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Conditions are derived of the existence of solutions of linear Fredholm’s boundary-value problems for systems of ordinary differential equations with constant coefficients and a single dela

Volume 2010, Article ID 593834, 20 pages

doi:10.1155/2010/593834

Research Article

Boundary Value Problems for

Delay Differential Systems

A Boichuk,1, 2 J Dibl´ık,3, 4 D Khusainov,5 and M R ˚u ˇziˇckov ´a1

1 Department of Mathematics, Faculty of Science, University of ˇ Zilina,

Univerzitn´a 8215/1, 01026 ˇ Zilina, Slovakia

2 Institute of Mathematics, National Academy of Sciences of Ukraine,

Tereshchenkovskaya Str 3, 01601 Kyiv, Ukraine

3 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering,

Brno University of Technology, Veveˇr´ ı 331/95, 60200 Brno, Czech Republic

4 Department of Mathematics, Faculty of Electrical Engineering and Communication,

Brno University of Technology, Technick´a 8, 61600 Brno, Czech Republic

5 Department of Complex System Modeling, Faculty of Cybernetics, Taras,

Shevchenko National University of Kyiv, Vladimirskaya Str 64, 01033 Kyiv, Ukraine

Correspondence should be addressed to A Boichuk,boichuk@imath.kiev.ua

Received 16 January 2010; Revised 27 April 2010; Accepted 12 May 2010

Academic Editor: A ˘gacik Zafer

Copyrightq 2010 A Boichuk 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

Conditions are derived of the existence of solutions of linear Fredholm’s boundary-value problems for systems of ordinary differential equations with constant coefficients and a single delay, assuming that these solutions satisfy the initial and boundary conditions Utilizing a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical form of a criterion for the existence of solutions in a relevant space and, moreover,

to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditionsdefined by a linear vector functional not coinciding with the number of unknowns of a differential system with a single delay As an example of application of the results derived, the problem of bifurcation of solutions of boundary-value problems for systems of ordinary differential equations with a small parameter and with a finite number of measurable delays of argument is considered

1 Introduction

First we mention auxiliary results regarding the theory of differential equations with delay Consider a system of linear differential equations with concentrated delay

assuming that

the degree p; the delay ht ≤ t is a function h : a, b → R measurable on a, b;

denotations

⎧

⎨

⎩

⎧

⎨

⎩

where ϕ is an n-dimensional column vector defined by the formula

add the initial vector function ψs, s < a to nonhomogeneity, thus generating an additive and homogeneous operation not depending on ψ, and without the classical assumption regarding the continuous connection of solution zt with the initial function ψt at t  a.

A solution of differential system 1.5 is defined as an n-dimensional column vector

differential system and the corresponding boundary value problem in the sense of the above definition

Such treatment makes it possible to apply the well-developed methods of linear

the space D p a, b for an arbitrary right-hand side ϕ ∈ L p a, b and has an n-dimensional

a

where the kernel Kt, s is an n × n Cauchy matrix defined in the square a, b × a, b which

where Kt, s ≡ Θ if a ≤ t < s ≤ b, and Θ is the n × n null matrix A fundamental n × n matrix

A serious disadvantage of this approach, when investigating the above-formulated

only be found numerically Therefore, it is important to find systems of differential equations with delay such that this problem can be solved directly Below, we consider the case of a

the Cauchy matrix is solved analytically thanks to a delayed matrix exponential, as defined

below

2 A Delayed Matrix Exponential

Consider a Cauchy problem for a linear nonhomogeneous differential system with constant coefficients and with a single delay τ

consider a related homogeneous problem

Denote by e At

defined as

e At

τ :

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

I A t

I A t

· · ·

I A t

· · ·

2.5

This definition can be reduced to the following expression:

e At

τ t/τ 1

n0

single delay

it is a matrix polynomial, depending on the time interval in which it is considered It is easy

˙

By integrating the delayed matrix exponential, we get

t

0

e τ As ds  I t

t

0

e As

τ ds  A−1·e τ A t−τ − e Aτ

τ

Delayed matrix exponential e At

lim

t → kτ−0



e At τ k 1

t → kτ 0



e At τ k 1

τ

as well hold

Theorem 2.1 A The solution of a homogeneous system 2.3 with a single delay satisfying the

initial condition2.4 where ψs is an arbitrary continuously differentiable vector function can be

represented in the form

−τ e A τ t−τ−s ψ sds. 2.11

B A particular solution of a nonhomogeneous system 2.1 with a single delay satisfying the

zero initial condition z s  0 if s ∈ −τ, 0 can be represented in the form

0

C A solution of a Cauchy problem of a nonhomogeneous system with a single delay 2.1

satisfying a constant initial condition

has the form

τ c

t

0

3 Main Results

where, in accordance with1.3, 1.4,

⎧

⎨

⎩

⎧

⎨

⎩

3.2

single delay satisfying a constant initial condition

0

the initial data X0  I, and the Cauchy matrix Kt, s has the form

Obviously,

constant coefficients and a single delay, satisfying a constant initial condition, has an n-parametric family of linearly independent solutions

