Abstract and Applied Analysis

Volume 2014 (2014), Article ID 627295, 37 pages

http://dx.doi.org/10.1155/2014/627295

Research Article

General Explicit Solution of Planar Weakly Delayed Linear Discrete Systems and Pasting Its Solutions

Josef Diblík1,2 and Hana Halfarová1,2

¹Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic

²Department of Mathematics, Faculty of Electrical Engineering, Brno University of Technology, 616 00 Brno, Czech Republic

Received 3 September 2013; Accepted 21 October 2013; Published 29 April 2014

Academic Editor: Miroslava Růžičková

Copyright © 2014 Josef Diblík and Hana Halfarová. 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.

Abstract.

Planar linear discrete systems with constant coefficients and delays are considered where , are constant integer delays, , are constant matrices, and . It is assumed that the considered system is weakly delayed. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and special delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.

1. Introduction

1.1. Preliminary Notions and Properties

We use the following notation: for integers , , , we define , where or is admitted, too. Throughout this paper, using notation , we always assume . In the paper, we deal with the discrete planar system

where are constant integer delays, , , are constant 2 × 2 matrices, , , , , and . Throughout the paper, we assume that

where and is 2 × 2 zero matrix. Together with (1), we consider an initial (Cauchy) problem

where with . The existence and uniqueness of the solution of the initial problem (1), (3) on are obvious. We recall that the solution of (1), (3) is defined as an infinite sequence

such that, for any , equality (1) holds.

The space of all initial data (3) with is obviously -dimensional. Below, we describe the fact that, among system (1), there are such systems that their space of solutions, being initially -dimensional, on a reduced interval turns into a space having a dimension less than . The problem under consideration (pasting property of solutions) is exactly formulated in Section 1.4.

1.2. Weakly Delayed Systems

We consider system (1) and look for a solution having the form , where , with , and is a nonzero constant vector. The usual procedure leads to a characteristic equation

where is the unit 2 × 2 matrix. Together with (1), we consider a system with the terms containing delays omitted:

and its characteristic equation

Definition 1. System (1) is called a weakly delayed system if characteristic equations (5), (7) corresponding to systems (1) and (6) are equal, that is, if, for every ,

We consider a linear transformation

with a nonsingular 2 × 2 matrix , then the discrete system for is

with , , where . We show that a system’s property of being one weakly delayed is preserved by every nonsingular linear transformation.

Lemma 2. If system (1) is a weakly delayed system, then its arbitrary linear nonsingular transformation (9) again leads to a weakly delayed system (10).

Proof. It is easy to show that

holds since

that is, equality (8) is assumed.

1.3. Necessary and Sufficient Conditions Determining Weakly Delayed Systems

In the next theorem, we give conditions, in terms of determinants, indicating whether a system is weakly delayed.

Theorem 3. System (1) is a weakly delayed system if and only if the following conditions hold simultaneously:

where and .

Proof. We start with computing determinant defined by (5). We get

where

Expanding the determinant on the right-hand side along summands of the first column, we get

Expanding each of the above determinants along summands of the second column, we have

After simplification, we get

Now we see that for (8) to hold; that is,

conditions (13)–(16) are both necessary and sufficient.

Lemma 4. Conditions (13)–(16) are equivalent to

where and .

Proof. (I) We show that assumptions (13)–(16) imply (23)–(25). It is obvious that condition (23) is equivalent to (13), (14). Now we consider

Expanding the determinant on the right-hand side along summands of the first column and then expanding each of the determinants along summands of the second column, we have

Now we consider

Expanding the determinant on the right-hand side along summands of the first column and then expanding each of the determinants along summands of the second column, we have

(II) Now we prove that assumptions (23)–(25) imply (13) and (16). Due to equivalence of (13) and (14) with (23), it remains to be shown that (23)–(25) imply (15) and (16).

If (24) holds, then, from computations in (27), we see that

and because of (23) we get (15).

Finally, we show that (23) and (25) imply (16). From (29) (using (23)) we get

that is, (16) holds.

1.4. Problem under Consideration

The aim of this paper is to give explicit formulas for solutions of weakly delayed systems and to show that, after several steps, the dimension of the space of all solutions, being initially equal to the dimension of the space of initial data (3) generated by discrete functions , is reduced to a dimension less than the initial one on an interval of the form with an . In other words, we will show that the -dimensional space of all solutions of (1) is pasted to a less-dimensional space of solutions on . This problem is solved directly by explicitly computing the corresponding solutions of the Cauchy problems with each of the cases arising being considered. The underlying idea for such investigation is simple. If (1) is a weakly delayed system, then the corresponding characteristic equation has only two eigenvalues instead of eigenvalues in the case of systems with nonweak delays. This explains why the dimension of the space of solutions becomes less than the initial one. The final results (Theorems 10–13) provide the dimension of the space of solutions. Our results generalize the results in [1, 2], where system (1) with and was analyzed.

