Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices
We represent a solution of a nonhomogeneous second-order differential equation with two delays using matrix functions under the assumption that the linear parts are given by permutable matrices.
Gespeichert in:
| Datum: | 2013 |
|---|---|
| Hauptverfasser: | , , , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2013
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2404 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508283700248576 |
|---|---|
| author | Diblik, J. Fečkan, M. Pospíšil, M. Діблик, Й. Фечкан, М. Поспісіль, М. |
| author_facet | Diblik, J. Fečkan, M. Pospíšil, M. Діблик, Й. Фечкан, М. Поспісіль, М. |
| author_sort | Diblik, J. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:14:46Z |
| description | We represent a solution of a nonhomogeneous second-order differential equation with two delays using matrix functions under the assumption that the linear parts are given by permutable matrices. |
| first_indexed | 2026-03-24T02:22:45Z |
| format | Article |
| fulltext |
UDC 517.9
J. Diblı́k ∗ (Brno Univ. Technology, Czech Republic),
M. Fečkan∗∗ (Comenius Univ. and Math. Inst. Slovak Acad. Sci., Bratislava, Slovakia),
M. Pospı́šil∗∗∗ (Brno Univ. Technology, Czech Republic)
REPRESENTATION OF A SOLUTION OF THE CAUCHY PROBLEM
FOR AN OSCILLATING SYSTEM WITH TWO DELAYS
AND PERMUTABLE MATRICES
ЗОБРАЖЕННЯ РОЗВ’ЯЗКУ ЗАДАЧI КОШI ДЛЯ КОЛИВНОЇ СИСТЕМИ
З ДВОМА ЗАПIЗНЮВАННЯМИ ТА ПЕРЕСТАВНИМИ МАТРИЦЯМИ
We represent a solution of a nonhomogeneous second-order differential equation with two delays using matrix functions
under the assumption that the linear parts are given by permutable matrices.
Отримано зображення розв’язку неоднорiдного диференцiального рiвняння другого порядку з двома запiзнювання-
ми iз використанням матричних функцiй за припущення, що лiнiйнi частини задано переставними матрицями.
1. Introduction. Representation of a solution of system of first-order differential equations with
single delay using matrix polynomial derived in [9] led to many new results in theory of ordinary dif-
ferential equations with delay, such as controllability, exponential stability, boundary-value problems,
etc. [1 – 3, 10, 13], but also in theory of boundary-value problems in partial differential equations [4].
On the other side, solutions of differential equations with multiple fixed or variable delays and dif-
ference equations with single and more delays were represented in similar ways [7, 11, 12, 15]. Also
their asymptotic properties and controllability were investigated [6, 11, 12, 14, 15].
In [8], any solution of a system of differential equations of second order with single delay is
represented using matrix polynomials. This representation was used in control theory [5].
In the present paper, by the use of permutable matrices, we are able to construct matrix polyno-
mials which solve a linear homogeneous system of differential equations of second order with two
different delays and linear terms given by these matrices. Later, using these functions, we represent
a solution of initial problem of the corresponding nonhomogeneous system. So, the assumption of
permutability leads to extension of [8] to systems with two different delays. Without the assumption,
the matrix functions would not solve the homogeneous equation.
Let us recall the result from [8].
Theorem 1.1. Let τ > 0, ϕ ∈ C2([−τ, 0],Rn), B be a nonsingular n × n matrix and
f : [0,∞)→ Rn be a given function. Solution x : [−τ,∞)→ Rn of equation
ẍ(t) = −B2x(t− τ) + f(t) (1.1)
satisfying initial condition
x(t) = ϕ(t),
ẋ(t) = ϕ̇(t),
−τ ≤ t ≤ 0, (1.2)
∗ Supported by the Grant GAČR P201/11/0768 and by the project No. CZ.1.05/2.1.00/03.0097.
∗∗ Supported in part by the Grants VEGA-MS 1/0507/11, VEGA-SAV 2/7140/27 and APVV-0134-10.
∗∗∗ Supported by the project No. CZ.1.07/2.3.00/30.0005 funded by European Regional Development Fund.
c© J. DIBLÍK, M. FEČKAN, M. POSPÍŠIL, 2013
58 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
OSCILLATING SYSTEM WITH TWO DELAYS 59
has the form
x(t) = CosτBtϕ(−τ) +B−1SinτBt ϕ̇(−τ)+
+B−1
0∫
−τ
SinτB(t− τ − s) ϕ̈(s)ds+B−1
t∫
0
SinτB(t− τ − s) f(s)ds
for t ∈ [−τ,∞), where
CosτBt =
Θ, −∞ < t < −τ,
E, −τ ≤ t < 0,
E −B2 t
2
2!
