Representation of the Solutions of Linear Discrete Systems with Constant Coefficients and Two Delays

Diblík, Josef; Morávková, Blanka

doi:https://doi.org/10.1155/2014/320476

Abstract and Applied Analysis

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Functional Differential and Difference Equations with Applications 2013

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Volume 2014 | Article ID 320476 | https://doi.org/10.1155/2014/320476

Representation of the Solutions of Linear Discrete Systems with Constant Coefficients and Two Delays

Josef Diblík^1,2and Blanka Morávková^1,2

Academic Editor: Miroslava Růžičková

Received05 Sept 2013

Accepted04 Oct 2013

Published16 Apr 2014

Abstract

The purpose of this paper is to develop a method for the construction of solutions to initial problems of linear discrete systems with constant coefficients and with two delays where , are fixed, , , are constant matrices, is a given vector, and is an unknown vector. Solutions are expressed with the aid of a special function called the discrete matrix delayed exponential for two delays. Such approach results in a possibility to express an initial Cauchy problem in a closed form. Examples are shown illustrating the results obtained.

1. Introduction

Throughout the paper, we will use the following notation. For integers , , , we define the set . Similarly, we define the sets and . The function used below is the floor integer function. We will employ the following property of the floor integer function: where .

In this paper, we deal with the discrete system where , , are fixed, , , are constant matrices, is a given vector, and is an unknown vector.

Together with (2), we consider an initial (Cauchy) problem

Define binomial coefficients as customary; that is, for and , We recall that, for a well-defined discrete function , the forward difference operator is defined as . In the paper, we also adopt the customary notation if . In the case of double sums, we set if at least one of the inequalities , holds.

In [1, 2], a discrete matrix delayed exponential for a single delay was defined.

Definition 1. For an constant matrix , , and fixed , one defines the discrete matrix delayed exponential as follows: where is an null matrix and is an unit matrix.

Such discrete matrix delayed exponential was used in [1] to construct solutions of the initial problems (2), (3) with , where is an zero matrix. In these constructions, the main property (Theorem 2) of discrete matrix delayed exponential for a single delay was utilized in [1].

Theorem 2. Let be a constant matrix. Then, for ,

The properties of delayed matrix exponential functions for their continuous and discrete variants and their applications are the topic of recent papers [1–18]. We note that the definition of the delayed matrix exponential was first defined for the continuous case in [4] and, for the discrete case, in [1, 2].

The paper is organized as follows. Discrete matrix delayed exponentials for two delays and their main property are considered in Section 2. A representation of the solution to problem (2), (3) is given in Section 3 and examples illustrating the results obtained are shown in Section 4.

2. Discrete Matrix Delayed Exponential for Two Delays and Its Main Property

In order to extend the results proved in [1, 2] to problems (2), (3), a discrete matrix delayed exponential for two delays was proposed in [3]. There is a discrete matrix delayed exponential for two delays , , defined as follows.

Definition 3. Let , be constant matrices with and let , , be fixed integers. One defines a discrete matrix function called the discrete matrix delayed exponential for two delays , and for two constant matrices , : where

Let us show an example illustrating this special exponential function.

Example 4. For we will construct the matrix if and . Computing particular matrices generating for , we get The main property of was proved in [3].

Theorem 5. Let , be constant matrices with and let , , be fixed integers. Then for .

The analysis of applicability to a representation of the solution to initial problem (2), (3) unfortunately does not lead to satisfactory results because, as we will see below, an additional condition is necessary. A small difference in the definition results in representations of solutions of initial problems without this assumption. Now we give a second definition of a discrete matrix delayed exponential for two delays .

Definition 6. Let , be constant matrices with and let , , be fixed integers. One defines a discrete matrix function called the discrete matrix delayed exponential for two delays , and for two constant matrices , as follows: where

Remark 7. For , it is easy to deduce that .

In order to compare both types of discrete delayed matrices for two delays and see the difference between both definitions, we consider the following example where delays are the same as in Example 4.