1.5. Auxiliary Formula

For the reader’s convenience, we recall one explicit formula (see, e.g., [3]) for the solutions of linear scalar discrete nondelayed equations used in this paper. We consider initial-value problem for the first order linear discrete nonhomogeneous equation

with and . Then, it is easy to verify that unique solution of this problem is

Throughout the paper, we adopt the customary notation for the sum: , where is an integer, is a positive integer, and “” denotes the function considered independently of whether it is defined for indicated arguments or not.

Note that the formula (33) is used many times in recent literature to analyze asymptotic properties of solutions of various classes of difference equations, including nonlinear equations. We refer, for example, to [4–8] and to relevant references therein.

2. General Solution of Weakly Delayed System

If (8) holds, then (5) and (7) have only two (and the same) roots simultaneously. In order to prove the properties of the family of solutions of (1) formulated in the introduction, we will discuss each combination of roots, that is, the cases of two real and distinct roots, a pair of complex conjugate roots, and, finally, a double real root.

Although computations in Sections 1.2 and 1.3 were performed under assumption that , results of this part remain valid also if one or both roots of characteristic equation (7) are zero.

2.1. Jordan Forms of the Matrix and Corresponding Solutions of Problem (1) and (3)

It is known that, for every matrix , there exists a nonsingular matrix transforming it to the corresponding Jordan matrix form . This means that

where has the following four possible forms (denoted below as ), depending on the roots of the characteristic equation (7), that is, on the roots of

If (35) has two real distinct roots , , then

if the roots are complex conjugate, that is, with , then

and, finally, in the case of one double real root , we have either

or

The transformation transforms (1) into a system

with , , , and . Together with (40), we consider an initial problem

with where is the initial function corresponding to the initial function in (3).

Next, we consider all four possible cases (36)–(39) separately.

We define

Assuming that (1) is a weakly delayed system, by Lemma 2, the system (40) is weakly delayed system again.

2.1.1. Case (36) of Two Real Distinct Roots

In this case, we have and . The necessary and sufficient conditions (13)–(16) for (40) turn into

Since , (43) and (45) yield , then, from (44), we get , so that either or . In view of assumptions , , we conclude that only the following cases I, II are possible:

(I), , ,(II), , .

In Theorem 5 both cases I, II are analyzed.

Theorem 5. Let (1) be a weakly delayed system and (35) has two real distinct roots , . If case (I) holds, then the solution of the initial problem (1), (3) is , , where has the form

If case (II) is true, then the solution of initial problem (1), (3) is , , where has the form

Proof. If case (I) is true, then the transformed system (40) takes the form

and if case (II) holds, then (40) takes the form

We investigate only the initial problem (49), (50), (41) since the initial problem (51), (52), (41) can be examined in a similar way.

From (50), (41), we get

then (49) becomes

First, we solve this equation for . This means that we consider the problem

With the aid of formula (33), we get

Now we solve (54) for with initial data deduced from (56); that is, we consider the problem

Applying formula (33) we get (for )

Now we solve (54) for with initial data deduced from (58); that is, we consider the problem

Applying formula (33) yields (for )

From (56), (58), and (60) we deduce that expected form of the solution of the initial problem for with initial data derived from the solution of previous equation for is

We solve (54) for with initial data deduced from (61); that is, we consider the problem

Applying formula (33) yields (for )

In the end we solve (54) for with initial data deduced from (63); that is, we consider the problem

Applying formula (33) yields (for )

Summing up all particular cases (56)–(65) we have

Now, taking into account (42), formula (47) is a consequence of (53) and (66). Formula (48) can be proved in a similar way.

Finally, we note that both formulas (47), (48) remain valid for . In this case, the transformed system (1) reduces to a system without delays. This possibility is excluded by condition (2).

2.1.2. Case (37) of Two Complex Conjugate Roots

The necessary and sufficient conditions (13)–(16) take the forms (43), (44), (46), and

where and .

The system of conditions (43), (44), and (67) gives , and admits only one possibility; namely,

Consequently, , .

The initial problem (1), (3) reduces to a problem without delay

and, obviously,

From this discussion, the next theorem follows.

Theorem 6. There exists no weakly delayed system (1) if has the form (37).

Finally, we note that the assumption (2) alone excludes this case.