+ . . .+ (−1)kB2k (t− (k − 1)τ)2k
(2k)!
, (k − 1)τ ≤ t < kτ, k ∈ N,
(1.3)
SinτBt =
Θ, −∞ < t < −τ,
B(t+ τ), −τ ≤ t < 0,
B(t+ τ)−B3 t
3
3!
+ . . .+ (−1)kB2k+1 (t− (k − 1)τ)2k+1
(2k + 1)!
,
(k − 1)τ ≤ t < kτ, k ∈ N.
(1.4)
Here, we used the notation Θ and E for the n×n zero and identity matrix, respectively. Moreover,
N denotes the set of all positive integers. The above stated matrix functions CosτBt, SinτBt have
the following properties (see [8])
d
dt
CosτBt = −BSinτB(t− τ),
d2
dt2
CosτBt = −B2CosτB(t− τ),
d
dt
SinτBt = BCosτBt,
d2
dt2
SinτBt = −B2SinτB(t− τ)
(1.5)
for any t ∈ R, considering one-sided derivatives at −τ and 0.
2. Systems with two delays. In this section, we derive the representation of a solution of
equation (1.1) with two delays, using matrix functions analogical to (1.3) and (1.4). More precisely,
we consider equation
ẍ(t) = −B2
1x(t− τ1)−B2
2x(t− τ2) + f(t) (2.1)
with τ1, τ2 > 0 and permutable matrices B1, B2, i.e., B1B2 = B2B1. But first, we slightly improve
Theorem 1.1 to C1-smooth initial function.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
60 J. DIBLÍK, M. FEČKAN, M. POSPÍŠIL
Proposition 2.1. Let τ > 0, ϕ ∈ C1([−τ, 0],Rn), B be a nonsingular n × n matrix and
f : [0,∞) → Rn be a given function. Solution x : [−τ,∞) → Rn of equation (1.1) satisfying initial
condition (1.2) has the form
x(t) =
ϕ(t), −τ ≤ t < 0,
CosτB(t− τ)ϕ(0) +B−1SinτB(t− τ) ϕ̇(0)−
−B
∫ 0
−τ
SinτB(t− 2τ − s)ϕ(s)ds+
+B−1
∫ t
0
SinτB(t− τ − s) f(s)ds, 0 ≤ t.
(2.2)
Proof. Obviously, x(t) satisfies the initial condition on [−τ, 0) and x(0) = ϕ(0) due to (1.3),
(1.4). Furthermore, if 0 ≤ t < τ, then
x(t) = ϕ(0) + tϕ̇(0)−B2
t−τ∫
−τ
(t− τ − s)ϕ(s)ds+
t∫
0
(t− s)f(s)ds
since
SinτB(t− 2τ − s) =
B(t− τ − s), s ∈ [−τ, t− τ ],
Θ, s ∈ (t− τ, 0],
for 0 ≤ t < τ and s ∈ [−τ, 0]. Accordingly,
ẋ(t) = ϕ̇(0)−B2
t−τ∫
−τ
ϕ(s)ds+
t∫
0
f(s)ds,
ẍ(t) = −B2ϕ(t− τ) + f(t). (2.3)
Hence, one can see that limt→0+ ẋ(t) = ϕ̇(0), i.e., x ∈ C1([−τ,∞),Rn).
Next, using properties (1.5) for t ≥ τ,
ẋ(t) = −BSinτB(t− 2τ)ϕ(0) + CosτB(t− τ) ϕ̇(0)−
−B2
0∫
−τ
CosτB(t− 2τ − s)ϕ(s)ds+
t∫
0
CosτB(t− τ − s) f(s)ds,
ẍ(t) = −B2CosτB(t− 2τ)ϕ(0)−BSinτB(t− 2τ) ϕ̇(0)+
+B3
0∫
−τ
SinτB(t− 3τ − s)ϕ(s)ds−B
t∫
0
SinτB(t− 2τ − s) f(s)ds+ f(t) =
= −B2x(t− τ) + f(t). (2.4)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
OSCILLATING SYSTEM WITH TWO DELAYS 61
In the last step, we applied the identity
t∫
0
SinτB(t− 2τ − s) f(s)ds =
t−τ∫
0
SinτB(t− 2τ − s) f(s)ds.
