Construction of the General Solution of Planar Linear Discrete Systems with Constant Coefficients and Weak Delay

doi:10.1155/2009/784935

Research Article

Construction of the General Solution of Planar

Linear Discrete Systems with Constant Coefficients

and Weak Delay

J. Dibl´ık,

_{D. Ya. Khusainov,}

_{and Z. ˇSmarda}

1_{Brno University of Technology, Brno, Czech Republic}

2_{Kiev University, Kiev, Ukraine}

Correspondence should be addressed to J. Dibl´ık,[email protected]

Received 19 January 2009; Accepted 30 March 2009

Recommended by Ulrich Krause

Planar linear discrete systems with constant coefficients and weak delay are considered. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, the space of solutions with a given starting dimension is pasted after several steps into a space with dimension less than the starting one. In a sense this situation copies an analogous one known from the theory of linear differential systems with constant coefficients and weak delay when the initially infinite dimensional space of solutions on the initial interval on a reduced interval, turnsafter several stepsinto a finite dimensional set of solutions. For every possible case, general solutions are constructed and, finally, results on the dimensionality of the space of solutions are deduced.

Copyrightq2009 J. Dibl´ık et al. 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.

1. Introduction

1.1. Preliminary Notions and Properties

We use the following notation: for integerss,q,s≤q, we defineZqs :{s, s1, . . . , q}where

s −∞orq ∞are admitted, too. Throughout this paper, using notation Zqs, we always

assumes≤q. In this paper we deal with the discrete planar systems

xk1 Axk Bxk−m, 1.1

wherem≥ 0 is a fixed integer,k ∈ Z∞₀,A aijandB bijare constant 2×2 matrices,

nondelayed discrete system ifm0 and as a delayed discrete system ifm >0. Together with 1.1, we consider an initialCauchyproblem

xk ϕk, 1.2

wherek −m,−m1, . . . ,0 withϕ : Z0₋m → R2. We will investigate only the casem > 0

since the solution of1.1form0 is given by the known formulaxk ABkϕ0for

k∈Z∞

1.

Theexistenceanduniquenessof the solution of the initial problems1.1and1.2on

Z∞

−mare obvious. We recall that thesolutionx:Z∞−m → R2 of1.1and1.2is defined as an

infinite sequence

x−m ϕ−m, x−m1 ϕ−m1, . . . , x0 ϕ0, x1, x2, . . . , xk, . . ., 1.3

such that, for anyk∈Z∞₀ , equality1.1holds.

The space of all initial data1.2withϕ:Z0₋m → R2is obviously 2m1-dimensional.

Below we describe the fact that, among the systems1.1, there are such systems that their space of solutions, being initially 2m1-dimensional, on a reduced interval turns into a space having dimension less than 2m1.

1.2. Systems with Weak Delay

We consider the system1.1and we look for a solution having the formxk λk,k ∈Z∞_−m,

λconst with aλ /0. The usual procedure leads to a characteristic equation

detAλ−mB−λI0, 1.4

whereI is the unit 2×2 matrix. Together with1.1, we consider a system with the terms containing delays omitted

xk1 Axk, 1.5

and the characteristic equation

detA−λI 0. 1.6

Definition 1.1. The system 1.1 is called a system with weak delay if the characteristic equations 1.4 and 1.6 corresponding to systems1.1 and1.5 are equal, that is, if for everyλ∈C\ {0}

We consider a linear transformation

xk Syk, 1.8

with a nonsingular 2×2 matrixS. Then the discrete system foryis

yk1 A_Syk B_Syk−m, 1.9

withA_S S−1_AS,B

SS−1BS. We show that the property of a system to be the system with weak delay is preserved by every nonsingular linear transformation.

Lemma 1.2. If the system1.1is a system with weak delay, then its arbitrary linear nonsingular transformation1.8again leads to a system with the weak delay1.9.

Proof. It is easy to show that

detASλ−mBS−λIdetAS−λI 1.10

holds since

detASλ−mBS−λIdet

S−1_ASλ−m_S−1_{BS −λI}

detS−1Aλ−mB−λIS

detAλ−mB−λI,

detA_S−λI detS−1_{AS −λI}

detS−1_A−λIS

detA−λI,

1.11

and the equality

detAλ−mB−λIdetA−λI 1.12

is assumed.

1.3. Necessary and Sufficient Conditions Determining the Weak Delay

Theorem 1.3. System1.1is a system with weak delay if and only if the following three conditions hold simultaneously:

b11b22 0,

b11 b12 b21 b22

0,

a11 a12 b21 b22

b11 b12 a21 a22

0.

