Solutions of Generalized Linear Matrix Differential Equations which Satisfy Boundary Conditions at Two Points

In this article, we study a boundary value problem of a class of generalized linear matrix differential equations whose coefficients are square constant matrices. By using matrix pencil theory we obtain formulas for the solutions and we give necessary and sufficient conditions for existence and uniqueness of solutions. Moreover we provide some numerical examples. These kinds of systems are inherent in many physical and engineering phenomena.


Introduction
Many authors have studied generalized continuous & discrete time systems, see [3-6, 10-12, 14-16, 19, 23, 24, 29-34, 37-39, 44-46], and their applications, see [2, 3, 20-22, 25, 26, 48-50, 53, 54].Many of these results have already been extended to systems of differential & difference equations with fractional operators, see [13,17,18,20,24,25,[33][34][35][36][40][41][42].In this article, our purpose is to study the solutions of a generalized boundary value problem of Linear Matrix Differential Equations (LMDEs) into the mainstream of matrix pencil theory.A boundary-value problem consists of finding solutions which satisfies an ordinary matrix differential equation and appropriate boundary conditions at two or more points.Thus we consider the boundary value problem where F, G, A, B ∈ M(m × m; F) and Y (t), D, E ∈ M(m × 1; F), (i.e. the algebra of square matrices with elements in the field F).For the sake of simplicity we set M m = M(m × m; F) and M nm = M(n × m; F).Systems of type (1) are more general, including the special case when F = I n , where I n is the identity matrix of M n .

Mathematical Background and Notation
This brief subsection introduces some preliminary concepts and definitions from matrix pencil theory, which are being used throughout the paper.Linear systems of type (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. 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, see [10,28,38,43,47].In any other case, the pencil will be called singular.
In addition, the pencil sF − G is said to be strictly equivalent to the pencil s F − G if and only if there exist non-singular P ∈ M m and Q ∈ M m such as In this article, we consider the case that pencil is regular.The class of sF −G is characterized by a uniquely defined element, known as a complex Weierstrass canonical form, sF w − G w , see [10,28,38,43,47], specified by the complete set of invariants of the pencil sF − G.This is the set of elementary divisors (e.d.) obtained by factorizing the invariant polynomials f i (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: • e.d. of the type s p are called zero finite elementary divisors (z.f.e.d.) • e.d. of the type (s − a) π , a = 0 are called nonzero finite elementary divisors (nz.f.e.d.) • e.d. of the type ŝq are called infinite elementary divisors (i.e.d.).
Let B 1 , B 2 , . . ., B n be elements of M n .The direct sum of them denoted by Then, the complex Weierstrass form sF w − G w of the regular pencil sF − G is defined by sF w − G 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.ŝq 1 , . . ., ŝqσ , σ j=1 q j = q of sF − G and has the form Thus, H q is a nilpotent element of M n with index q = max{q j : j = 1, 2, . . ., σ}, then H q q = 0 q,q .We denote with O q,q the zero matrix.I p j , J p j (a j ), H q j are defined as 2 Main Results-Solution space form of a consistent boundary value problem In this section, the main results for a consistent boundary value problem of type (1) are analytically presented.Moreover, it should be stressed out that these results offer the necessary mathematical framework for interesting applications.
Definition 2.1.The boundary value problem ( 1) is said to be consistent if it possesses at least one solution.
Consider the problem (1).From the regularity of sF − G, there exist nonsingular M(m × m, F ) matrices P and Q such that (see also Section 1), Where I p j , J p j (a j ), H q j are defined in Section 1.
Lemma 2.1.System (1) is divided into two subsystems: and the subsystem Proof.Consider the transformation Substituting the previous expression into (1) we obtain where by multiplying by P and using ( 2) we arrive at Moreover, we can write Z(t) as Z(t) = Z p (t) Z q (t) , where Z p (t) ∈ M p1 and Z q (t) ∈ M q1 .Taking into account the above expressions, we arrive easily at (3) and (4).
Proposition 2.1.The subsystem (4) has the unique solution Proof.Let q * be the index of the nilpotent matrix H q , i.e. (H q * q = 0 q,q ), we obtain the following equations and H q Z q (t) = Z q (t), H 2 q Z q (t) = H q Z q (t), H 3 q Z q (t) = H 2 q Z q (t), . . .
The sum of the above relations gives, (note H q * q = 0 qq ) is the solution (6).

The boundary value problem
A necessary and sufficient condition for the boundary value problem to be consistent is given by the following result Theorem 2.1.The boundary value problem ( 1) is consistent, if and only if Where Q p has column vectors all the linear independent eigenvectors of the finite generalized eigenvalues of sF − G.

This solution exists if and only if
The columns of Q p are the p eigenvectors of the finite elementary divisors (eigenvalues) of the pencil sF − G. (see [9,25,35,40,44,48] for algorithms for the computation of Q p ) It is obvious that, if there is a solution of the boundary value problem, it needs not to be unique.The necessary and sufficient conditions, for uniqueness, when the problem is consistent, are given by the following theorem.
Theorem 2.2 Assume the boundary value problem (1).Then when it is consistent, it has a unique solutions if and only and the linear system gives a unique solution for constant column C. Then the unique solution is given by Proof Let the boundary value problem (1) be consistent, then from Theorem 2.1 the solution is and It is clear that for given A, B, D, E the problem (1) has a unique solution if and only if the system (9) has a unique solution.Since AQ p , BQ p e Jp(t−t 0 ) ∈ M mp , the solution is unique for the system (9) if and only if the matrices AQ p , BQ p e Jp(t−t 0 ) are left invertible (rank[AQ p ] = rank[BQ p e Jp(t−t 0 ) ] = p) and both equations give the same unique solution for the constant column C.
and A the identity matrix and and B a matrix that satisfies the equation BQ p e Jp(t N −t 0 = Q p .The invariants of sF − G are s − 1, s − 2, s − 3 (finite elementary divisors) and s3 (infinte elementary divisor of degree 3).Then and the unique solution of the system by substituting in (10) is