Taking the second derivatives at τ, we get
lim
t→τ−
ẍ(t) = −B2ϕ(0) + f(τ) = lim
t→τ+
ẍ(t)
by (2.3) and (2.4). Summarizing, x(t) given by (2.2) solves equation (1.1) on [0,∞), satisfies condi-
tion (1.2) on [−τ, 0] and
x ∈ C1([−τ,∞),Rn) ∩ C2([0,∞),Rn).
Proposition 2.1 is proved.
From now on, we assume the property of empty sum, i.e.,∑
i∈∅
f(i) = 0,
∑
i∈∅
F (i) = Θ
for any function f and matrix function F, whether they are defined or not. Define functions XB2
τ ,
YB2
τ : R→ L(Rn) as
XB2
τ (t) :=
∑
i≥0
iτ≤t
(−1)iB2i (t− iτ)2i
(2i)!
,
YB2
τ (t) :=
∑
i≥0
iτ≤t
(−1)iB2i (t− iτ)2i+1
(2i+ 1)!
for any t ∈ R. Note that
CosτB(t− τ) = XB2
τ (t), SinτB(t− τ) = BYB2
τ (t).
Function XB2
τ inherits its properties from function CosτB(· − τ). As the next lemma explains, so
does the function YB2
τ from SinτB(· − τ).
Lemma 2.1. Let τ > 0 and B be n× n complex matrix. Function YB2
τ (t) solves equation
ÿ(t) = −B2y(t− τ) (2.5)
for t ≥ 0 with initial condition
y(t) = Θ, −τ ≤ t ≤ 0, ẏ(t) =
Θ, −τ ≤ t < 0,
E, t = 0.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
62 J. DIBLÍK, M. FEČKAN, M. POSPÍŠIL
Proof. The initial condition is immediately verified. If the matrix B is regular, then YB2
τ (t) =
= B−1SinτB(t−τ) and equation (2.5) is fulfilled by (1.5) (clearly, this is valid also if B is complex).
Also the other case can be proved easily:
Let t ≥ 0. Then
YB2
τ (t) =
∑
i≥0
iτ≤t
(−1)iB2i (t− iτ)2i+1
(2i+ 1)!
= tEχ[0,∞)(t) +
∑
i≥1
iτ≤t
(−1)iB2i (t− iτ)2i+1
(2i+ 1)!
,
where χM is a characteristic function of a set M given by
χM (t) =
1, t ∈M,
0, t /∈M.
Hence,
ŸB2
τ (t) =
∑
i≥1
iτ≤t
(−1)iB2i (t− iτ)2i−1
(2i− 1)!
=
= −B2
∑
i−1≥0
(i−1)τ≤t−τ
(−1)iB2(i−1) (t− τ − (i− 1)τ)2(i−1)+1
(2(i− 1) + 1)!
= −B2YB2
τ (t− τ). (2.6)
Lemma 2.1 is proved.
Define functions XB
2
1 ,B
2
2
τ1,τ2 ,YB
2
1 ,B
2
2
τ1,τ2 : R→ L(Rn) as
XB
2
1 ,B
2
2
τ1,τ2 (t) :=
∑
i,j≥0
iτ1+jτ2≤t
(−1)i+j
(
i+ j
i
)
B2i
1 B
2j
2
(t− iτ1 − jτ2)2(i+j)
(2(i+ j))!
,
YB
2
1 ,B
2
2
τ1,τ2 (t) :=
∑
i,j≥0
iτ1+jτ2≤t
(−1)i+j
(
i+ j
i
)
B2i
1 B
2j
2
(t− iτ1 − jτ2)2(i+j)+1
(2(i+ j) + 1)!
(2.7)
for any t ∈ R. Some properties of functions XB
2
1 ,B
2
2
τ1,τ2 and YB
2
1 ,B
2
2
τ1,τ2 are concluded in the next lemma.
Lemma 2.2. Let τ1, τ2 > 0, B1, B2 be permutable matrices. Then the following holds for any
t ∈ R :
(1) if B1 = Θ, then XB
2
1 ,B
2
2
τ1,τ2 (t) = XB
2
2
τ2 (t),
(2) if B2 = Θ, then XB
2
1 ,B
2
2
τ1,τ2 (t) = XB
2
1
τ1 (t),
(3) if τ := τ1 = τ2, then XB
2
1 ,B
2
2
τ1,τ2 (t) = XB
2
1+B
2
2
τ (t),
(4) XB
2
1 ,B
2
2
τ1,τ2 (t) = XB
2
2 ,B
2
1
τ2,τ1 (t),
(5) taking the one-sided derivatives at 0, τ1, τ2
ẌB
2
1 ,B
2
2
τ1,τ2 (t) = −B2
1X
B2
1 ,B
2
2
τ1,τ2 (t− τ1)−B2
2X
B2
1 ,B
2
2
τ1,τ2 (t− τ2), (2.8)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
OSCILLATING SYSTEM WITH TWO DELAYS 63
(6) considering the one-sided derivatives at 0 (they both equal Θ)
ẎB
2
1 ,B
2
2
τ1,τ2 (t) = XB
2
1 ,B
2
2
τ1,τ2 (t).