1.13

Proof. We start with computing the determinant1.4. We get

detAλ−mB−λIa11b11λ

−m_{−λ a}

12b12λ−m a21b21λ−m a22b22λ−m−λ

a11−λ a12 a21 a22−λ

−λ−m1b11b22

λ−m

a11 a12 b21 b22

b11 b12 a21 a22

λ−2m

b11 b12 b21 b22

.

1.14

Now we see that, for1.7to hold, that is,

detAλ−mB−λIdetA−λI

a11−λ a12 a21 a22−λ

, 1.15

conditions1.13are both necessary and sufficient.

Remark 1.4. It is easy to see that conditions1.13are equivalent to

trBdetB0, detAB detA. 1.16

1.4. Problem under Consideration

The aim of this paper is to show that the dimension of the space of all solutions, being initially equal to the dimension 2m1of the space of initial data1.2generated by discrete functions

delay. This explains why the dimension of the space of solutions becomes less than the initial one. The final resultsTheorems2.5–2.8provide the dimension of the space of solutions.

1.5. Auxiliary Formula

For the reader’s convenience we recall one explicit formulasee, e.g.,3 for the solutions of linear scalar discrete nondelayed equations used in this paper. We consider the first-order linear discrete nonhomogeneous equation

wk1 awk gk, wk0 w0, k∈Z∞k0, 1.17

witha∈Candg :Z∞_k

0 → C. Then it is easy to verify that

wk ak−k0w

0

k−1

rk0

ak−1−r_{gr, k∈Z}∞

k01. 1.18

Throughout the paper, we adopt the customary notation for the sum:_isFi 0 where is an integer,sis a positive integer and, “F” denotes the function considered independently of whether it is defined for indicated arguments or not.

2. Results

If 1.7 holds, then 1.4 and 1.6 have only twoand the sameroots simultaneously. In order to prove the properties of the family of solutions of1.1formulated inSection 1.4, we will separately discuss all the possible combinations of roots, that is, the cases of two real and distinct roots, a couple of complex conjugate roots, and, finally, a two-fold real root.

2.1. Jordan Forms of Matrix

A

and Corresponding Solutions of

the Problem

1.1

,

1.2

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

Λ S−1_{AS, 2.1}

where Λ has the following possible forms, depending on the roots of the characteristic equation1.6, that is, on the roots of

λ2_−a

If2.2has two real distinct rootsλ1,λ2, then

Λ

λ1 0

0 λ2

, 2.3

if the roots are complex conjugate, that is,λ1,2p±iqwithq /0, then

Λ

p q

−q p

, 2.4

and, finally, in the case of one two-fold real rootλ1,2λ, we have either

Λ

λ 0 0 λ

2.5

or

Λ

λ 1

0 λ

. 2.6

The transformationyk S−1_{xktransforms1.1into a system}

yk1 Λyk B∗yk−m, k∈Z∞₀ , 2.7

withB∗S−1_BS,B∗ _b∗

ij,i, j1,2. Together with2.7, we consider an initial problem

yk ϕ∗_{k, 2.8}

k ∈Z0

−mwithϕ∗ :Z0−m → R2whereϕ∗k S−1ϕkis the initial function corresponding to

the initial functionϕin1.2.

Below we consider all four possible cases2.3–2.6separately. We define

Φ1k:

0, ϕ∗₁kT, Φ2k:

ϕ∗

2k,0

T_{, k∈Z}0

−m. 2.9

Assuming that the system 1.1 is a system with weak delay, the system 2.7, due to

2.1.1. The Case

2.3

of Two Real Distinct Roots

takes either the form

y1k1 λ1y1k b∗12y2k−m, k∈Z∞0 , 2.15

ifb∗₂₁0 or the form

y1k1 λ1y1k, k∈Z∞0, 2.17

y2k1 λ2y2k b∗21y1k−m, k∈Z∞0 , 2.18

ifb∗₁₂0. We investigate only the initial problem2.15,2.16,2.8since the initial problem 2.17,2.18,2.8can be examined in a similar way. From2.16and2.8, we get

y2k

⎧ ⎨ ⎩

ϕ∗

2k, ifk∈Z0−m, λk

2ϕ∗20, ifk∈Z∞1 .

2.19

Then2.15becomes

y1k1

⎧ ⎨ ⎩

λ1y1k b∗₁₂ϕ∗₂k−m, ifk∈Zm₀,

λ1y1k b∗₁₂λk₂−mϕ∗₂0, ifk∈Z∞_m1.

2.20

First we solve this equation fork∈Zm₀. This means that we consider the problem

y1k1 λ1y1k b₁₂∗ ϕ∗2k−m, k∈Z∞0,

y10 ϕ∗10.

2.21

With the aid of formula1.18, we get

y1k λk1ϕ∗10 b12∗

k−1

r0 λk−1−r

1 ϕ∗2r−m, k∈Zm11. 2.22

Now we solve2.20fork∈Z∞_m1, that is, we consider the problemwith initial data deduced from2.22

y1k1 λ1y1k b₁₂∗ λ₂k−mϕ∗20, k∈Z∞m1,

y1m1 λ₁m1ϕ∗₁0 b∗₁₂

m

r0 λm−r

1 ϕ∗2r−m.

