Research

# On a boundary value problem of a class of generalized linear discrete-time systems

Ioannis K Dassios

Author Affiliations

Department of Mathematics, University of Athens, Panepistimioupolis, Athens, Greece

Advances in Difference Equations 2011, 2011:51  doi:10.1186/1687-1847-2011-51

 Received: 14 June 2011 Accepted: 7 November 2011 Published: 7 November 2011

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this article, we study a boundary value problem of a class of generalized linear discrete-time 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

Linear matrix difference equations (LMDEs) are systems in which the variables take their values at instantaneous time points. Discrete time systems differ from continuous time ones in that their signals are in the form of sampled data. With the development of the digital computer, the discrete time system theory plays an important role in control theory. In real systems, the discrete time system often appears when it is the result of sampling the continuous-time system or when only discrete data are available for use. LMDEs are inherent in many physical, engineering, mechanical, and financial/actuarial models. In this article, our purpose is to study the solutions of generalized linear discrete-time boundary value problems into the mainstream of matrix pencil theory. A boundary value problem consists of finding solutions which satisfies an ordinary matrix difference equation and appropriate boundary conditions at two or more points. Thus, we consider

(1)

with known boundary values of type

(2)

where (i.e., the algebra of square matrices with elements in the field ). For the sake of simplicity, we set and .

Systems of type (1) are more general, including the special case when F = In, where In is the identity matrix of .

The matrix pencil theory has extensively been used for the study of linear difference equations with time invariant coefficients, see for instance [1-5]. A matrix pencil is a family of matrices sF - G, parametrized by a complex number s. When G is square and F = In, where In is the identity matrix, the zeros of the function det (sF - G) are the eigenvalues of G. Consequently, the problem of finding the nontrivial solutions of the equation

(3)

is called the generalized eigenvalue problem. Although the generalized eigenvalue problem looks like a simple generalization of the usual eigenvalue problem, it exhibits some important differences. In the first place, it is possible for det (sF - G) to be identically zero, independent of s. Second, it is possible for F to be singular, in which case the problem has infinite eigenvalues. To see this, write the generalized eigenvalue problem in the reciprocal form

(4)

If F is singular with a null vector X, then , 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 Mnm 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 if and only if there exist nonsingular and such as

(5)

The class of sF - G is characterized by a uniquely defined element, known as a complex Weierstrass canonical form, sFw - Qw, 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 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 sp 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 are called infinite elementary divisors (i.e.d.).

Let B1, B2, ..., Bn be elements of . The direct sum of them denoted by B1 B2 ⊕ ··· ⊕ Bn is the block diag {B1, B2, ..., Bn}.

Then, the complex Weierstrass form sFw - Qw of the regular pencil sF - G is defined by sFw - Qw := sIp - Jp sHq - Iq, where the first normal Jordan-type element is uniquely defined by the set of f.e.d.

(6)

of sF - G and has the form

(7)

and also the q blocks of the second uniquely defined block sHq - Iq correspond to the i.e.d.

(8)

of sF - G and has the form

(9)

Thus, Hq is a nilpotent element of with index , then

We denote with O the zero matrix. are defined as

(10)

(11)

(12)

### 3 Main results-Solution 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.

Consider the problem (1) with known boundary conditions (2). From the regularity of sF - G, there exist nonsingular matrices P and Q such that (see also Section 2),

(13)

and

(14)

where are defined by (10), (11), (12) and moreover

(15)

Note that and , where p + q = n.

Lemma 3.1. System (1) is divided into two subsystems:

(16)

and the subsystem

(17)

Proof. Consider the transformation

(18)

Substituting the previous expression into (1) we obtain

whereby, multiplying by P, we arrive at

Moreover, we can write Zk as , where and . Taking into account the above expressions, we arrive easily at (16) and (17).

Proposition 3.2. The subsystem (16) has general solution

(19)

where and constant.

Proof. See [2,3].

Proposition 3.3. The subsystem (17) has the unique solution

(20)

Proof. Let q* be the index of the nilpotent matrix Hq, i.e. (), we obtain the following equations

and

The conclusion, i.e., , is obtained by repetitive substitution of each equation in the next one, and using the fact that .

#### The boundary value problem

A necessary and sufficient condition for the boundary value problem to be consistent is given by the following result

Theorem 3.1. The boundary value problem (1), (2) is consistent, if and only if

(21)

Where . The matrix Qp has column vectors the p linear independent eigenvectors of the finite generalized eigenvalues of sF-G (see [1] for an algorithm of the computation of Qp).

Proof. Let Q = [QpQq], where and ; Combining propositions (3.2) and (3.3), we obtain

or

(22)

The solution exists if and only if

or

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 3.2. Assume the boundary value problem (1), (2). Then when it is consistent, it has a unique solutions if and only if

(23)

Then the formula of the unique solution is

where C is the solution of the equation

(24)

Proof. Let the boundary value problem (1), (2) be consistent, then from Theorem 3.1 and (22) the solution is

with

and

It is clear that for given A, B, D the problem (1), (2) has a unique solution if and only if the system (24) has a unique solution. Since , the solution is unique for system (24) if and only if the matrix is left invertible. This fact is equivalent to:

Then the formula of the unique solution is

where C is the solution of the equation

#### Other type of boundary conditions

Assume that the matrix difference equation (1) has a different type of boundary conditions. Let the boundary conditions be

(25)

where . Then we can state the following theorem.

Theorem 3.3. The boundary value problem (1), (25) is consistent, if and only if

(26)

Moreover when it is consistent, it has a unique solution if and only

(27)

and the linear system

(28)

gives a unique solution for the constant column C.

Proof. From (22) and (25) the solution exists if and only if

or

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 , the solution is unique if and only if the matrices are left invertible or .

### 4 Numerical example

Consider the boundary value problem (1), (2), where

and A, B the identity and zero matrices, respectively. The invariants of sF - G are s - 1, s - 2, s - 3 (finite elementary divisors) and (infinite elementary divisor of degree 3). Then

and the columns of Qp are the eigenvectors of the generalized eigenvalues 1, 2, 3, respectively. Then

(29)

where ()T is the transpose tensor.

4.1 Example 1

Let

Then

and by calculating C from (24) we get

and the unique solution of the system by substituting in (22) is

4.2 Example 2

Let

Then

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 discrete-time 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 sub-systems. 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.

### Competing interests

The author declares that they have no competing interests.

### 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.

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