Example 8. For we will construct the matrix if and . Computing particular matrices generating for , we get
The main property of is given by the following theorem.

Theorem 9. Let , be constant matrices with and let , , be fixed integers. Then for .

Proof. Let . From (1) and (13), we can see easily that, for an integer satisfying the equation holds by Definition 6 of . Since , we have
By the definition of the forward difference, that is, we conclude that it is reasonable to divide the proof into four parts given by the four values of integer .
In the first case, is such that in the second case in the third case and in the fourth case We see that the above four cases cover all the possible relations between , , and .
In the proof, we use obvious identities where and where , derived from (4) and (24).
Now we consider (in parts (I)–(IV) below) all four cases and perform auxiliary computations. The proof will be finished in part (V).
(I) . From (1) and (13), we get
Therefore, and, by Definition 6,
Similarly, omitting details, we get, using (1) and (13), and
Let . We show that
By (1), or From the last inequalities, we get and (29) holds by (4). For that reason and since , we can replace by in (27). Thus, we have
It is easy to see that, due to (5), formula (33) can be used instead of (27) if too. Let . Similarly, we can show that and, since , we can replace by in (28). Thus, we have It is easy to see that, due to (5), formula (35) can be used instead of (28) if too By Definition 6,
Due to (1), we also conclude that because The second formula can be proved similarly.
Then,
Now we are able to prove that
(II) . In this case, and . In addition to this (see the relevant computations performed in case (I)), we have and .
Then,
For , , and , we have and, for , , and , we have Thus, we can substitute for in (42) and for in (43).
Like with the computations performed in the previous part of the proof, (29), (34) hold. So we can substitute for in (43) and for in (44).
Accordingly, we have
It is easy to see that, due to (5), formula (48) can also be used instead of (43) if and formula (49) can also be used instead of (44) if . Therefore, we see that (like in part (I)) the relation (40) must be proved.
(III) . In this case, we have (see the relevant computations in cases (I) and (II)) Then, For , , and , we have and, for , , and , we get Thus we can replace by in (51) and by in (53).
Like with the computations performed in cases (I) and (II), formulas (29), (34) hold and we can substitute for in (52) and for in (53). This means that It is easy to see that, due to (5), formula (57) can also be used instead of (52) if and formula (58) can also be used instead of (53) if . Therefore, we see that (as in parts (I), (II)) (40) must be proved.
(IV) . In this case, we have (see similar combinations in the cases (II) and (III)) Then, As in part (II), for , , and , formulas (45) hold and, for , , and , formulas (46) hold. Thus we can substitute for in (60) and for in (61).
As in part (III), for , , and , formulas (54) hold and, for , , and , formulas (55) hold. Thus we can replace by in (60) and by in (62).
As before, (29), (34) hold and we can substitute for in (61) and for in (62). Thus, we have It is easy to see that, due to (5), formula (64) can also be used instead of (61) if and formula (65) can also be used instead of (62) if . Therefore, we see that (as in all the previous parts) (40) must be proved.
(V) The Proof of Formula (40). Now we prove (40). With the aid of (18), (19), (24), and (36), we get By (25), we have
Now, in the first and third sum, we replace the summation index by and, in the second and fourth sum, we replace the summation index by . Then,
Due to (33) and (35), we conclude that formula (40) is valid.
We proved that formula (15) holds in each of the considered cases (I), (II), (III), and (IV) for . If , the proof can be done directly because , ,
Formula (15) holds again. Theorem 9 is proved.

3. Representing the Solution of an Initial Problem by Discrete Matrix Delayed Exponential for Two Delays

In this part, we prove the main results of the paper. With the aid of both discrete matrix delayed exponentials we give formulas for the solution of the homogeneous and nonhomogeneous initial problem (2), (3).

3.1. Representing the Solution of a Homogeneous Initial Problem

Consider the homogeneous initial problem First we derive formulas for the solution of (70), (71) with the aid of discrete matrix delayed exponential and then with the aid of discrete matrix delayed exponential .