Applying formula1.18yieldsfork∈Z∞_m2 formula2.14can be proved in a similar way.

Finally, we note that both formulas2.13and2.14remain valid forb∗₁₂ b∗₂₁ 0 as well. In this case, the transformed system2.7reduces to a system without delay.

2.1.2. The Case

2.4

of Two Complex Conjugate Roots

The necessary and sufficient conditions1.13for2.7take the forms2.10and2.11and only one possibility, namely,

and, obviously,

2.1.3. The Case

2.5

of Two-Fold Real Root

formulas2.31and2.32since it can be treated as system2.28considered previouslywith

Then the system2.7reduces to

We can see2.40as a homogeneous equation with respect to the unknown expressiony1k

It is easy to see that the system 2.42is decomposed into two separate equations. Solving each of them in a similar way as in the proof ofTheorem 2.1using1.18 details are omitted, we conclude

2.1.4. The Case

2.6

of Two-Fold Real Root

Then2.45turns into

y1k1

⎧ ⎨ ⎩

λy1k λkϕ∗₂0 b∗₁₂ϕ∗₂k−m, ifk∈Zm₀,

λy1k λkϕ∗₂0 b∗₁₂λk−mϕ∗₂0, ifk∈Z∞_m1.

2.48

Equation2.48can be solved in a similar way as in the proof ofTheorem 2.1using1.18 we omit details. We get

y1k

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ϕ∗

1k, if k∈Z0−m,

λk_ϕ∗

10

k−1

r0

λk−1−r_λr_ϕ∗

20 b∗12ϕ∗2r−m

, if k∈Zm₁1,

λk_ϕ∗

10 kλk−1ϕ∗20 b∗12k−1−mλk−1−mϕ∗20

b∗

12

m

r0

λk−1−r_ϕ∗

2r−m, if k∈Z∞m2.

2.49

Formulas2.47,2.49can be used in the caseb∗₁₂0 as well. In this way, the ensuing result is proved.

Theorem 2.4. Let1.1be a system with weak delay,2.2admit two repeated rootsλ1,2 λ, and the

matrixΛhas the form2.6. Thenb∗₁₁ b∗₂₂ b₂₁∗ 0and the solution of the initial problems1.1

and1.2isxk Syk,k∈ Z∞_−mwhereyk y1k, y2kT,y1kis defined by2.49and y2kby2.47.

2.2. Dimension of the Set of Solutions

Since all the possible cases of the planar system1.1with weak delay have been analysed, we are ready to formulate results concerning the dimension of the space of solutions of1.1

assuming that initial conditions1.2are variable.

Theorem 2.5. Let1.1be a system with weak delay, and2.2has both roots different from zero. Then the space of solutions, being initially2m1-dimensional, becomes onZ∞_m2only

1 m2-dimensional if 2.2has

atwo real distinct roots andb∗₁₂2 b∗₂₁2 >0;

ba two-fold real root,b∗₁₂b₂₁∗ 0andb∗₁₂2 b₂₁∗ 2 >0; ca two-fold real root andb∗₁₂b∗₂₁/0,

22−dimensional if 2.2has

atwo real distinct roots andb∗₁₂b∗₂₁0; ba pair of complex conjugate roots;

Proof. We will carefully trace all theorems consideredTheorems2.1–2.4together with the case of a pair of complex conjugate roots uncovered by a theorem and our conclusion will hold onZ∞_m2some of the statements hold on a greater interval.

aAnalysing the statement ofTheorem 2.1the case2.3of two real distinct roots we obtain the following subcases.

a1Ifb∗₁₁ b∗₂₂ b∗₂₁ 0,b∗₁₂/0, then the dimension of the space of solutions onZ∞_m2 equalsm2 since the last line in2.13uses onlym2 arbitrary parameters

ϕ∗

10, ϕ∗2−m, ϕ∗2−m1, . . . , ϕ∗20. 2.50

a2Ifb∗₁₁ b∗₂₂ b∗₁₂ 0,b∗₂₁/0, then the dimension of the space of solutions onZ∞_m2 equalsm2 since the last line in2.14uses onlym2 arbitrary parameters

ϕ∗

1−m, ϕ∗1−m1, . . . , ϕ∗10, ϕ∗20. 2.51

a3Ifb∗₁₁ b₂₂∗ b₁₂∗ b₂₁∗ 0, then the dimension of the space of solutions onZ∞_m2 equals 2 since the last line in2.13and in2.14uses only 2 arbitrary parameters

