020003_𝐽3𝑡⎤⎥⎥⎥⎦,=⎡⎢⎢⎢⎣𝑒^𝑡0𝑒0_2𝑡000𝑒0_3𝑡⎤⎥⎥⎥⎦.(5.9) The columns of 𝑄_𝑝 are the eigenvectors of the eigenvalues 1,2,3 𝑄^𝑇𝑝=⎡⎢⎢⎢⎣311−−−511323−−−523349−−−549⎤⎥⎥⎥⎦^𝑇.(5.10) Let the initial values of the system be 𝑋(0)=−13𝑌(,0)_𝑇𝑋^(0)=−02,𝑋^(0)=−810,=1−3−20−108^𝑇.(5.11) Then 𝑌(0)∈colspan𝑄_𝑝 (consistent initial conditions) and the solution of the system is 𝑌(𝑡)=𝑄₃𝑒^𝐽3𝑡𝑍₃(0)(5.12) and by calculating 𝑍_𝑝(0) we get 𝑍𝑌(0)=𝑄₃𝑍₃(0),₃(0)^𝑇=1−1−1^𝑇(5.13) and the solution of the system is 𝑌(𝑡)𝑌=(𝑡)⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣=3𝑄𝑒_3𝑡𝑒^𝐽3𝑡𝑍₃(0),−𝑒^2𝑡−𝑒^3𝑡−5𝑒^𝑡+𝑒^2𝑡+𝑒^3𝑡3𝑒^𝑡−2𝑒^2𝑡−3𝑒^3𝑡−5𝑒^𝑡+2𝑒^2𝑡+3𝑒^3𝑡3𝑒^𝑡−4𝑒^2𝑡−9𝑒^2𝑡−5𝑒^𝑡+4𝑒^2𝑡+9𝑒^3𝑡𝑋(𝑡)=3𝑒_𝑡⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,−𝑒^2𝑡−𝑒^3𝑡−5𝑒^𝑡+𝑒^2𝑡+𝑒^3𝑡.(5.14) Next assume the initial conditions 𝑋(0)=00𝑌,(0)_𝑇𝑋^(0)=00,𝑋^(0)=11,=000011^𝑇.(5.15) Then 𝑌(0)∉colspan𝑄_𝑝 nonconsistent initial conditions and the solution is 𝑋(𝑡)=𝑓(𝑡)𝑄¹³𝑒^𝐽3𝑡𝐶+𝑔(𝑡)²_𝑖=0𝑡_𝑖𝑖!𝑋^(𝑖)𝑋(𝑡)=𝑓(𝑡)3𝑒_𝑡(0),𝑐₁+𝑒^2𝑡𝑐₂+𝑒^3𝑡𝑐₃−5𝑒^𝑡𝑐₁−𝑒^2𝑡𝑐₂−𝑒^3𝑡𝑐₃+𝑔(𝑡)𝑡2²11,𝑡≥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 [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.

Acknowledgment

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.