Statements (1) – (5) hold with Y instead of X .
Proof. (1) and (2) are obtained easily from definition of XB
2
1 ,B
2
2
τ1,τ2 , because Θ2i = E if i = 0 and
Θ2i = Θ whenever i > 0. Since ∑
i,j≥0
(i+j)τ≤t
f(i, j) =
∑
l≥0
lτ≤t
∑
i,j≥0
i+j=l
f(i, j)
for any (matrix) function f, for (3) we get
XB
2
1 ,B
2
2
τ1,τ2 (t) =
∑
l≥0
lτ≤t
(−1)l(B2
1 +B2
2)l
(t− lτ)2l
(2l)!
= XB
2
1+B
2
2
τ (t).
Property (4) is trivial.
Next, if τ := τ1 = τ2, then
ẌB
2
1 ,B
2
2
τ1,τ2 (t) = ẌB
2
1+B
2
2
τ (t) = −(B2
1 +B2
2)XB
2
1+B
2
2
τ (t− τ) =
= −B2
1X
B2
1 ,B
2
2
τ1,τ2 (t− τ1)−B2
2X
B2
1 ,B
2
2
τ1,τ2 (t− τ2)
for any t ∈ R, by (3) and the property of Cosτ
√
B2
1 +B2
2t (see (1.5)). Note that CosτBt solves
equation (2.5) even for complex matrix B.
Without any loss of generality, we assume that τ1 < τ2 and consider two cases: t < τ2 and
t ≥ τ2.
If t < τ2, then t − τ2 < 0 and XB
2
1 ,B
2
2
τ1,τ2 (t) = XB
2
1
τ1 (t). Thus (2.8) is verified for t < τ2 by the
property of Cosτ1B1(t− τ1).
Now, let t ≥ τ2. We decompose
XB
2
1 ,B
2
2
τ1,τ2 (t) = Eχ[0,∞)(t) + S1(t) + S2(t) + S3(t),
where
S1(t) =
∑
i≥1
iτ1≤t
(−1)iB2i
1
(t− iτ1)2i
(2i)!
,
S2(t) =
∑
j≥1
jτ2≤t
(−1)jB2j
2
(t− jτ2)2j
(2j)!
,
S3(t) =
∑
i,j≥1
iτ1+jτ2≤t
(−1)i+j
(
i+ j
i
)
B2i
1 B
2j
2
(t− iτ1 − jτ2)2(i+j)
(2(i+ j))!
.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
64 J. DIBLÍK, M. FEČKAN, M. POSPÍŠIL
Similarly to (2.6), we obtain
S′′1 (t) =
∑
i≥1
iτ1≤t
(−1)iB2i
1
(t− iτ1)2(i−1)
(2(i− 1))!
=
= −B2
1
∑
i≥0
iτ1≤t−τ1
(−1)iB2i
1
(t− τ1 − iτ1)2i
(2i)!
= −B2
1 −B2
1S1(t− τ1),
S′′2 (t) =
∑
j≥1
jτ2≤t
(−1)jB2j
2
(t− jτ2)2(j−1)
(2(j − 1))!
=
= −B2
2
∑
j≥0
jτ2≤t−τ2
(−1)jB2j
2
(t− τ2 − jτ2)2j
(2j)!
= −B2
2 −B2
2S2(t− τ2).
Using the property of binomial numbers(
i+ j
i
)
=
(
i− 1 + j
i− 1
)
+
(
i+ j − 1
j − 1
)
,
for i, j ≥ 1 we derive
S′′3 (t) =
∑
i,j≥1
iτ1+jτ2≤t
(−1)i+j
(
i+ j
i
)
B2i
1 B
2j
2
(t− iτ1 − jτ2)2(i+j−1)
(2(i+ j − 1))!
=
= −B2
1
∑
i−1≥0,j≥1
(i−1)τ1+jτ2≤t−τ1
(−1)i−1+j
(
i− 1 + j
i− 1
)
×
×B2(i−1)
1 B2j
2
(t− τ1 − (i− 1)τ1 − jτ2)2(i−1+j)
(2(i− 1 + j))!