τ c

0

e A τ t−τ−s ϕ sds, ∀c ∈ R n 3.8

3.1 Fredholm Boundary Value Problem

if the number m of boundary conditions does not coincide with the number n of unknowns

in a differential system with a single delay

We consider a boundary value problem

˙zt − Azt − τ  gt, if t ∈ 0, b,

assuming that

functional It is well known that, for functional differential equations, such problems are of

the form

τ c

0

In the algebraic system

0

where, obviously, n1≤ minm, n Adopting the well-known notation e.g., 9, we define an

property

P Q∗

d α − 

0

the paper a d-dimensional column zero vector If such condition is true, system 3.14 has a solution

c  P Q r c r Q α − 

0

whereGϕt is a generalized Green operator If the vector functional  satisfies the relation

9, page 176



0

0

which is assumed throughout the rest of the paper, then the generalized Green operator takes the form

0

where

and the Cauchy matrix Kt, s has the form of 3.6 Therefore, the following theorem holds

see 10

Theorem 3.1 Let Q be defined by 3.15 and rank Q  n1 Then the homogeneous problem

corresponding to the problem3.11, 3.12 has exactly r  n − n1linearly independent solutions

Nonhomogeneous problem3.11, 3.12 is solvable if and only if ϕ ∈ Lp 0, b and α ∈ R m satisfy

d linearly independent conditions3.21 In that case, this problem has an r-dimensional family of

linearly independent solutions represented in an explicit analytical form3.23

problem is overdetermined, the number of boundary conditions is more than the number of

Corollary 3.2 If rank Q  n, then the homogeneous problem 3.27 has only the trivial solution

Nonhomogeneous problem3.11, 3.12 is solvable if and only if ϕ ∈ Lp 0, b and α ∈ R m satisfy d linearly independent conditions3.21 where d  m − n Then the unique solution can be represented

as

The case of rank Q  m is interesting as well Then the inequality m ≤ n, holds If

m < n the boundary value problem is not fully defined In this case,Theorem 3.1has the following corollary

Corollary 3.3 If rank Q  m, then the homogeneous problem 3.27 has an r-dimensional r 

Nonhomogeneous problem3.11, 3.12 is solvable for arbitrary ϕ ∈ Lp 0, b and α ∈ R m and has an r-parametric family of solutions

noncritical case

Corollary 3.4 If rank Q  m  n (i.e., Q  Q−1), then the homogeneous problem3.27 has only

the trivial solution The nonhomogeneous problem3.11, 3.12 is solvable for arbitrary ϕ ∈ Lp 0, b

and α ∈ R n and has a unique solution

where

0

is a Green operator, and

is a related Green matrix, corresponding to the problem3.11, 3.12

4 Perturbed Boundary Value Problems

differential equations

i1

ε is a small parameter, delays h i : 0, b → R are measurable on 0, b, h i t ≤ t, t ∈ 0, b,

4.2 can be rewritten as

is an N-dimensional column vector, and ϕt, ε is an n-dimensional column vector given by

i1

L N

k times

,

4.6

following representation:

S h i z t 

0

where

χ h i t, s 

⎧

⎨

⎩

4.8

is the characteristic function of the set

Assume that nonhomogeneities ϕt, 0 ∈ L p 0, b and α ∈ R m are such that the shortened boundary value problem

at the following question

Is it possible to make the problem4.10 solvable by means of linear perturbations and, if this

is possible, then of what kind should the perturbations B i and the delays h i , i  1, 2, , k be for the

boundary value problem4.3 to be solvable?

B0:

0

0

H sk

i1

where

constructed by using the coefficients of the problem 4.3

solution of a boundary value problem was specified in part 1

Theorem 4.1 Consider system

i1

where A is n × n constant matrix, Bt  B1t, , B k t is an n × N matrix, N  nk, consisting

delays h i :0, b → R are measurable on 0, b, h i t ≤ t, t ∈ 0, b, g : 0, b → R, g ∈ L p 0, b,

with the initial and boundary conditions

where α ∈ Rm , ψ : R \ 0, b → R n is a given vector function with components in L p a, b, and

 : D p 0, b → R m is a linear vector functional, and assume that

(satisfying ϕ ∈ L p 0, b) and α are such that the shortened problem

does not have a solution If

then the boundary value problem4.13, 4.14 has a set of ρ : n − m linearly independent solutions

in the form of the series

i−1

ε i z i

t, c ρ ,

4.18

converging for fixed ε ∈ 0, ε∗, where ε∗is an appropriate constant characterizing the domain of the convergence of the series4.18, and zi t, c ρ  are suitable coefficients.