2.1.3. Case (38) of Double Real Root

In this case we have and . For (40), the necessary and sufficient conditions (13)–(16) are reduced to (43), (44), (46), and

where .

From (43), (44), and (71), we get . From the condition (46) we get

where and . Multiplying (72) by , we have

Substituting , into (73) and using (43) we obtain

The equation (74) can be written as

Now we will analyse the two possible cases: and .

For the case , we have from (43), (44) that and or . For and , condition (46) gives , where and . Then, from (43), (44) for , we get and .

For and , condition (46) gives , where and , then, from (43), (44) for , we get and .

Now we discuss the case . From conditions (43), (44), we have and . This yields , and, from (75), we have , . By conditions (43), (44) for , we get , .

From the assumptions , we conclude that only the following cases ((I), (II), (III)) are possible:

(I), ,(II), ,(III),

where .

2.1.4. Case

Theorem 7. Let (1) be a weakly delayed system, (35) has a twofold root , and the matrix has the form (38). Then the solution of the initial problem (1), (3) is , , where in case , has the form

If is true then the solution of initial problem (1), (3) is , , where has the form

Proof. Case (I) means that . Then (40) turns into the system

and, if , (40) turns into the system

System (78) can be solved in much the same way as the systems (49), (50) if we put , and the discussion of the system (79) goes along the same lines as that of the systems (51), (52) with . Formulas (76) and (77) are consequences of (47), (48).

2.1.5. Case

For , we define

Theorem 8. Let system (1) be a weakly delayed system, (35) admits two repeated roots , and the matrix has the form (38). Then the solution of the initial problem (1), (3) is given by , , where has the form

Proof. In this case, all the entries of are nonzero and, from (43), (44), and (71), we get

where , then, the system (40) reduces to

where . It is easy to see (multiplying (84) by and summing both equations) that

Equation (85) is a homogeneous equation with respect to the unknown expression

then, using (33), we obtain

With the aid of (87), we rewrite the systems (83), (84) as follows:

First, we solve this system for and consider the problems

With the aid of formula (33), we get

Now we solve system (88) for ; that is, we consider the problem (with initial data deduced from (90), (91))

Formula (33) yields (for )

Now we solve (88) for ; that is, we consider the problem (with initial data deduced from (93), (94))

Applying formula (33) yields (for )

From (93)–(97) we deduce that expected form of the solution of the initial problem for with initial data derived from the solution of previous equation for is

We solve (88) for with initial data deduced from (98); that is, we consider the problem

Applying formula (33) yields (for )

In the end, we solve (88) for with initial data deduced from (100) and (101); that is, we consider the problem

Applying formula (33) yields (for )

Summing up all particular cases (90), (93), (96), (100), and (103) we have

and from cases (91), (94), (97), (101), and (104) we conclude that

Formula (81) is now a direct consequence of (105), (106), and (80).

2.1.6. Case (39) of a Double Real Root

If the matrix has the form (39), the necessary and sufficient conditions (13)–(16), for (40), are reduced to (43), (44), (46), and

Then (43), (44), and (107) give .

Theorem 9. Let (1) be a weakly delayed system, (35) has a double root and the matrix has the form (39). Then and the solution of the initial problem (1), (3) is , , and

Proof. The system (40) can be written as

Solving (111), we get

then (110) turns into

Equation (113) can be solved in a way similar to that of (54) in the proof of Theorem 5 using (33).

First we solve (113) for . This means that we consider the problem

With the aid of formula (33), we get

Now we solve (113) for with initial data deduced from (115); that is, we consider the problem

Applying formula (33) we get (for )

Now we solve (113) for with initial data deduced from (117); that is, we consider the problem

Applying formula (33) yields (for )

From (115), (117), and (119) we deduce that expected form of the solution of the initial problem for with initial data derived from the solution of previous equation for is

We solve (113) for with initial data deduced from (120); that is, we consider the problem

Applying formula (33) yields (for )

In the end, we solve (113) for with initial data deduced from (122); that is, we consider the problem

Applying formula (33) yields (for )

Summing up all particular cases (115)–(124), we get

Formulas (108) and (109) are consequences of (125), (112).

3. Dimension of the Set of Solutions

Since all the possible cases of the planar system (1) with weak delay have been analysed, we are ready to formulate results concerning the dimension of the space of solutions of (1) assuming that initial condition (3) is variable. Although case does not lead to a weakly delayed system and is excluded by (2), for completeness of analysis we incorporate such possibility in our analysis as well (such a case can be considered as a degenerated weakly delayed system). Before formulation we remark that if an assumption in the following theorem is assumed to be valid for a fixed index , it is easy to see that it must be valid for all indices .