−
−B2
2
∑
i≥1,j−1≥0
iτ1+(j−1)τ2≤t−τ2
(−1)i+j−1
(
i+ j − 1
j − 1
)
×
×B2i
1 B
2(j−1)
2
(t− τ2 − iτ1 − (j − 1)τ2)
2(i+j−1)
(2(i+ j − 1))!
.
Rewrite i− 1→ i in the first sum and j − 1→ j in the second one:
S′′3 (t) = −B2
1
∑
i≥0,j≥1
iτ1+jτ2≤t−τ1
(−1)i+j
(
i+ j
i
)
B2i
1 B
2j
2
(t− τ1 − iτ1 − jτ2)2(i+j)
(2(i+ j))!
−
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
OSCILLATING SYSTEM WITH TWO DELAYS 65
−B2
2
∑
i≥1,j≥0
iτ1+jτ2≤t−τ2
(−1)i+j
(
i+ j
j
)
B2i
1 B
2j
2
(t− τ2 − iτ1 − jτ2)2(i+j)
(2(i+ j))!
.
Now, we split the first sum to i = 0 and i ≥ 1, and the second sum to j = 0 and j ≥ 1:
S′′3 (t) = −B2
1
∑
j≥1
jτ2≤t−τ1
(−1)jB2j
2
(t− τ1 − jτ2)2j
(2j)!
−B2
1S3(t− τ1)−
−B2
2
∑
i≥1
iτ1≤t−τ2
(−1)iB2i
1
(t− τ2 − iτ1)2i
(2i)!
−B2
2S3(t− τ2) =
= −B2
1S2(t− τ1)−B2
1S3(t− τ1)−B2
2S1(t− τ2)−B2
2S3(t− τ2).
In conclusion (adding formulae for S′′1 (t), S′′2 (t) and S′′3 (t)), we get exactly formula (2.8) for t ≥ τ2.
For YB
2
1 ,B
2
2
τ1,τ2 , statements (1) – (4) are proved as for XB
2
1 ,B
2
2
τ1,τ2 . Next, if τ := τ1 = τ2, we apply (3)
and Lemma 2.1 with B =
√
B2
1 +B2
2 to get
ŸB
2
1 ,B
2
2
τ1,τ2 (t) = ŸB
2
1+B
2
2
τ (t) = −(B2
1 +B2
2)YB
2
1+B
2
2
τ (t− τ) =
= −B2
1Y
B2
1 ,B
2
2
τ1,τ2 (t− τ1)−B2
2Y
B2
1 ,B
2
2
τ1,τ2 (t− τ2).
For τ1 < τ2, t < τ2, we have YB
2
1 ,B
2
2
τ1,τ2 (t) = YB
2
1
τ1 (t) and (2.8) is verified with the aid of Lemma 2.1.
If t ≥ τ2, we split the sum
YB
2
1 ,B
2
2
τ1,τ2 (t) = tEχ[0,∞)(t) + S1(t) + S2(t) + S3(t)
with
S1(t) =
∑
i≥1
iτ1≤t
(−1)iB2i
1
(t− iτ1)2i+1
(2i+ 1)!
,
S2(t) =
∑
j≥1
jτ2≤t
(−1)jB2j
2
(t− jτ2)2j+1
(2j + 1)!
,
S3(t) =
∑
i,j≥1
iτ1+jτ2≤t
(−1)i+j
(
i+ j
i
)
B2i
1 B
2j
2
(t− iτ1 − jτ2)2(i+j)+1
(2(i+ j) + 1)!
.
The rest proceeds analogically to XB
2
1 ,B
2
2
τ1,τ2 (t).
Property (6) follows immediately from definition (2.7).
Lemma 2.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
66 J. DIBLÍK, M. FEČKAN, M. POSPÍŠIL
Remark 2.1. Using statement (5) of the previous lemma, statements (1) – (4) can be proved
easily by the uniqueness of a solution of the corresponding initial value problem. For instance in the
statement (2), both XB
2
1 ,B
2
2
τ1,τ2 (t) and XB
2
1
τ1 (t) are matrix solutions of equation
ẍ(t) = −B2
1x(t− τ1)
with initial condition
x(t) =
Θ, −τ1 ≤ t < 0,
E, t = 0,
ẋ(t) = Θ, −τ1 ≤ t ≤ 0.
