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On a boundary value problem of a class of generalized linear discretetime systems
 Ioannis K Dassios^{1}Email author
https://doi.org/10.1186/16871847201151
© Dassios; licensee Springer. 2011
 Received: 14 June 2011
 Accepted: 7 November 2011
 Published: 7 November 2011
Abstract
In this article, we study a boundary value problem of a class of generalized linear discretetime systems 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.
Keywords
 linear difference equations
 boundary value problem
 matrix pencil
 discrete time system
 matrix difference equations
1 Introduction
where $F,G,A,B,\in \mathcal{M}\left(m\times m;\mathcal{F}\right),{Y}_{k},D\in \mathcal{M}\left(m\times 1;\mathcal{F}\right)$ (i.e., the algebra of square matrices with elements in the field $\mathcal{F}$). For the sake of simplicity, we set ${\mathcal{M}}_{m}=\mathcal{M}\left(m\times m;\mathcal{F}\right)$ and ${\mathcal{M}}_{nm}=\mathcal{M}\left(n\times m;\mathcal{F}\right)$.
Systems of type (1) are more general, including the special case when F = I _{ n }, where I _{ n }is the identity matrix of ${\mathcal{M}}_{n}$.
If F is singular with a null vector X, then $GX=O$, so that X is an eigenvector of the reciprocal problem corresponding to eigenvalue s ^{1} = 0; i.e., s = ∞. It might be thought that infinite eigenvalues are special, unhappy cases to be ignored in our perturbation problem but that is a misconception (see also [6–9]).
2 Mathematical background and notation
This brief section introduces some preliminary concepts and definitions from matrix pencil theory, which are being used throughout the article. 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.
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.
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 } Q _{ w }, see [5], 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 ${f}_{i}\left(s,\hat{s}\right)$ 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 ${\hat{s}}^{q}$ are called infinite elementary divisors (i.e.d.).
Let B _{1}, B _{2}, ..., B _{ n }be elements of ${\mathcal{M}}_{n}$. The direct sum of them denoted by B _{1} ⊕ B _{2} ⊕ ··· ⊕ B _{ n }is the block diag {B _{1}, B _{2}, ..., B _{ n }}.
3 Main resultsSolution space form of a consistent boundary value problem
In this section, the main results for a consistent boundary value problem of types (1) and (2) are analytically presented. Moreover, it should be stressed that these results offer the necessary mathematical framework for interesting applications.
Definition 3.1. The boundary value problem (1) and (2) is said to be consistent if it possesses at least one solution.
Note that ${\sum}_{j=1}^{\nu}{p}_{j}=p$ and ${\sum}_{j=1}^{\sigma}{q}_{j}=q$, where p + q = n.
Moreover, we can write Z _{ k }as ${Z}_{k}=\left[\begin{array}{c}\hfill {Z}_{k}^{p}\hfill \\ \hfill {Z}_{k}^{q}\hfill \end{array}\right]$, where ${Z}_{k}^{p}\in {\mathcal{M}}_{p1}$ and ${Z}_{k}^{q}\in {\mathcal{M}}_{q1}$. Taking into account the above expressions, we arrive easily at (16) and (17).
where ${\sum}_{j=1}^{\nu}{p}_{j}=p$ and $C\in {\mathcal{M}}_{m1}$ constant.
The conclusion, i.e., ${Z}_{k}^{q}=O$, is obtained by repetitive substitution of each equation in the next one, and using the fact that ${H}_{q}^{{q}_{*}}=O$.
The boundary value problem
A necessary and sufficient condition for the boundary value problem to be consistent is given by the following result
Where ${Q}_{p}\in {\mathcal{M}}_{mp}$. The matrix Q _{ p }has column vectors the p linear independent eigenvectors of the finite generalized eigenvalues of sFG (see [1] for an algorithm of 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.
Other type of boundary conditions
where $K,L,S,T\in \mathcal{M}\left(m\times m;\mathcal{F}\right)$. Then we can state the following theorem.
gives a unique solution for the constant column C.
It is obvious that a consistent solution of the boundary value problem (1), (25), is unique if and only if the system (28) gives a unique solution for C. Since $K{Q}_{p},L{Q}_{p}{J}_{p}^{{k}_{N}{k}_{0}}\in {\mathcal{M}}_{mp}$, the solution is unique if and only if the matrices $K{Q}_{p},L{Q}_{p}{J}_{p}^{{k}_{N}{k}_{0}}$ are left invertible or $rank\left[K{Q}_{p}\right]=rank\left[L{Q}_{p}{J}_{p}^{{k}_{N}{k}_{0}}\right]=p$.
4 Numerical example
where ()^{ T }is the transpose tensor.
4.1 Example 1
4.2 Example 2
and the problem is not consistent.
5 Conclusions
The aim of this article was to give necessary and sufficient conditions for existence and uniqueness of solutions for generalized linear discretetime boundary value problems of a class of linear rectangular matrix difference equations whose coefficients are square constant matrices. By taking into consideration that the relevant pencil is regular, we use the Weierstrass canonical form to decompose the difference system into two subsystems. Afterwards, we provide analytical formulas when we have a consistent problem. Moreover, as a further extension of this article, we can discuss the case where the pencil is singular. Thus, the Kronecker canonical form is required. For all these, there is some research in progress.
Declarations
Acknowledgements
The author would like to express his sincere gratitude to Professor G. I. Kalogeropoulos for his fruitful discussion that improved the article. The author would also like to thank the anonymous referees for their comments.
Authors’ Affiliations
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Copyright
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