Remark 4.2 Coe fficients z i t, c ρ , i  −1, , ∞, in 4.18 can be determined The procedure

give their form as well

Proof Substitute4.18 into 4.3 and equate the terms that are multiplied by the same powers

z0t.

the form

P Q∗

d α−

0

from which we receive, with respect to c−1∈ Rr, an algebraic system

B0c−1 P Q∗d α−

0

B0and

c−1 −B

0 P Q∗

d α−

0

P B0 I r − B

This solution can be rewritten in the form

where

c−1 −B

0 P Q∗d α−

0

z−1

family of solutions

0



z−1·, c−1 X·P Q r P B ρ c ρ

4.28

For ε1, we get the boundary value problem

i1

argument denoted by ” ”

0

H sk

i1

0

0

G , s1ϕ s1, 0  BsS h



z−1·, c−1 X·P Q r P B ρ c ρ

s1ds1

sds  0

4.30

or, equivalently, the form

B0c0 −

0

H sk

i1

−

0

0

G , s1ϕ s1, 0  Bs1S h



z−1·, c−1 X·P Q r P B ρ c ρ

s1ds1

sds.

4.31

c0 c0



I r − B

0

0

b

0

G , s1Bs1S h X ·P Q r s1ds1

sds



P B ρ c ρ ,

4.32

c0 −B

0

0

H sk

i1

0

0

0

G , s1ϕ s1, 0  Bs1S h z−1·, c−1s1ds1

sds.

4.33

formula:

z0

where

0



I r − B 0

0

b

0

G , s1Bs1S h X ·P Q r s1ds1

sds



0

4.35

family of solutions

0



z0·, c0 X0·P B ρ c ρ

0

0



z0·, c0 X0·P B ρ c ρ

s1ds1

sds  0

4.38

or, equivalently, the form

B0c1 −

0

b

0



z0·, c0 X0·P B ρ c ρ

s1ds1

sds.

4.39

c1 c1



I r − B

0

0

b

0

G , s1Bs1S h X0· s1ds1

sds



P B ρ c ρ ,

4.40 where

c1 −B

0

0

b

0

G , s1Bs1S h z0·, c0s1ds1

z1

where

0



I r − B 0

0

b

0

G , s1Bs1S h X0· s1ds1

sds



0

4.43

boundary value problems as follows:

z i

z i t, c i   XtP Q r c1

0

c i  −B

0

0

b

0

G , s1Bs1S h z i−1·, c i−1s1ds1

sds, i  2, ,



I r − B 0

0

b

0

G , s1Bs1S h X i−1· s1ds1

sds



0

4.45

4.14 has a unique solution

Example 4.3 Consider the linear boundary value problem for the delay differential equation

i1

4.46

system:

i1

Under the condition that the generating boundary value problem has no solution, we

see that, in this case, Xt  e I t−τ

unperturbed system ˙zt  zt − τ, and

τ − e I T−τ

τ  0,

⎧

⎨

⎩

e I τ t−τ−s  I, if 0 ≤ s ≤ t ≤ T,

S h i I t  χ h i t, 0I  I ·

⎧

⎨

⎩

4.49

B0

0

0

k



i1

i1

0

4.50

χ h i t, 0 

⎧

⎨

⎩

or, equivalently,

χ h i t, 0 

⎧

⎨

⎩

B0 −k

i1

0

i1

and the boundary value problem4.46 is uniquely solvable if

det



i1

Δi



/

Acknowledgments

The authors highly appreciate the work of the anonymous referee whose comments and suggestions helped them greatly to improve the quality of the paper in many aspects The first

and Project APVV-0700-07 of Slovak Research and Development Agency The second author was supported by Grant 201/08/0469 of Czech Grant Agency and by the Council of Czech Government MSM 0021630503, MSM 0021630519, and MSM 0021630529 The third author was supported by Project M/34-2008 of Ukrainian Ministry of Education The fourth author

project APVV-0700-07 of Slovak Research and Development Agency

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4.28

For ε1, we get the boundary value problem

i1...

4.43

boundary value problems as follows:

z i

z... condition that the generating boundary value problem has no solution, we