Definition 2.1. Let τ1, τ2 > 0, τ := max{τ1, τ2}, ϕ ∈ C1([−τ, 0],Rn), B1, B2 be n × n
matrices and f : [0,∞) → Rn be a given function. Function x : [−τ,∞) → Rn is a solution of
equation (2.1) and initial condition (1.2), if x ∈ C1([−τ,∞),Rn) ∩ C2([0,∞),Rn) (taken the
second right-hand derivative at 0), satisfies equation (2.1) on [0,∞) and condition (1.2) on [−τ, 0].
We are ready to state and prove our main result.
3. Main result.
Theorem 3.1. Let τ1, τ2 > 0, τ := max{τ1, τ2}, ϕ ∈ C1([−τ, 0],Rn), B1, B2 be n × n
permutable matrices and f : [0,∞) → Rn be a given function. Solution x(t) of equation (2.1)
satisfying initial condition (1.2) has the form
x(t) =
ϕ(t), −τ ≤ t < 0,
X (t)ϕ(0) + Y(t)ϕ̇(0)−B2
1
0∫
−τ1
Y(t− τ1 − s)ϕ(s)ds−
−B2
2
0∫
−τ2
Y(t− τ2 − s)ϕ(s)ds+
t∫
0
Y(t− s)f(s)ds, 0 ≤ t,
(3.1)
where X = XB
2
1 ,B
2
2
τ1,τ2 , Y = YB
2
1 ,B
2
2
τ1,τ2 .
Proof. We consider only the case τ1 6= τ2, since if τ1 = τ2, one can use (3) in Lemma 2.2 to
show that this theorem coincides with Proposition 2.1.
Obviously, x(t) satisfies the initial condition on [−τ, 0) and x(0) = ϕ(0). For the derivative, it
holds limt→0− ẋ(t) = ϕ̇(0). Moreover, if 0 ≤ t < min{τ1, τ2}, then
x(t) = ϕ(0) + tϕ̇(0)−B2
1
t−τ1∫
−τ1
(t− τ1 − s)ϕ(s)ds−
−B2
2
t−τ2∫
−τ2
(t− τ2 − s)ϕ(s)ds+
t∫
0
(t− s)f(s)ds (3.2)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
OSCILLATING SYSTEM WITH TWO DELAYS 67
since
Y(t− τi − s) =
(t− τi − s)E, s ∈ [−τi, t− τi],
Θ, s ∈ (t− τi, 0],
for 0 ≤ t < min{τ1, τ2}, s ∈ [−τi, 0] and i = 1, 2. Thus
ẋ(t) = ϕ̇(0)−B2
1
t−τ1∫
−τ1
ϕ(s)ds−B2
2
t−τ2∫
−τ2
ϕ(s)ds+
t∫
0
f(s)ds (3.3)
and limt→0+ ẋ(t) = ϕ̇(0). Clearly,
x ∈ C1((0,∞),Rn) ∩ C2([0,∞)\{τ1, τ2},Rn).
We show, that although X (t) is not C2 at τ1, τ2, function x(t) is C2 at these points, and, therefore,
in [0,∞). At once, we show that x(t) is a solution of equation (2.1).
For instance, let τ1 < τ2. Assume that 0 ≤ t < τ1. Then equalities (3.2), (3.3) are valid and we
obtain
ẍ(t) = −B2
1ϕ(t− τ1)−B2
2ϕ(t− τ2) + f(t).
Now, let τ1 ≤ t < τ2. Then
x(t) = X (t)ϕ(0) + Y(t)ϕ̇(0)−B2
1
0∫
−τ1
Y(t− τ1 − s)ϕ(s)ds−
−B2
2
t−τ2∫
−τ2
Y(t− τ2 − s)ϕ(s)ds+
t∫
0
Y(t− s)f(s)ds
since Y(t− τ2 − s) = Θ if s ∈ (t− τ2, 0]. By (6) of Lemma 2.2,
ẋ(t) = Ẋ (t)ϕ(0) + Ẏ(t)ϕ̇(0)−B2
1
0∫
−τ1
Ẏ(t− τ1 − s)ϕ(s)ds−
−B2
2
t−τ2∫
−τ2
X (t− τ2 − s)ϕ(s)ds+
t∫
0
Ẏ(t− s)f(s)ds.