Theorem 10. Let (1) be a weakly delayed system and let (35) having both roots different from zero and be fixed. Then the space of solutions, being initially -dimensional, becomes on only

(1)-dimensional if (35) has

(a)two real distinct roots and ,(b)a double real root, , and .(c)a double real root and ,

(2)-dimensional if (35) has

(a)two real distinct roots and ,(b)a pair of complex conjugate roots,(c)a double real root and .

Proof. We will carefully go through all the theorems considered (Theorems 5–9) adding the case of a pair of complex conjugate roots and our conclusion will hold at least on (some of the statements hold on a larger interval).

(a) Analysing the statement of Theorem 5 (case (36) of two real distinct roots), we obtain the following subcases.

(a1)If , , then the dimension of the space of solutions on equals since the last formula in (47) uses only arbitrary parameters:

(a2)If , , then the dimension of the space of solutions on equals since the last formula in (48) uses only arbitrary parameters:

(a3)If , then and Theorem 5 is not applicable. The dimension of the space of solutions on equals 2 since the solution is determined only by 2 arbitrary parameters

 This means that all the cases considered are covered by conclusions (1)(a) and (2)(a) of Theorem 10.

(b) In case (37) of two complex conjugate roots, we have (i.e., we deal not with a weakly delayed system, as noted previosly) and the formula (70) uses only 2 arbitrary parameters

for every . This is covered by case (2)(b) of Theorem 10.

(c) Analysing the statement of Theorems 7 and 8 (case (38) of a double real root), we obtain the following subcases.

(c1)If , , then the dimension of the space of solutions on equals since the last formula in (76) uses only arbitrary parameters:

(c2)If , , then the dimension of the space of solutions on equals since the last formula in (77) uses only arbitrary parameters:

(c3)If (degenerated weakly delayed system), then the dimension of the space of solutions on equals 2and solutions are determined only by 2 arbitrary parameters:

(c4)If , then the dimension of the space of solutions on equals since the last formula in (81) uses only arbitrary parameters:

 where

 The parameter cannot be seen as independent since it depends on the independent parameters and .

All the cases considered are covered by conclusions (1)(b), (1)(c), and (2)(c) of Theorem 10.

(d) Analysing the statement of Theorem 9 (case (39) of a double real root), we obtain the following subcases:

(d1)If , , then the dimension of the space of solutions on equals since the last formula in (108) uses only arbitrary parameters:

 and the last formula in (109) provides no new information.(d2)If (degenerated weakly delayed system), then the dimension of the space of solutions on equals since solutions are determined only by 2 arbitrary parameters

 Both cases are covered by conclusions (1)(b) and (2)(c) of Theorem 10.

Since there are no cases other than cases (a)–(d), the proof is finished.

Theorem 10 can be formulated simply as follows.

Theorem 11. Let (1) be a weakly delayed system and let (35) have both roots different from zero, then the space of solutions, being initially -dimensional, is on only

(1)-dimensional if ,(2)-dimensional if .

We omit the proofs of the following two theorems since, again, they are much the same as those of Theorems 5–9.

Theorem 12. Let (1) be a weakly delayed system and let (35) have a simple root , then the space of solutions, being initially -dimensional, is either -dimensional or -dimensional on .

Theorem 13. Let (1) be a weakly delayed system and let (35) have a double root , then the space of solutions, being initially -dimensional, turns into a -dimensional space on , namely, into the zero solution.

4. Concluding Remarks

To our best knowledge, weakly delayed systems were firstly defined in [9] for systems of linear delayed differential systems with constant coefficients and in [1] for planar linear discrete systems with a single delay (in these papers such systems are called systems with a weak delay). The weakly delayed systems analyzed in this paper can be simplified and their solutions can be found in explicit analytical forms (results obtained generalize those in [1, 2]). Consequently, analytical forms of solutions can be used directly to solve several problems for weakly delayed systems, for example, problems of asymptotical behavior of their solutions, boundary-value problems, or some problems of control theory (using different methods, such problems have recently been investigated e.g., in [10–18]). For an alternative approach to differential-difference equations using the variational iteration method and new analytical and asymptotic methods see, for example, [19–21].

In the case of discrete systems of two equations investigated in this paper, to obtain the corresponding eigenvalues, it is sufficient to solve only a second-order polynomial equation rather than a polynomial equation of order .

Conflict of Interests

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

Acknowledgments

The first author was supported by Grant no. P201/10/1032 of the Czech Grant Agency (Prague). The second author was supported by Grant no. FEKT-S-14-2200 of the Faculty of Electrical Engineering and Communication, Brno University of Technology.

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