ϕ∗

10, ϕ∗20. 2.52

This means that all the cases considered are covered by conclusions 1a and 2a of

Theorem 2.5.

bIn the case2.4of two complex conjugate roots, we haveb∗₁₁ b∗₂₂ b∗₁₂ b∗₂₁0 and the formula2.29uses only 2 arbitrary parameters

ϕ∗

10, ϕ∗20, 2.53

for everyk∈Z∞₁ . This is covered by case2bofTheorem 2.5.

c Analysing the statement of Theorems2.2and 2.3the case 2.5of two-fold real root, we obtain the following subcases.

c1Ifb∗₂₁0,b₁₂∗ /0, then the dimension of the space of solutions onZ∞_m2equalsm2 since the last line in2.31uses onlym2 arbitrary parameters

ϕ∗

10, ϕ∗2−m, ϕ∗2−m1, . . . , ϕ∗20. 2.54

c2Ifb∗₁₂0,b₂₁∗ /0, then the dimension of the space of solutions onZ∞_m2equalsm2 since the last line in2.32uses onlym2 arbitrary parameters

ϕ∗

1−m, ϕ∗1−m1, . . . , ϕ∗10, ϕ∗20. 2.55

c3Ifb∗₁₂b₂₁∗ 0,then the dimension of the space of solutions onZ∞_m2 equals 2 since the last line in2.31and in2.32uses only 2 arbitrary parameters

ϕ∗

c4Ifb₁₂∗ b∗₂₁/0, then the dimension of the space of solutions onZ∞_m2equalsm2 since the last line in2.36uses onlym2 arbitrary parameters

C−m, C−m1, . . . , C0, ϕ∗₁0, 2.57

where

Ck:

ϕ∗

1k b∗

12 b∗

11 ϕ∗

2k

, k∈Z0

−m. 2.58

The parameter ϕ∗₂0 cannot be seen as independent since it depends on the independent parametersϕ∗₁0andC0.

All the cases considered are covered by conclusions1b,1c, and2cofTheorem 2.5. dAnalysing the statement ofTheorem 2.4The case2.6of two-fold real root, we obtain the following subcases.

d1Ifb∗₁₁ b∗₂₂ b∗₂₁ 0,b∗₁₂/0, then the dimension of the space of solutions onZ∞_m2 equalsm2 since the last line in2.49uses onlym2 arbitrary parameters

ϕ∗

10, ϕ∗2−m, ϕ∗2−m1, . . . , ϕ∗20, 2.59

and the last line in2.47provides no new information.

d2If b∗₁₁ b₂₂∗ b∗₂₁ b∗₁₂ 0, then the dimension of the space of solutions on

Z∞

m2 equals 2 since, as follows from2.49 and 2.47, there are only 2 arbitrary

parameters

ϕ∗

10, ϕ∗20. 2.60

Both cases are covered by conclusions1band2cofTheorem 2.5.

Since there are no cases other than the above casesa–d, the proof is finished.

Theorem 2.5can be formulated simply as follows.

Theorem 2.6 Main result. Let 1.1 be a system with weak delay and let2.2have both roots different from zero. Then the space of solutions, being initially2m1-dimensional, is onZ∞_m2only

1 m2-dimensional ifb∗₁₂2 b∗₂₁2>0. 22-dimensional ifb∗₁₂b∗₂₁ 0.

We omit the proofs of the following two theorems since again, they can be done in much the same way as Theorems2.1–2.4.

Theorem 2.8. Let1.1be a system with weak delay and let2.2have a two-fold rootλ 0. Then the space of solutions, being initially2m1-dimensional, turns into0-dimensional space onZ∞₁ , namely, into the zero solution.

3. Concluding Remarks

To our best knowledge, weak delay was first defined in 4 for systems of linear delayed differential systems with constant coefficients. Nevertheless, separate particular examples can be found in various books concerning delayed differential equations. Let us summarize the advantage of investigating “weak” delayed systems in the plane. Such systems can be simplified and their solutions can be found in a simple explicit analytical form. In the case of ordinary differential systems with delay, to obtain the corresponding eigenvalues, it is sufficient to solve only a polynomial equation rather than a quasipolynomial one. In the case of discrete systems of two equations investigated in this paper in the “weak” case, to obtain the corresponding eigenvalues, it is sufficient to solve only polynomial equation of the second order rather than a polynomial equation of 2m1th order. Note that results obtained can be directly used to investigate such asymptotic problems as boundedness or convergence of solutionsusing different methods, such problems have recently been investigated, e.g., in 5–11 .

Acknowledgments

The first author was supported by the Grant 201/07/0145 of Czech Grant AgencyPrague, by the Council of Czech Government MSM 00216 30503 and MSM 00216 30519. The third author was supported by the Council of Czech Government MSM 00216 30503 and MSM 00216 30529.

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