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

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.


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 where Y t Y T 1 t Y T 2 t · · · Y T n t T where T is the transpose tensor and the coefficient matrices F, G are given by 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 wellknown 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 t 0 . Hence, it is derived that 1.1 should adopt a generalized solution, in the sense of Dirac δ-solutions.

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 s F − G if and only if there exist nonsingular P ∈ M n and Q ∈ M m such as 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 n g : P, Q : P, Q ∈ M n , P, Q nonsingular 2.2 and a composition rule * defined on g as follows: * : g × g such that P 1 , Q 1 * P 2 , Q 2 : P 1 · P 2 , Q 2 · Q 1 .

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 This group has the following properties: 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 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.
of sF − G and has the form And also the q blocks of the second uniquely defined block sH q − I q correspond to the i.e.d.
of sF − G and has the form Thus, H q is a nilpotent element of M n with index q max{q j : j

2.11
In the last part of this section, some elements for the analytic computation of e A t−t 0 , t ∈ t 0 , ∞ 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.

2.12
Another expression for the exponential matrix of Jordan block, see 2.11 , is provided by the following lemma.
where the f i t − t 0 's are given analytically by the following p j equations:

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 with known initial conditions X t 0 , X t 0 , . . . , X n−1 t 0 .

3.2
Analytically, we consider the system 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 where I p , J p , H q , and I q are given by 2.11 where I p I p 1 ⊕ · · · ⊕ I p ν , J p J p 1 a 1 ⊕ · · · ⊕ J p ν a ν , H q H q 1 ⊕ · · · ⊕ H q σ , I q I q 1 ⊕ · · · ⊕ I q σ .

3.5
Note that ν j 1 p j p and σ j 1 q j q, where p q n. Whereby, multiplying by P , we arrive at Moreover, we can write Z t as Z t Z p t Z 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
where v j 1 p j p.

3.14
The conclusion, that is,

Form on Nonconsistent Initial Condition
In this short section, we describe the impulse behavior of the original system 1.1 , at time t 0 .
In that case, we reformulate Theorem 3.4, so the impulse solution is finally obtained. Proof. Let Q p , Q q be the matrices defined in Theorem 3.4. If the initial conditions are nonconsistent then Y t 0 / ∈ colspanQ p and Z q t 0 / 0. Moreover Y t 0 Q p Z p t 0 Q q Z q t 0 . This means 3.3 is defined for t / t 0 because if t t 0 then FY t 0 GY t 0 and Z q t 0 O which is a contradiction. Let H t − t 0 be the Heaviside function and

4.1
Then the system can be written as This is a linear matrix differential equation of first order with Y h t f t Q p e J p t−t 0 C being the solution of the homogeneous and Y p t g t Y t 0 a partial solution. And we obtain the general solution where C C 1 C 2 · · · 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.

ISRN Mathematical Analysis
Proof. We rewrite the system 3.1 in the following form: 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 The dimension of the solution vector space is p.
For t > t 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.

Numerical Example
where X t X 1 t T X 2 t T T . We adopt the following notations:

5.2
Or in Matrix form where T is the transpose tensor and the coefficient matrices F, G are given by s − 1, s − 2, s − 3 finite elementary divisors and s 3 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

5.6
For consistent initial conditions the solution is and for nonconsistent initial conditions the solution is

5.9
The columns of Q p are the eigenvectors of the eigenvalues 1,2,3

5.10
Let the initial values of the system be

5.11
Then Y 0 ∈ colspanQ p consistent initial conditions and the solution of the system is Y t Q 3 e J 3 t Z 3 0 5.12 and by calculating Z p 0 we get Y 0 Q 3 Z 3 0 ,

5.13
ISRN Mathematical Analysis 13 and the solution of the system is −5e t e 2t e 3t .

5.14
Next assume the initial conditions

5.15
Then Y 0 / ∈ colspanQ p nonconsistent initial conditions and the solution is , t ≥ 0.

5.16
The dimension of the domain that describes the solutions of the system is 3.

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 subsystems. 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 nonhomogeneous case has also a special interest, since it appears often in applications. For all these, there is some research in progress.