With the aid of (5) of Lemma 2.2 and since X (t) = Y(t) = Θ for t < 0, we derive
ẍ(t) = −B2
1X (t− τ1)ϕ(0)−B2
1Y(t− τ1)ϕ̇(0)+
+B4
1
0∫
−τ1
Y(t− 2τ1 − s)ϕ(s)ds−B2
2ϕ(t− τ2)+
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
68 J. DIBLÍK, M. FEČKAN, M. POSPÍŠIL
+B2
1B
2
2
t−τ1−τ2∫
−τ2
Y(t− τ1 − τ2 − s)ϕ(s)ds+ f(t)−
−B2
1
t−τ1∫
0
Y(t− τ1 − s)f(s)ds =
= −B2
1x(t− τ1)−B2
2ϕ(t− τ2) + f(t).
Finally, if τ2 ≤ t, we have
x(t) = X (t)ϕ(0) + Y(t)ϕ̇(0)−B2
1
0∫
−τ1
Y(t− τ1 − s)ϕ(s)ds−
−B2
2
0∫
−τ2
Y(t− τ2 − s)ϕ(s)ds+
t∫
0
Y(t− s)f(s)ds.
So, using t∫
0
Y(t− s)f(s)ds
′′ =
t∫
0
X (t− s)f(s)ds
′ = f(t) +
t∫
0
Ÿ(t− s)f(s)ds,
we get directly the second derivative
ẍ(t) = Ẍ (t)ϕ(0) + Ÿ(t)ϕ̇(0)−B2
1
0∫
−τ1
Ÿ(t− τ1 − s)ϕ(s)ds−
−B2
2
0∫
−τ2
Ÿ(t− τ2 − s)ϕ(s)ds+
t∫
0
Ÿ(t− s)f(s)ds+ f(t)
and after applying (5) of Lemma 2.2, relation (2.1) results. Hence, one can see, that function x(t)
given by (3.1) really solves equation (2.1), satisfies initial condition (1.2) and, moreover, that x ∈
∈ C2((0,∞),Rn). Indeed, taking limits at τ1, τ2 in the computed second derivatives, one gets
lim
t→τ−1
ẍ(t) = −B2
1ϕ(0)−B2
2ϕ(τ1 − τ2) + f(τ1) = lim
t→τ+1
ẍ(t),
lim
t→τ−2
ẍ(t) = −B2
1x(τ2 − τ1)−B2
2ϕ(0) + f(τ2) = lim
t→τ+2
ẍ(t).
Cases τ1 = τ2 and τ1 > τ2 can be proved analogically.
Theorem 3.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
OSCILLATING SYSTEM WITH TWO DELAYS 69
1. Boichuk A., Diblı́k J., Khusainov D., Růžičková M. Boundary value problems for delay differential systems // Adv.
Different. Equat. – 2010. – Article ID 593834. – 20 p.
2. Boichuk A., Diblı́k J., Khusainov D., Růžičková M. Fredholm’s boundary-value problems for differential systems with
a single delay // Nonlinear Anal.-Theor. – 2010. – 72. – P. 2251 – 2258.
3. Boichuk A., Diblı́k J., Khusainov D., Růžičková M. Boundary-value problems for weakly nonlinear delay differential
systems // Abstr. Appl. Anal. – 2011. – Article ID 631412. – 19 p.
4. Diblı́k J., Khusainov D., Kukharenko O., Svoboda Z. Solution of the first boundary-value problem for a system of
autonomous second-order linear PDEs of parabolic type with a single delay // Abstr. Appl. Anal. – 2012. – Article ID
219040. – 27 p.
5. Diblı́k J., Khusainov D. Ya., Lukáčová J., Růžičková M. Control of oscillating systems with a single delay // Adv.
Different. Equat. – 2010. – Article ID 108218. – 15 p.
6. Diblı́k J., Khusainov D. Ya., Růžičková M. Controllability of linear discrete systems with constant coefficients and
pure delay // SIAM J. Contr. Optimiz. – 2008. – 47. – P. 1140 – 1149.
7. Khusainov D. Ya., Diblı́k J. Representation of solutions of linear discrete systems with constant coefficients and pure
delay // Adv. Different. Equat. – 2006. – Article ID 80825. – P. 1 – 13.
8. Khusainov D. Ya., Diblı́k J., Růžičková M., Lukáčová J. Representation of a solution of the Cauchy problem for an
oscillating system with pure delay // Nonlinear Oscillations. – 2008. – 11, № 2. – P. 276 – 285.
9. Khusainov D. Ya., Shuklin G. V. Linear autonomous time-delay system with permutation matrices solving // Stud.
Univ. Žilina. Math. Ser. – 2003. – 17. – P. 101 – 108.
10. Khusainov D. Ya., Shuklin G. V. Relative controllability in systems with pure delay // Int. Appl. Mech. – 2005. – 41.