Theorem 10. Let , be constant matrices such that and let , , be fixed integers. Then, the solution of the initial Cauchy problem (70), (71) can be expressed in the form where and

Proof. We are going to find the solution of the problem (70), (71) in the form with unknown constant vectors . Due to linearity (taking into account that varies), we have, for ,
Using formula (11),
Now we conclude that, for any and , the equation holds. We will try to satisfy initial conditions (71). Due to (75), we have, for ,
Due to Definition 3, for . So we have Subtracting the neighbouring equations , ,,, we get By Definition 3, we have and, from the foregoing equations, we get The previous formulas can be shortened as where . Finally, from , we get
Theorem 10 is proved.

Now we express the solution of the homogeneous Cauchy problem by . In this case, the condition is not necessary.

Theorem 11. Let , be constant matrices with and let , , be fixed integers. Then the solution of the initial Cauchy problem (70), (71) can be expressed in the form where and

Proof. We are going to find the solution of the problem (70), (71) in the form with unknown constant vectors . Due to linearity (taking into account that varies), we have
We use formula (15) and we get
Now we conclude that, for any and , the equation holds. We will try to satisfy initial conditions (71). Due to (83), we have, for ,
By Definition 6, we have for and for . Thus, we have
We see directly that . Subtracting the neighbouring equations , ,,, we immediately get the formulas for as follows:
Further, subtracting the neighbouring equations , ,,, we get
The previous formulas can be written as
Theorem 11 is proved.

3.2. Representing the Solution of a Nonhomogeneous Initial Problem

We consider a nonhomogeneous initial Cauchy problem By the theory of linear equations, we can obtain its solution as the sum of a solution of adjoint homogeneous problem (70), (71) (satisfying the same initial data) and a particular solution of (90) being zero on an initial interval. Let us, therefore, find such a particular solution.

We need an auxiliary lemma the proof of which is omitted.

Lemma 12. Let a function of two discrete variables be given. Then,

Now we are ready to find a particular solution , of the initial Cauchy problem:

Theorem 13. The solution of the initial Cauchy problem (93), (94) can be represented on in the form

Proof. We are going to find a particular solution of problem (93), (94) in the form (95). We substitute (95) into (93). Then, we get We modify the left-hand side of (96). With the aid of Lemma 12, we obtain and, applying Theorem 9, we get By Definition 6, we have , for and for . Thus, we get and (96) holds.

Combining the results of Theorems 10, 11, and 13, we get immediately the following two theorems, which describe the solution of (90), (91). The first theorem uses the delayed matrix exponential and the second one uses the delayed matrix exponential .

Theorem 14. Let , be constant matrices with and let , , be fixed integers. Then, the solution of the initial Cauchy problem (90), (91) can be expressed in the form where and

Theorem 15. Let , be constant matrices with and let , , be fixed integers. Then, the solution of the initial Cauchy problem (90), (91) can be expressed in the form where and

4. Examples

Below, we show four examples to demonstrate the results achieved.

Example 16. Let us represent the solution of the scalar () problem (70), (71) where we put , , , , , , , and , using Theorem 10. We get By Theorem 10, the solution of problem (105), (106) is where Thus, we get
We give values of for as follows:

Example 17. Let us represent the solution of the scalar () problem (90), (91) where we put , , , , , , , , and , using Theorem 11. Thus, we have By Theorem 11, the solution of problem (111), (112) is where Thus, we get The first eight values of the homogeneous problem are given in Example 16. Now, we compute the first eight values of a particular solution as follows: Together, we get

Example 18. Let us represent the solution of the scalar () problem (70), (71) where we put , , , , , , , and , using Theorem 10. Thus, we have By Theorem 10, the solution of problem (118), (119) is where Thus, we get

Example 19. Let us represent the solution of the scalar () problem (90), (91) where we put , , , , , , , , and , using Theorem 11. Thus, we have By Theorem 11, the solution of the problem (123), (124) is where Thus, we get

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The first author was supported by the Grant 201/10/1032 of the Czech Grant Agency (Prague).

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Copyright

Copyright © 2014 Josef Diblík and Blanka Morávková. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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