Reducibility and Stability Results for Linear System of Difference Equations

Abstract

We first give a theorem on the reducibility of linear system of difference equations of the form . Next, by the means of Floquet theory, we obtain some stability results. Moreover, some examples are given to illustrate the importance of the results.

1. Introduction

Consider the homogeneous linear system of difference equations

(1.1)

where is a nonsingular matrix with real entries and

If for some

(1.2)

is specified, then (1.1) is called an initial value problem (IVP). The solution of this IVP is given by

(1.3)

where is the fundamental matrix defined by

(1.4)

However, (1.1) is called reducible to equation

(1.5)

if there is a nonsingular matrix with real entries such that

(1.6)

Let be a matrix function whose entries are real-valued functions defined for . Consider the system

(1.7)

Let be a fundamental matrix of (1.7) satisfying . This can be used to transform (1.1) into (1.5).

Stability properties of (1.1) can be deduced by considering the reduced form (1.5) under some additional conditions. In this study, we first give a theorem on the reducibility of (1.1) into the form of (1.5) and then obtain asymptotic stability of the zero solution of (1.1).

2. Reducible Systems

In this section, we give a theorem on the structure of the matrix , and provide an example for illustration. The results in this section are discrete analogues of the ones given in [1].

Theorem 2.1.

The homogeneous linear difference system (1.1) is reducible to (1.5) under the transformation (1.6) if and only if there exists a regular real matrix such that

(2.1)

hold.

Proof.

Let and be defined as above. Under the transformation (1.6), (1.1) becomes

(2.2)

and after reorganizing, we get

(2.3)

Thus, (1.1) is reducible to (1.5) with

(2.4)

Clearly, is the unique solution of the IVP:

(2.5)

where

This problem is equivalent to solving (2.1). □

Corollary 2.2.

The homogeneous linear system of difference equation (1.1) is reducible to

(2.6)

with a constant matrix under transformation (1.6) if and only if there exists a regular real matrix defined for such that

(2.7)

(2.8)

hold.

Below, we give an example for Corollary 2.2 in the special case . To obtain the matrix , we choose a suitable form of the matrix .

Example 2.3.

Consider the system

(2.9)

where

• (i) are real-valued functions defined for such that for all

• (ii) for all

• (iii)

We also assume that for all ,

(2.10)

It is easy to verify that if we take

(2.11)

where

(2.12)

(2.13)

(2.14)

then (2.7) holds. Moreover, from (2.8) we have

(2.15)

In case for every , that is,

(2.16)

the relations (2.10), (2.12), and (2.13) take the form

(2.17)

where is a real constant and , are arbitrary real constants such that

Corollary 2.4.

If there exists a regular constant matrix such that

(2.18)

then (1.1) reduces to (2.6) with

It should be noted that in case the constant matrices and commute, that is, , then must be a constant matrix as well.

3. Stability of Linear Systems

It turns out that to obtain a stability result, one needs take , a periodic matrix [2]. Indeed, this allows using the Floquet theory for linear periodic system (1.7).

We need the following three well-known theorems [3–5].

Theorem 3.1.

Let be the fundamental matrix of (1.1) with

The zero solution of (1.1) is

• (i) stable if and only if there exists a positive constant M such that

  

  (3.1)

• (ii) asymptotically stable if and only if

  

  (3.2)

  where is a norm in.

Theorem 3.2.

Consider system (1.1) with a constant regular matrix. Then its zero solution is

• (i) stable if and only if and the eigenvalues of unit modulus are semisimple;

• (ii) asymptotically stable if and only if , where is an eigenvalue of is the spectral radius of

Consider the linear periodic system

(3.3)

where , for some positive integer N.

From the literature, we know that if with is a fundamental matrix of (3.3), then there exists a constant matrix, whose eigenvalues are called the Floquet exponents, and periodic matrix with period N such that

Theorem 3.3.

The zero solution of (3.3) is

• (i) stable if and only if the Floquet exponents have modulus less than or equal to one; those with modulus of one are semisimple;

• (ii) asymptotically stable if and only if all the Floquet exponents lie inside the unit disk.

In view of Theorems 3.1, 3.2, and 3.3, we obtain from Corollary 2.2 the following new stability criteria for (1.1).

Theorem 3.4.

The zero solution of (1.1) is stable if and only if there exists a regular periodic matrix satisfying (2.8) such that

• (i) the Floquet exponents of have modulus less than or equal to one; those with modulus of one are semisimple;

• (ii) ; those eigenvalues of of unit modulus are semisimple.

Theorem 3.5.

The zero solution of (1.1) is asymptotically stable if and only if there exists a regular periodic matrix satisfying (2.8) such that either

• (i) all the Floquet exponents of lie inside the unit disk and ; those eigenvalues of of unit modulus are semisimple; or

• (ii) the Floquet exponents of have modulus less than or equal to one; those with modulus of one are semisimple; and

Remark 3.6.

Let be periodic with period N. The Floquet exponents mentioned in Theorem 3.3 are the eigenvalues of where

Example 3.7.

Consider the system

(3.4)

Note that the conditions of Example 2.3 are all satisfied. It follows that

(3.5)

Now,

(3.6)

for which the eigenvalues are

On the other hand, for

(3.7)

if and if .

Applying Theorems 3.4 and 3.5, we see that the zero solution of (3.4) is asymptotically stable if and is stable if

In fact, the unique solution of (3.4) satisfying is

(3.8)

where , , , , and .

It is easy to see that if and is bounded if

Remark 3.8.

In the computation of , is calculated by using Example 2.3, and is obtained by the method given in [6, 7].