– P. 210 – 221.
11. Medved’ M., Pospı́šil M. Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations
with linear parts defined by pairwise permutable matrices // Nonlinear Anal.-Theor. – 2012. – 75. – P. 3348 – 3363.
12. Medved’ M., Pospı́šil M. Representation and stability of solutions of systems of difference equations with multiple
delays and linear parts defined by pairwise permutable matrices // Commun. Appl. Anal. – 2012 (to appear).
13. Medved’ M., Pospı́šil M., Škripková L. Stability and the nonexistence of blowing-up solutions of nonlinear delay
systems with linear parts defined by permutable matrices // Nonlinear Anal.-Theor. – 2011. – 74. – P. 3903 – 3911.
14. Medved’ M., L. Škripková L. Sufficient conditions for the exponential stability of delay difference equations with
linear parts defined by permutable matrices // Electron. J. Qual. Theory Different. Equat. – 2012. – 22. – P. 1 – 13.
15. Pospı́šil M. Representation and stability of solutions of systems of functional differential equations with multiple
delays // Electron. J. Qual. Theory Different. Equat. – 2012. – 4. – P. 1 – 30.
Received 17.12.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
|
| id | umjimathkievua-article-2404 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:45Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e6/08d777e0ec7ec01c0bf87943df8478e6.pdf |
| spelling | umjimathkievua-article-24042020-03-18T19:14:46Z Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices Зображення розв’язку задачi кошi для коливної системи з двома запiзнюваннями та переставними матрицями Diblik, J. Fečkan, M. Pospíšil, M. Діблик, Й. Фечкан, М. Поспісіль, М. We represent a solution of a nonhomogeneous second-order differential equation with two delays using matrix functions under the assumption that the linear parts are given by permutable matrices. Отримано зображення розв’язку неоднорiдного диференцiального рiвняння другого порядку з двома запiзнюваннями iз використанням матричних функцiй за припущення, що лiнiйнi частини задано переставними матрицями. Institute of Mathematics, NAS of Ukraine 2013-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2404 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 1 (2013); 58-69 Український математичний журнал; Том 65 № 1 (2013); 58-69 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2404/1575 https://umj.imath.kiev.ua/index.php/umj/article/view/2404/1576 Copyright (c) 2013 Diblik J.; Fečkan M.; Pospíšil M. |
| spellingShingle | Diblik, J. Fečkan, M. Pospíšil, M. Діблик, Й. Фечкан, М. Поспісіль, М. Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices |
| title | Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices |
| title_alt | Зображення розв’язку задачi кошi для коливної системи з двома запiзнюваннями та переставними матрицями |
| title_full | Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices |
| title_fullStr | Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices |
| title_full_unstemmed | Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices |
| title_short | Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices |
| title_sort | representation of a solution of the cauchy problem for an oscillating system with two delays and permutable matrices |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2404 |
| work_keys_str_mv | AT diblikj representationofasolutionofthecauchyproblemforanoscillatingsystemwithtwodelaysandpermutablematrices AT feckanm representationofasolutionofthecauchyproblemforanoscillatingsystemwithtwodelaysandpermutablematrices AT pospisilm representationofasolutionofthecauchyproblemforanoscillatingsystemwithtwodelaysandpermutablematrices AT díblikj representationofasolutionofthecauchyproblemforanoscillatingsystemwithtwodelaysandpermutablematrices AT fečkanm representationofasolutionofthecauchyproblemforanoscillatingsystemwithtwodelaysandpermutablematrices AT pospísílʹm representationofasolutionofthecauchyproblemforanoscillatingsystemwithtwodelaysandpermutablematrices AT diblikj zobražennârozvâzkuzadačikošidlâkolivnoísistemizdvomazapiznûvannâmitaperestavnimimatricâmi AT feckanm zobražennârozvâzkuzadačikošidlâkolivnoísistemizdvomazapiznûvannâmitaperestavnimimatricâmi AT pospisilm zobražennârozvâzkuzadačikošidlâkolivnoísistemizdvomazapiznûvannâmitaperestavnimimatricâmi AT díblikj zobražennârozvâzkuzadačikošidlâkolivnoísistemizdvomazapiznûvannâmitaperestavnimimatricâmi AT fečkanm zobražennârozvâzkuzadačikošidlâkolivnoísistemizdvomazapiznûvannâmitaperestavnimimatricâmi AT pospísílʹm zobražennârozvâzkuzadačikošidlâkolivnoísistemizdvomazapiznûvannâmitaperestavnimimatricâmi |